M.G.Goman and A.V.Khramtsovsky (1997 draft) - Textbook for KRIT Toolbox users

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M.G.Goman and A.V.Khramtsovsky "Methodology of the qualitative investigation. Theory and numerical methods. Application to Aircraft Flight Dynamics", draft 1997 version of the textbook for KRIT Toolbox users, 53 p.

The textbook covers theory of qualitative analysis of nonlinear systems, basic numerical methods supported by KRIT package and some aircraft flight dynamics applications.

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M.G.Goman and A.V.Khramtsovsky (1997 draft) - Textbook for KRIT Toolbox users

  1. 1. The Textbook (draft) Dr Mikhail Goman Dr Andrew Khramtsovsky June 1997
  2. 2. Contents 1 Investigation of aircraft nonlinear dynamics problems 2 2 Qualitative methods and bifurcation analysis of aircraft dynamics 6 2.0.1 Equilibrium solutions : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.0.2 Periodic solutions : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.0.3 Poincare mapping : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 Bifurcational analysis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.1.1 Bifurcations of the equilibrium points : : : : : : : : : : : : : : : : 13 2.1.2 Bifurcations of the periodic trajectories : : : : : : : : : : : : : : : 16 3 Numerical algorithms and methods 19 3.1 Continuation method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 3.2 Systematic search for solutions of nonlinear system of equations : : : : : 23 3.3 Computing manifolds of equilibrium solutions and bifurcation sets : : : : 28 3.4 Computation of the periodic solutions : : : : : : : : : : : : : : : : : : : : 29 3.4.1 Time{advance mapping. : : : : : : : : : : : : : : : : : : : : : : : 29 3.4.2 Poincare mapping. : : : : : : : : : : : : : : : : : : : : : : : : : : 32 3.5 Algorithms for stability regions computation : : : : : : : : : : : : : : : : 35 3.5.1 Computation of stability region boundaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 3.5.2 Direct computation of stability region cross-sections : : : : : : : : 37 3.6 Example: Global stability analysis in roll-coupling problem : : : : : : : : 39 1
  3. 3. Chapter 1 Investigation of aircraft nonlinear dynamics problems The desire to advance aircraft maneuverability results in expansion of the region of the allowed ight regimes. Inevitably critical situations connected with the loss of stability and/or controllability, become more probable. This is especially true for high angle of attack ight regimes where stall and spin entry are possible. Critical ight regimes signi cantly di er from normal ight regimes in aircraft dy- namics and control. Auto-rotation regimes also belong to the class of critical regimes, these regimes can be entered during active spatial maneuvering with fast rotation in roll. Thus investigation of aircraft dynamics at high angles of attack and fast rotation is closely coupled with aircraft combat e ciency and ight safety problems. It is neces- sary to use nonlinear equations of motion, taking into account nonlinear dependencies of aerodynamic characteristics on motion parameters as well as nonlinearities in con- trol system. The behavior of the aircraft can be described by an autonomous nonlinear dynamic system depending on a number of parameters. The parameters include con- trol inputs (i.e. de ections of control surfaces or control levers), aircraft parameters, parameters of the ight regime etc. Since aircraft is su ciently nonlinear dynamic system, it is impossible to get all the qualitative pattern of aircraft dynamics by means of mathematical modeling only. Aircraft behavior may be very di erent depending on initial conditions and control applied. Methods of the qualitative theory of dynamical systems and bifurcation analysis are very e cient for the investigation of aircraft dynamics they and direct mathematical modeling complement each other. Basic results of qualitative theory can be found in many publications 1, 2, 3, 49, 50, 4, 5, 6, 7, 8, 9, 37]. Unfortunately, mathematical models of aircraft dynamics are too complex to be investigated analytically. That's why major advances in the investigation of aircraft dynamics at high angles of attack are achieved using powerful computers and numerical methods of qualitative analysis. A great e ort was devoted lately for the development of numerical methods and software packages for qualitative analysis of nonlinear dynamic systems 43, 44, 45, 10, 11, 12, 13, 14, 48, 15, 16, 17, 47]. Most frequently used for aircraft applications are the software packages BISTAB, AUTO, 2
  4. 4. ASDOBI, KRIT 43, 44, 45, 47]. Methods for the investigation of nonlinear aircraft dynamics problems can be divided into two groups. The rst one comprises approximate approaches and methods from earlier research, the second one includes new numerical methods based on bifurcation analysis and theory of nonlinear dynamic systems. Approximate methods Since the dawn of aviation, many authors investigated stall and spin problems for ev- ery new generation of the aircraft. With the advent of jet planes a new critical regime appeared, the regime was caused by a loss of motion stability due to inertia coupling between longitudinal and lateral motions at fast rotation in roll. Both spin 23, 24, 25] and inertia roll-coupling problem 18, 19, 21, 20, 22] were investigated using approximate equations of motion and simpli ed (usually linear) aerodynamic models. Approximate methods were used as well. Often for simplicity some state variables (roll rate, angle of attack) were considered as problem parameters. Such methods allowed to a certain de- gree to avoid solving nonlinear problems. Partial analytical investigation was performed, and nal solution was received by graphical methods. Revealed peculiarities of aircraft motion (arising of spin and auto-rotation regimes, departures in roll) were not directly linked with bifurcations of steady-state solutions of equations of motion. Thus general results obtained by qualitative theory of dynamic systems couldn't be applied. It's worth noting that in some works a program of qualitative investigation of object's dynamics was formulated and even realized to the extent achievable with graphical- analytical methods, see for example 21]. Application of continuation methods Nowadays bifurcation analysis methods and theory of dynamic systems are often applied towards nonlinear aircraft dynamic problems, and there are enough many publications 26, 28, 29, 30, 31, 32, 33, 34, 35, 46]. At the same time these methods are seldom used at early stages of aircraft development. Common feature of the above-mentioned papers is an application of continuation method for the investigation of nonlinear problems. Carroll and Mehra were the rst who used the continuation method in ight dynam- ics and connected the main types of an aircraft instabilities with bifurcation phenomena. For example, the occurrence of wing rock motion was connected with a Hopf bifurcation the examples of chaotic motions were presented, etc. Later the existence of new types of bifurcations in an aircraft dynamics was explored, such as origination of stable torus manifold 31] global bifurcation of a closed orbit related with the appearance of homo- clinical trajectory 42] so-called ip or period doubling and pitchfork bifurcations for closed orbits 32, 33]. Continuation methodology and bifurcation analysis can be used for determining the recovery technique from critical regimes and for control law design for improving dynamical behavior 27, 31, 28, 33]. While rst works were devoted mainly to theory and corresponding numerical algo- rithms 26, 31], in recent publications the results obtained for real aircraft were presented 3
  5. 5. ( 26] { F-4, 29] { German-French Alpha-Jet, 35] { F-14, 33] { F-15). For a number of aircraft the results of theoretical analysis were compared with ight test results, and good quantitative as well as qualitative agreement was found (see 29]). Methodology of investigation For a dynamic system describing an aircraft behavior, its dependence on a number of parameters is investigated. The parameters can include ight regime values, mass and inertia data, de ections of the control surfaces etc. Qualitative investigation of dynamics is performed at xed values of parameters so that the system can be considered as autonomous one. The behavior of the system can be predicted knowing qualitative changes in the structure of system's solutions and type and character of bifurcations. The Lyapunov's concept of stability suggesting unlimited time variation is used. This validity of such an approach is limited because of usage of approximate math- ematical models of motion. For energetic spatial maneuvers, mathematical model is valid only for nite time period. At t !+1 the solutions of equations may di er from genuine motion of the object. Nevertheless the eigenvalues can describe the local character of motion near steady- state conditions within nite period of time. In that case the ratio between characteristic times of the aircraft motion, maneuver duration and "validity time" of the mathematical model, is important for the estimation of the motion character. For example, in case of very fast maneuver an instability may have no time to manifest itself or else an approach to a weak attractor may be very slow because of its weakness. Therefore the usage of Lyapunov's concept of stability is a su cient simpli cation of the problem. That's why it is necessary to verify the results obtained by means of mathematical modeling the veri cation is especially important in the case of critical and near-critical conditions. Relying on experience achieved, the following three-stage investigation plan is sug- gested for the investigation of nonlinear dynamics of an aircraft, the plan is based on bifurcation analysis and continuation technique : At rst stage the parameters of the problem are considered as xed. Main goal is to nd out all possible steady-state solutions (equilibrium, periodic and more complex ones), and to determine their local stability. Continuation method and systematic search method are of great help here. Global structure of the state space can be revealed after determining of asymptotic stability regions of all the existing attractors. Convenient graphical representation of phase portrait and steady-state solutions is very important for e cient data processing. At the second stage the changes of the behaviour of the system due to variations of some parameters are investigated. Prediction of the system response is based on the knowledge of possible bifurcations and dominating attractors that determine system's dynamics immediately after bifurcations. 4
  6. 6. Rate of parameter change is of importance for the prediction too. The higher the rate, the greater might be the di erence between transient and steady-state motions. At the last stage mathematical modeling is used for veri cation of the qualitative results obtained. The situations to be checked include di erent nite disturbances of state variables, and di erent variations of the parameters of the system. In the next chapters methods of qualitative investigation and bifurcation analysis are described together with corresponding numerical algorithms. 5
  7. 7. Chapter 2 Qualitative methods and bifurcation analysis of aircraft dynamics In this chapter the basic concepts and ideas of the qualitative theory of dynamical systems are described. They will be useful further for the investigation of nonlinear dynamics of the aircraft 2, 49]. Consider a nonlinear autonomous dynamical system depending on parameters, the system is described by a set of di erential equations dx dt = F(x c) x 2Rn c 2M Rm (2.1) where F is a smooth vector function. The vector eld F de nes a map Rn+m !Rm. The system (2.1) satis es to the conditions of the existence and uniqueness of the solution x(t xo) with initial condition x(0 xo) = xo. The solution 't(xo) = x(t xo) is called a trajectory or the ow of the dynamical system . The set of all the trajectories constitute the phase portrait of the dynamical system . The success of qualitative analysis of a dynamical system is closely coupled with a success in nding singular trajectories or critical elements . It is also extremelyimportant to understand the structure of the state space in the neighbourhood of the critical elements. The data gathered then allow to understand the global structure of the state space of the system (2.1). In Fig.2.1 the examples of stable singular trajectories of the system (2.1) are shown. There are equilibrium point, closed trajectory (periodical solution), invariant toroidal manifold (non-regular oscillations) and strange attractor (chaotic motion). Complex behaviour of the nonlinear dynamical system is often due to the existence of a number of isolated attracting sets, each attractor has its own region of attraction. All that brings about the strong dependence of motion on initial conditions and on the sequence of variations of parameters. The following notation will be used 49]: 6
  8. 8. Figure 2.1: Singular trajectories of nonlinear dynamic system. O - equilibrium point, ; - closed orbit or phase trajectory (i.e. periodic motion ), T - toroidal manifold of trajectories in the state space. Qualitative analysis pays special attention to the simplest attractors (i.e. equilibrium points and closed orbits) and their dependence on the parameters. Stable equilibrium points and periodic motions are the simplest attractors determin- ing steady{state regimes of motion of the system. Saddle{type solutions and their stable and unstable invariant manifolds of trajectories are the decisive factors in forming global structure of the state space since they de ne the boundaries of the regions of attraction 49, 39]. 2.0.1 Equilibrium solutions Equilibrium points (or else equilibrium solutions) x = xe of the system (2.1) are the solutions of the system of equations F(xe c) = 0 x 2Rn c 2M Rm (2.2) At any given values of the parameters c one can have a set of di erent isolated equilibrium points, the number of equilibrium points may change while the values of the parameters c are varied. On linearizing in the neighbourhood of the equilibrium point xe, the system (2.1) takes the form dx dt = Ax (2.3) where 7
  9. 9. x = x;xe { vector of deviations with respect to equilibrium xe, A = @F @x x=xe { Jacoby matrix. If there are complex pairs among eigenvalues f igi=1 n with the indexes and +1: c +1 = i! c +1 = i +1 then general solution of (2.3) can be written as x = xe + a1e 1t 1 + + (a cos! t ;a +1 sin! t)e t + (a sin! t+ a +1 cos! t)e t +1 + + ane nt n (2.4) The values of a1 ::: an are de ned by the initial deviation from the equilibrium point xe: a1 1 + a2 2 + ::: + an n = xo ;xe (2.5) or, using a matrix Q in which eigenvectors are the columns Q = jj 1 2 ::: njj one may express the vector a = (a1 a2 ::: an) as follows a = Q;1(xo ;xe) (2.6) Using matrix notation, the solution of (2.4) is x = xe + Q (t)Q;1(xo ;xe) (2.7) where (t) is the matrix of fundamental solutions in the eigenvectors' basis. (t) is a solution of matrix equation d dt = (0) = E where E is a unit diagonal matrix, = Q;1AQ is a Jordan block{diagonal matrix of the form = 1 0 0 0 0 2 0 0 0 0 ::: ;! ::: 0 0 ::: ! ::: 8
  10. 10. If real parts of all the eigenvalues are negative Ref igi=1 n < 0, then the trajectory 't(xo) will go towards the equilibrium point xe for any xo belonging to small neigh- bourhood of the equilibrium. The equilibrium point becomes unstable if any of the eigenvalues has a positive real part. The placement of the eigenvalues on the complex plane de nes the structure of the state space in the vicinity of the equilibrium point xe. Suppose that q eigenvalues lie to the right of the imaginary axis and p ones lie to the left, the sum p + q = n equals to the dimension of the state space. Then two integral manifolds of trajectories Ws p and Wu q connected with the equilibrium point xe can be de ned as follows : Ws p = fx : 't(x) !xe as t !+1g Wu q = fx : 't(x) !xe as t !;1g The stable p-dimensional manifold Ws p comprises all the trajectories in the state space that go to xe while t ! +1. Similarly, the unstable q-dimensional manifold Wu q comprises all the trajectories in the state space that go to xe while t !;1. For linear approximation (2.3), the manifolds Ws p and Wu q lie in the hyperplanes Lp and Lq. The hyperplane Lp is determined by p stable eigenvectors f igi=k1 ::: kp, while Lq is determined by q unstable ones f igi=l1 ::: lp: Lp = span( k1 k2 ::: kp ) Lq = span( l1 l2 ::: lq ) When nonlinear terms are taken into account, then the surfaces Ws p and Wu q will deviate from hyperplanes Lp and Lq (the farther from the equilibrium, the more the deviations). But the surfaces remain tangent to hyperplanes Lp and Lq at xe (see g.2.2). The notation Op q will be used for such an equilibrium point 49]. 2.0.2 Periodic solutions As for equilibrium points, one can write the equations for deviations with respect to an arbitrary trajectory 't(xo) dx dt = F(x c) x(0) = xo d dt = A(t) (0) = E (2.8) where A(t) = @F @xjx(t)='t(xo) { Jacoby matrix calculated along the trajectory 't(xo), (t) { matrix of fundamental solutions of the linearized system. It de nes the evolution of the small per- turbations. 9
  11. 11. Figure 2.2: Invariant manifolds of trajectories Ws p, Wu q for equilibrium and periodic solutions. Consider closed trajectory or periodic solution with the period . The trajectory is closed when the following condition is satis ed ' (x ) = x where x is a point belonging to the closed trajectory ;. If xo in (2.8) is such that xo 2 ;, then the matrix A(t) is one with periodic coe - cients. The value of (t) at time de nes stability properties of the periodic trajectory. The matrix ( ) is called a monodromy matrix . The eigenvalues of ( ) are called characteristic multipliers. The periodic solution is stable if all the multipliers lie inside the unit circle on the complex plane (note that one multiplier always equals to 1). 2.0.3 Poincare mapping Poincare mapping technique is an e ective tool for investigation of the state space in the vicinity of the closed trajectory ; or the integral invariant manifolds of the higher dimension. The equivalent n ;1-dimensional system with discrete time may be studied instead of original dynamical system. Poincare mapping can be set using a n ;1-dimensional hyperplane transversal to the closed trajectory ; at the point x . The trajectories in the neighbourhood of ; also cross . Thus every point xk 2 (belonging to some neighbourhood of x ) can be mapped into some other point of the hyperplane xk+1 2 which corresponds to the second intersection of the trajectory 't(xk) with hyperplane . The type of the mapping P depends on the intersection condition. The mapping can be one{sided (as above) or two{sided when all the intersection points are taken into 10
  12. 12. account. Poincare mapping generates the sequence xk+1 = P(xk) (2.9) where xk+1 xk 2 . Fixed point x of the mapping P P(x ) = x corresponds to the periodic trajectory ; (see g.2.3). Figure 2.3: Poincare mapping. The behaviour of the mapping P near x is described by discrete linear system derived from (2.9) xk = Axk+1 (2.10) where A = @P @x x=x { constant matrix, x = x;x { deviation vector with respect to xed point of the map- ping The orbit of the mapping P in the small neighbourhood of the xed point O with coordinates x is determined from the relationship xk = x + a1 k 1 1 + + an;1 k n;1 n;1 (2.11) where f igi=1 n;1 and f igi=1 n;1 are the eigenvalues and eigenvectors of the Jacoby 11
  13. 13. matrix A calculated at the xed point of the mapping. The parameters faigi=1 n;1 are determined using initial condition xo a1 1 + a2 2 + + an;1 n;1 = xo ;x The eigenvalues i are the characteristic multipliers for the xed point. The mapping is compressing in the direction of the eigenvector i if j ij< 1 . If there are complex pairs e i' among eigenvalues then there will be a rotation of the points belonging to the orbit of the mapping in the plane de ned by the eigenvectors and +1. The equation (2.11) will look like that: xk = x + a1 k 1 1 + + (a cosk' ;a +1 sink' ) k + (a sink' + a +1 cosk' ) k +1 + + an;1 k n;1 n;1 (2.12) The multipliers f igi=1 n;1 do not depend on the orientation of the secant hyperplane as far as it remains transversal to the trajectory. The multipliers coincide with the eigenvectors of the monodromy matrix. The behaviour of the trajectories in the state space near the closed orbit ; is in one-to-one relation with the behaviour of the mapping (2.10) in the vicinity of the xed point O. The type of the xed point can be determined provided the placement of the characteristic multiplies with respect to the unit circle in the complex plane is known. As for an equilibrium solution (2.3) for xed point, one can nd the p { dimensional stable and q { dimensional unstable invariant manifolds Ws p and Wu q of mapping points. Here p is a number of multipliers inside the unit circle, and q is the number of multipliers outside unit circle. The stable manifold Ws p is composed of the points being the results of successive mappings, each sequence of these points converges to the xed point O. This manifold corresponds to the stable multipliers j ij< 1. The unstable manifold Wu q is also composed of the points being the results of succes- sive mappings, but each sequence of these points converges to the xed point O using inverse mappings. The manifold corresponds to the unstable multipliers j ij> 1. Fixed point O corresponds to the closed trajectory ; in the state space. Invariant manifolds of mapping points Ws p and Wu q for the xed point O correspond to the invariant manifolds of the trajectories Ws p+1 and Wu q+1 (the di erence in the dimension of the manifolds is 1). That's why the notation ;p+1 q+1 will be used to describe the type of the closed trajectory. Trajectories from Ws p+1 manifold go closer to ; while t !1, and those from Wu q do so while t ! ;1. Other trajectories from the neighbourhood of ; rst go closer to ; along Ws p+1, and then go away of it along the Wu q+1 manifold ( g. 2.2). 2.1 Bifurcational analysis Investigation of the possible changes in the structure of the state space due to the changes of parameters c is a vital part in the methodology of the qualitative analysis. 12
  14. 14. There are critical or bifurcational values of parameters cb when qualitative type of the state space structure changes. The most common is the so-called one{parameter bifurcation when in theparameter space near the point cb one can nd dynamical systems with only two di erent types of state space. Global changes of the state space structure are the result of successive local bifur- cations in the vicinity of the singular trajectories or critical elements (i.e. equilibrium points, closed orbits etc.) and bifurcations with the invariant manifolds Ws p and Wu q . 2.1.1 Bifurcations of the equilibrium points There is a condition that allows to distinguish bifurcational situations for the equilibrium point O of (2.1) with coordinates x = xe. The point c (in the space of parameters) is not a bifurcational one for the equilibrium O provided there are no eigenvalues (computed at the equilibrium point) on the imaginary axis in the complex plane. In that case, the dimensions p and q of the invariant manifolds Ws p and Wu q remain the same for all c close enough to the initial value, hence the type of the equilibrium Op q remains the same. Otherwise the point in space of parameters cb will be bifurcational one. The two simplest and the most common bifurcational situations for equilibrium so- lutions are there is one eigenvalue equal to zero, there is a complex pair on the imaginary axis. The surfaces No and N! of codimension 1 in the space of parameters M correspond to these situations. The equation for No is Det 0 @ @F @x x=xe 1 A = 0 (2.13) The surface N! is de ned by the equation Det 0 @@F @x x=xe ;i!E 1 A = 0 (0 < ! < 1) (2.14) Consider the characteristic equation of the linearized system in the vicinity of the equilibrium n + a1 n;1 + a2 n;2 + + an;1 + an = 0 (2.15) Using coe cients ai = ai (c) i = 1 ::: n, one can express the equations for the bifurcational surfaces No (2.13) and N! (2.14) as follows an = 0 and 13
  15. 15. n;1 = a1 1 0 0 a3 a2 a1 1 a5 a4 a3 a2 = 0 where n;1 is the last but one Raus determinant for the characteristic equation (2.15). When the point in the space of parameters crosses the bifurcational boundary No, then the following bifurcations may occur with the initially stable equilibrium point stable and saddle-type unstable equilibrium solutions merge and vanish. An abrupt loss of stability takes place. Bifurcation scheme is On o + On;1 1 ! stable point becomes saddle-type unstable, at the same time two more stable equilibrium points appear. The soft" loss of stability occurs. Bifurcation is On o !On;1 1 + 2On o the merge of two saddle-type and one stable equilibrium solutions. The loss of stability is abrupt. On o + 2On;1 1 !On;1 1 When the bifurcational boundary N! is crossed, the Andronov{Hopf bifurcation occurs. There are two forms of the Andronov{Hopf bifurcation with initially stable equilibrium point ( g. 2.4): the equilibrium becomes oscillatory unstable. At that moment stable periodic solution is detached from the equilibrium. Soft-type loss of stability takes place. On o !On;2 2 + ;n 1 the stable equilibrium solution merges with saddle-type unstable periodic trajec- tory and becomes oscillatory unstable. The abrupt loss of stability occurs. On o + ;n;1 2 !On;2 2 The type of the loss of stability mentioned above takes place when the point in the space of parameters slowly enough passes through the bifurcational boundary. Soft-type loss of stability means that the trajectory in the state space (started in the vicinity of the equilibrium) slowly goes away from the equilibrium point while the values of parameters are close to the critical ones. If the parameters return to the initial values, then the trajectory return to the neighbourhood of the equilibrium point. When the loss of stability is abrupt, then the trajectory in the state space quickly leaves the equilibrium point for some other attractor even if the deviations of the pa- rameters from critical values are very small. If the parameters then return to the initial values, the trajectory usually would not return to the initial equilibrium point. 14
  16. 16. The bifurcation points where two or more branches of the equilibrium solutions intersect ( g.2.4), has the codimension greater or equal to 2. So at least two conditions must be satis ed. Hence, in general case, such points are structurally unstable. Possible changes in the extended state space (the parameter axis is added) in the vicinity of the branching point are shown in g.2.5 Figure 2.4: Bifurcations of equilibrium solutions. 15
  17. 17. Figure 2.5: Changes of extended state space structure near branching point. 2.1.2 Bifurcations of the periodic trajectories The stability type of the xed point O of the mapping P and of the periodic trajectory ; may change when the characteristic multipliers cross the unit circle on the complex plane. That's why only those values of parameters cb are bifurcational when there exists a multiplier belonging to the unit circle. Consider the bifurcations of the stable closed orbits. Three simplest cases are multiplier crosses the unit circle at the point (+1.,0.), multiplier crosses the unit circle at the point ({1.,0.), a pair of multipliers crosses the unit circle at the point e i'. In these cases the simplest bifurcational surfaces N+1,N;1,N' of codimension 1 in the space of parameters are generated (see g. 2.6). Bifurcational surfaces N+1 and N' for the xed point are equivalent to the bifurca- tional surfaces No and N! for theequilibrium point, thesurface N;1 representssomething new. There are three more situations when periodic trajectory vanishes: 16
  18. 18. Figure 2.6: Bifurcations of periodic solutions. the periodic trajectory ; shrinks into a point, an equilibrium point emerges on the closed orbit ;. some point belonging to ; goes to in nity, thus the curve is no more closed. When the boundary N+1 is crossed, then two periodic trajectories either merge and vanish or emerge simultaneously (in the latter case the abrupt loss of stability occurs): ;n 1 + ;n;1 2 ! !;n 1 + ;n;1 2 When the boundary N' is crossed then stable closed orbit ;n 1 becomes unstable ;n;2 3. At the same time the stable 2{dimensional toroidal invariant manifold Tn 2 emerges. In the second possible pattern unstable periodic solution ;n;2 3 appears when unstable 2{dimensional toroidal invariant manifold Tn;2 4 and stable closed orbit ;n 1 merge. The bifurcations may be written as follows ;n 1 !Tn 2 + ;n;2 3 { soft loss of stability, ;n 1 + Tn;2 4 !;n;2 3 { abrupt loss of stability If the boundary N+1 is crossed, there may be bifurcations when branching of the periodic solutions takes place: ;n 1 !;n;1 2 + 2;n 1 { soft loss of stability ;n 1 + 2;n;1 2 !;n;1 2 { abrupt loss of stability 17
  19. 19. When the bifurcational boundary N;1 is crossed, then the changes in comparison with equilibrium bifurcations are somewhat unusual. Stable periodic trajectory with the period ;n 1 becomes unstable one ;n;1 2, and at the same time stable periodic trajectory with double period ;n 1 2 detaches from the initial closed orbit. ;n 1 !;n;1 2 + ;n 1 2 Similarly, the merge of the saddle-type unstable double period closed orbit ;n;1 2 2 with stable periodic solution ;n 1 is possible ;n 1 + ;n;1 2 2 !;n;1 2 18
  20. 20. Chapter 3 Numerical algorithms and methods 3.1 Continuation method Continuation method is intended for computing a dependence of a solution of a nonlinear system of equations on a parameter. Consider system of nonlinear equations F(x c) = 0 (3.1) where vector function F 2 Rn, x is a vector in n-dimensional state space, c is a scalar parameter. The system (3.1) de nes in extended state space (x, c) one-parameter set of solutions. In general case this set consists of a number of spatial curves (some of these curves may intersect), see Fig.3.1. Functions Fi i = 1 2 ::: n are continuous and have continuous partial derivatives with respect to all variables xi and the parameter c. In general, F is some smooth mapping (i.e. as many times di erentiable as needed) F : Rn+1 ! Rn, Poincare point mapping for example. If the parameter c is changed continuously, then the solution x(c) set implicitly by equation (3.1) will also change continuously. To establish the dependence x(c), the following equation can be considered: @F @xdx+ @F @c dc = 0 (3.2) At limit and branching points Jacoby matrix @F @x becomes singular. To overcome the singularity, the trajectory is parameterized using such "natural" parameter as curve length s 26, 41]. When moving along the trajectory of the solution z = (x c), the increments of the state vector and the parameter are de ned by the projections of the unit tangent vector s. The vector s is normal to the rows of the matrix @F @z = @F @x @F @c ! So, if the matrix 19
  21. 21. has full rank rank @F @z = n (that's true at all points except branching ones), then the rows of this matrix together with tangent vector s constitute the basis of subspace z and corresponding matrix is non-singular 41] : = 0 BBBBBBBBB@ @F1 @x1 @F1 @x2 @F1 @xn @F1 @c @Fn @x1 @Fn @x2 @Fn @xn @Fn @c s1 s2 sn sn+1 1 CCCCCCCCCA = 0 BB@ @F @z s0 1 CCA (3.3) Since the vector s is normal to the rows of the matrix @F @z, hence its components si are determined by the relationships : si = An+1 i det (3.4) where An+1 i are algebraic supplements of the elements of the bottom row of the matrix, and det = vuut n+1X j=1 A2 n+1 j Therefore the trajectory of the solution (x c) in the extended state space can be de ned in di erential form dz dt = k1s or 8 >>>>>>>>< >>>>>>>>: dxi dt = k1si i = 1 2 ::: n dc dt = k1sn+1 = k1 det @F @x ! det , (3.5) with initial conditions z = zo, F(zo) = 0. Here s is a scalar parameter along the curve. Integrating numerically equations (3.5), one can obtain a continuous branch of the solution curve (3.1) in the extended state space z. As seen from the last equation in (3.5), determinant of the matrix @F @x changes its sign in turning points. These points correspond to the bifurcation points of dynamical system _x = F(x c). When integrating the di erential equations (3.5) numerically, one inevitably makes some error and deviate from the genuine solution curve. To build a continuation method stabilizing the motion along the solution curve (3.5) for a system of nonlinear equations (3.1), one should transform the di erential system 20
  22. 22. Figure 3.1: One-parameter family of solutions of the system F(x c) = 0 in extended state space z = (x c). (3.5) such that trajectory curve becomes an attractor in modi ed system 41]. One can append the equations (3.5) with linear equation dF dt + k2F = 0 (3.6) The solution of this auxiliary equation is in the form F = Foe;k2t, where positive constant k2 sets the speed of decreasing the error kFk= vuut n+1X j=i F2 i while increasing t. Making substitutions in the left side of (3.6) and using z variables, we obtain @F @x dx dt + @F @c dc dt = @F @z dz dt = ;k2F Scalar product of this equation with s vector may be appended to eqs.(3.5) resulting in joint system of equations dz dt = ;k2F k1 ! (3.7) At the turning points on the curve when matrix @F @x becomes singular, matrix remains regular, and system of equations (3.7) also remains regular. Thus one can 21
  23. 23. rewrite (3.7) in the following form: dz dt = ;1 ;k2F k1 ! (3.8) When integrating equations (3.8) with k1 = 1, not only following the curve", but also an asymptotic convergence to the solution curve starting from rather wide region in z space is assured. If k1 = 0 then only convergence to the solution curve z(t) along subspace normal to a family" of curves F(x c) = Fo is done, Fo being a constant vector. Equations (3.8) enable the computation of the desired solution with automatic elim- ination of the error kFk6= 0 due to numerical integration procedure with xed step 4t, and the errors in the initial conditions. Assuming coe cient k1 = 1 4t, where 4t is a parameter increment (i.e. step of numerical integration along the curve), the correction vector 4z can be obtained from the system of linear equations @F @z4z = ;F s04z = 0 (3.9) Correction vector 4z is calculated using a condition that it is normal to unit tangent vector s, the condition ensures the uniqueness of correction vector. To calculate the solution curve x(c) one can use a numerical method which includes the following steps: 1. Start. Calculation of the initial point xo co : F(xo co) = 0 either using known methods, with xed c = co or using eqs (3.9). Iterations continue until error jjFjj becomes less than some prede ned value. In many cases convergence is much better if the parameter c is allowed to vary. 2. A unit tangent vector s is calculated using (3.4). 3. Finite step along trajectory 4t is obtained from a condition, that increments of all variables 4zi do not exceed some prede ned values 4zi : 4t = mink (4zi =sk). 4. The next point on a curve is calculated using equations (3.2). After that a correc- tion is made (see 3.9) until required accuracy is reached. 5. Go to 2. Problem of passing through branching point is due to non-uniqueness of the continuation direction s at such point. The matrix becomes singular (det = 0), and rankFz (n ;1). In the case of two intersecting branches, rankFz = (n ;1) and the intersecting curves touch (in linear approximation, lay in) some 2-dimensional plane passing through the branching point. The plane is de ned by two eigenvectors of the (n+1 n+1)-matrix FT z Fz corresponding to two zero eigenvalues. It is not too di cult to nd the second branch using 1-dimensional search method. 22
  24. 24. 3.2 Systematicsearchfor solutions of nonlinear sys- tem of equations Method for systematic search of the set of solutions of nonlinear system of equations (SSNE method, systematic search method) is a regular algorithm that looks for unknown solutions of nonlinear system of equations of the order N > 1. At least one solution should be known in advance. One cannot guarantee full success however, in many cases the method allows to nd all the solutions. The SSNE algorithm is based on continuation technique 41]. There are only a few papers devoted to systematic search algorithms 40, 47]. Consider a system of nonlinear equations : f (x) = 0 where f 2RN x 2RN (3.10) and suppose that xo is a known solution of the system (3.10) : f (xo) = 0 First one can choose an arbitrary unit vector ~n in the most general case ~n might depend on vector x. Then a set of mutually-normal unit vectors k, k = 1 ::: N ;1, normal to ~n is computed. Together ~n and k form basis B of N-dimensional linear space : (~n0 k) = 0 k = 1 ::: N ;1 (~n0 ~n) = 1 ( 0 i j) = ( 0 i 6= j 1 i = j B = jj 1 ::: N;1 ~njj B0 B = EN (3.11) Consider auxiliary system of equations G n] : G n] (x) = 0 where G k] 2RN;1 x 2RN (3.12) and scalar function g n] (x) obeying the following conditions : G n](x) g n](x) = B0 f(x) or g n](x) = (~n0 f(x)) G n] k (x) = ( 0 k f(x)) k = 1 ::: N ;1 (3.13) From (3.13) it follows that if xo is a solution of basic system (3.10), then it will be a solution of auxiliary system too : G n] (xo) = 0 g n] (xo) = 0 (3.14) 23
  25. 25. If basic system (3.10) is non-singular and has only isolated solutions, then the system G n](x) = 0 (consisting of (N ;1) equations and depending on N unknowns) determines some spatial curves of co-dimension 1. There may be only one curve or a number of them di erent curves can intersect at branching points. Now choose xo as initial point and apply continuation method towards auxiliary system (3.12) 41]. As a result 1-dimensional spatial curve Ln] 1 will be computed (or a number of them, since advanced continuation method allows to compute all the branches intersecting at branching points). The curve is a set of solutions of auxiliary system. At each point belonging to the curve the value of the function g n] (x) is determined. If there exists such a point x 2 Ln] 1 belonging to Ln] 1 , that the function g n] (x ) changes sign in its vicinity, then x is the solution of equation g n] (x ) = 0. Hence x is (might be, unknown) solution of basic system : f (x ) = 0. Computing numerically points belonging to the curve Ln] 1 and selecting the points where the function g n] changes sign, one can determine a number of new solutions of the basic system of equations (3.10). When is the SSNE algorithm able to nd all the solutions? The following Theorem is valid 39] : if there exists such an auxiliary system G n] that it de nes connected set of curves Ln] 1 , then 1) SSNE algorithm will nd all the solutions of the basic system, and 2) it will be enough to consider only this one auxiliary system G n]. If the set of the curves Ln] 1 is a connected one, continuation method easily computes all these curves. Since all the solutions of the basic system also belong to this set, all of them can be found. There is one important special case when known solution xo of the basic system (3.10) is close to bifurcation with some other solutions. In the small neighbourhood of xo basic equations are equivalent to a system consisting of one nonlinear and (N ; 1) linear equations provided proper system of co-ordinates is chosen. Selecting vector normal to the rows of (N ;1) N-matrix determining linear equations, as ~n vector, one can get linear auxiliary system. Hence, the set Ln] 1 consists of only one spatial straight line. That means that SSNE method will nd all the solutions belonging to close-to- bifurcation family. In practice it is reasonable to try a number of di erent ~n vectors. One simple ap- proach showed good results 47], when the vectors are selected from the following family : ~n 2fe1 ::: eNg where ek i = ( 1 i = k 0 i 6= k As an example, consider the system of equations : 8 < : fi6=k = ai 1x1 + ::: + ai NxN fk = '(x) where A = faijgis an (N-1) N-matrix of the rank N-1, and ' is an arbitrary nonlinear function. Linear auxiliary system G n] = Ax de ne straight line in state space, and again SSNE method will easily nd all the solutions determined by the equation '(x) = 0. 24
  26. 26. u u u u C D B A L2] 1 L2] 1 L1] 1 L1] 1 y x f1 (x y) = (x; 1)(x+ 1) f2 (x y) = (y ; 1)(y +1) 4 solutions: A : (x y) = (1 1) B : (x y) = (1 ;1) C : (x y) = (;1 1) C : (x y) = (;1 ;1) Figure 3.2: System with four connected solutions. Experience with SSNE applications shows that method works much more e ciently if a series of auxiliary systems are used consecutively. Ability of continuation method to detect and analyze branching points is very important. Another way to increase the e ciency of the method is to re-start it using each newly-found solution as the initial one. Three simple examples, presented in Fig.3.2, show the peculiarities of the SSNE algorithm. The top plot shows rather simple situation. The system of equations has four solutions, each solution is connected with another two solutions via Ln] 1 , ~n = ek curves. Working with only one auxiliary system, one cannot nd more that one new solution. To get all the solutions, it is necessary to consider two auxiliary systems and to re-start the algorithm from newly-found solutions. In the middle plot of Fig.3.2 only computation of all the branches intersecting at branching point allows to nd the second solution. In the case of bottom plot the algorithm will fail because the solutions are not connected by the curves Ln] 1 it is necessary to select better values of unit vector ~n. So, SSNE algorithm is not a universal one, it cannot guarantee a success. At the same time it is very useful and e cient in di erent applications. Example in Fig.3.5 shows computation of the set of solutions for the aircraft roll-coupling problem (5th order system) dashed lines are the projections of the curves Ln] 1 onto the plane of the plot. One more example, taken from 52], is shown in Fig.3.6. The steady-state conditions of 4-reactor chemical plant were investigated. This time SSNE algorithm managed to nd 14 new solutions. 25
  27. 27. u u B A L1 1 L2 1 L1 1 L2 1 y x f1 (x y) = (x;1)(y ; 1) f2 (x y) = (x+1)(y + 1) 2 solutions: A : (x y) = (1 ;1) B : (x y) = (;1 1) Figure 3.3: System with two solutions connected via branching point. u u B A L2 1 L2 1 L1 1 L1 1 y x f1 (x y) = x2 ; 1 f2 (x y) = xy + 1 2 solutions: A : (x y) = (1 ;1) B : (x y) = (;1 1) Figure 3.4: System with two non-connected solutions. 26
  28. 28. Figure 3.5: Example of SSNE method application Figure 3.6: Steady-state conditions of 4-reactor chemical plant 27
  29. 29. 3.3 Computing manifolds of equilibrium solutions and bifurcation sets A number of coexisting equilibrium solutions under the same conditions (i.e. values of parameters) may be di erent. One can consider regions in the space of the parameters where under the conditions de ned by each point of the region there exists the same number of equilibrium solutions. Boundary surfaces of these regions at the parameter space correspond to bifurcations of dynamical system. In most cases bifurcation results in birth" or death" of a pair of equilibrium solutions, one of these solutions always being unstable. It is possible to reconstruct qualitatively the dynamics of the system at the conditions of varied parameters if manifolds of equilibrium solutions and corresponding bifurcation sets can be determined. If a dynamical system is in the form dx dt = F(x c) x 2Rn c 2Rm (3.15) then a manifold of equilibrium solutions (equilibrium surface) Ee of the system 3.15 is de ned as follows Ee = fx : F(x c) = 0 x 2Rn F 2Rn c 2M Rmg When two parameters are varied, then it is convenient to express the dependence of the solution on parameters as a number of two-dimensional surfaces in 3-dimensional space (two parameters and one of state variables). A huge amount of information about system's behaviour can be gathered through analysis of the smooth maps of these surfaces onto the plane of parameters. The most common types of bifurcations are the so-called fold", cusp", butter- y" and swallow's tail" bifurcation patterns well-known from catastrophe theory for gradient systems 51]. The equilibrium surfaces can be calculated using either continuation method or by scanning across the plane of two selected parameters c1 and c2. A set of bifurcation points Be of the equilibrium surface Ee is de ned by a condition Be = n x : G(x c) = 0 x 2Rn G 2Rn+1 c 2M Rm o where G = 0 BBB@ F(x c) det @F @x 1 CCCA If a dependence of one state variable on two parameters c1 and c2 is studied, then Be is a manifold of dimension 1. The projection of Be onto plane of parameters gives boundaries N0, which separate regions with di erent number of equilibrium solutions. 28
  30. 30. Boundaries N0 are also called bifurcational portrait or diagrams. These diagrams may have cusp points. The curve Be and bifurcational diagrams N0 can be computed using continuation method. Continuation method employs the extended Jacoby matrix @G @z , where z = (x c), and thus requires evaluation of gradient of the determinant = detkFxk of matrix Fx. Partial derivatives of the determinant of Jacoby matrix @ @zk k = 1 2 ::: n+2 may be expressed using partial derivatives of F(x c) vector function @ @zk = nX i=1 nX j=1 @2Fi @zk@xj Aij k = 1 2 ::: n + 2 (3.16) where Aij are algebraic complements to the elements of the Jacoby matrix ( @Fi @xj )i=1 n j=1 n . If there are no analytic expressions for partial derivatives @ @zk , they can be computed using known multipoint numerical schemes. 3.4 Computation of the periodic solutions and their stability Periodic solutions of the system of nonlinear equations dx dt = F(x c) x 2Rn c 2M Rm (3.17) corresponding to closed orbits in the state space, can be found using di erent iterative numerical algorithms. The problem is reduced to a search for xed point of appropriate mapping. Time{ advance mapping 't(x) and mapping of hypersurface of codimension 1 onto itself may be used. 3.4.1 Time{advance mapping. The requirement for a trajectory in state space to be closed results in a search for zeros of nonlinear vector function H H(x ) = ' (x);x (3.18) where ' (x) is, for example, time-advance mapping of initial point x with time interval . Thus we need to nd a period of the solution and some point x belonging to the closed orbit ;. 29
  31. 31. The system (3.18) comprise n equations depending on n+1 unknowns. So it de nes a set of solutions x 2; depending on one parameter. If x =2 ; and H 6= 0, one can obtain equations in increments 4x and 4 in the vicinity of the closed orbit ( g. 3.7) @' @x ;E ! 4x+ F(' (x))4 = ;H(x ) (3.19) Matrix @' @x x2; , calculated in some point belonging to the curve ; while being the period, is called the monodromy matrix. Monodromy matrix always has a unit eigenvalue, and the corresponding eigenvector 1 coincides with tangent vector to a closed trajectory at x: 1 = F(x). Figure 3.7: Time-advance mapping Consider a small shift along the closed trajectory: x = F(x). The result of a mapping applied to a new point ' (x+ x) so we have x+ x = ' (x+ x) (3.20) hence 30
  32. 32. F(x) = @' @x F(x) (3.21) Thus the matrix @' @x ;E ! in the left hand side of the equation (3.19) is singular in the vicinity of the closed orbit ;, i.e. it has zero eigenvalue (with a corresponding eigenvector F(x)). For nite solutions x of the equations (3.19) to exist, it is necessary that a vector in the right hand side of the equations @' @x ;E ! x = ;H(x );F(' (x)) (3.22) is normal to vector F(x), which is eigenvector of @' @x matrix (3.21), i.e. F0 (x) H(x )+ F(' (x)) ] = 0 (3.23) where F0 denotes a transposed vector F. This condition may be used for calculation of the increment of time period = ; F0(x)H(x ) F0(x)F(' (x)) (3.24) Geometrically the relationship (3.24) means that a nal point on the trajectory ' (x) must belong to a hyperplane F, hyperplane F being a set of vectors normal to F(x) vector. If ' (x) 2 F , then period doesn't change during convergence to a closed orbit. Thus hyperplane F de nes mapping with a constant period in a vicinity of a closed orbit. It follows also that all other eigenvectors i i = 1 2 ::: n of the monodromy matrix @' @x lie in the hyperplane F and hence are normal to vector F(x). General solution x of the system of equations (3.22) will include a certain shift in the F hyperplane and some arbitrary displacement along vector F. In order to minimize absolute value of x, it is reasonable to append equations (3.23) with a condition that requires x to be normal to the tangent vector F0 x = 0 : @' @x ;E F(' (x)) x ;H(x ) = F0 0 0 (3.25) The resulting system of equations (3.25) is regular. Thus one can remain in the hyperplane normal to a closed orbit during convergence to the orbit. 31
  33. 33. Eigenvalues (1 1 ::: n;1) and eigenvectors F(x) 1 ::: n+1 of the matrix @' @x x2; de ned in a point belonging to a closed orbit provide an information about stability of the periodic solution and about the structure of the state space in the vicinity of the closed orbit. Monodromy matrix @' @x is computed by means of numerical integration of equations (3.18) with initial conditions being varied successively with xi i = 1 2 ::: n @' @x = ' (x+ xi) ;' (x; xi) 2 xi ! i=1 ::: n (3.26) One can check the accuracy of computation knowing that always there is an eigen- value equal to 1 with corresponding eigenvector F(x). 3.4.2 Poincare mapping. Poincare point mapping P: ! of secant (n;1){dimensional hyperplane (transver- sal to phase trajectories in the vicinity of closed orbit) can also be used for nding periodic solutions. Let's consider the properties of that mapping. Transversal intersection of the trajectory in the state space with secant hyperplane results in a sequence of points. One can distinguish these points according to direction of intersection with . It is possible to consider the subsequence which includes points with the same direction of intersection fx1 x3 x5 :::gwhen trajectory leaves half-space ; for half-space +. Full sequence form an orbit of bidirectional Poincare mapping fx1 x2 x3 x4 x5 :::g (see g. 2.3). Also K-fold iterated Poincare mapping can be considered. In that case point of the each K-th intersection in the chosen direction is included into the sequence. K-fold mapping Pk (x) is used to study periodical solutions with K-fold period. hyperplane is set using two vectors: = fx : h0(x;xo) = 0g (3.27) where vector xo is some point on hyperplane, and vector h is normal to hyperplane . When integrating the trajectory one can determine in what half-space current point x is located: + = fx : h0(x;xo) > 0g (3.28) or ; = fx : h0(x;xo) < 0g (3.29) 32
  34. 34. The point of intersection x with can be re ned using linear interpolation between two successive point on the trajectory x1 2 + and x2 2 ; x = x1 H2 H1 + H2 + x2 H1 H1 + H2 H1 = h0(x1 ;xo) > 0 H2 = ;h0(x2 ;xo) > 0 (3.30) To satisfy the required accuracy for intersection point h0(x ;xo) (3.31) iterative procedure can be used. It comprises a backward step along trajectory (in reverse time) of the size t = t2 ;t , where t = t1 H2 H1 + H2 + t2 H1 H1 + H2 (3.32) and once more linear interpolation (3.30). It is possible to look for xed points of the point mapping xk+1 = P(xk) or for zeros of vector function H(x) = P(x) ;x, x 2 using variations x that keep the vector x in the hyperplane . A basis i of linear independent vectors i i = 1 2 ::: n ;1 for is to be constructed rst, the orthogonalization procedure starting with normal to hyperplane vector h may be used. Matrix @P @x eigenvalues do not depend on the choice of hyperplane until it remains transversal to the closed orbit. Let i i = 1 2 ::: n ;1 be eigenvectors of the matrix @P @x. If xk = 1 1 + 2 2, where 1 2 are constants, and 1 2 are eigenvectors forming complex eigenvector c = 1 i 2 corresponding to a complex pair of eigenvalues e i', then the deviation will be de ned as follows xk+1 = ( 1 cos' + 2 sin') 1 + (; 1 sin' + 2 cos') 2 (3.33) Suppose that the other secant hyperplane h is de ned, vector h being normal to it. Let the trajectory to intersect with h in points xk and xk+1. If this trajectory intersects with F in xk and xk+1, then xk = xk ; x0 kh F0h F xk+1 = xk+1 ; x0 k+1h F0h F (3.34) 33
  35. 35. The map (3.34) is linear, hence xk = ( 1 cos' + 2 sin') 1 + (; 1 sin' + 2 cos') 2 (3.35) where i = i ; 0 ih F0h F It means that Jacoby matrix for the point mapping de ned on h hyperplane has the eigenvectors 1 , 2 (3.35) and eigenvalues as the matrix for mapping de ned on F hyperplane. These eigenvalues are the characteristic multipliers for the periodic solution. Jacoby matrix of the point mapping can also be computed using arbitrary variations x of initial point, so that the variations leaving the hyperplane h are allowed. The result of the mapping P(x+ x) belongs to hyperplane. In this case the matrix of linearized map @P @x can be linked with monodromy matrix @' @x @' @x = @P @x + Fh0 F0h (3.36) Let F 1 ::: n;1 be the eigenvectors and 1 1 ::: n;1 be corresponding eigen- values of @' @x matrix. One can check that F vector is an eigenvector of @' @x , but corresponds in this case to zero eigenvalue. Indeed, @' @x F = F @P @xF = F; Fh0 F0hF = 0 (3.37) All other eigenvalues of @P @x, as well as those of @' @x matrix, correspond to charac- teristic multipliers of the periodic solution. One may get the eigenvectors using linear transformation of this shift along trajectory @P @x i = i i i = i ; 1 i h0 i F0h (3.38) 34
  36. 36. Iterative search for a periodic solution is performed in accordance with ( x = xi+1; xi) @P @x ;E ! x = ;H(x) = ;P(x) + x (3.39) Unlike (3.19), the matrix in the left hand side (3.39) is regular. The value of x can be obtained directly solving linear system of equations (3.39). Continuation algorithm may be e ciently used to investigate the dependence of the periodic solutions and their stability on parameters, zeros of the vector function H(x c) = P(x c) ;x = 0 (3.40) can be studied. 3.5 Algorithms for stability regions computation Every stable attractor (equilibrium point, periodic trajectory, etc.) has its own domain of attraction, also referred to as stability region. Domain of attraction is a set of initial points in state space, trajectories starting from these initial conditions approach the attractor as t !1. In many cases boundaries of stability regions are formed by stable invariant manifolds generated by saddle equilibrium and saddle periodic trajectories 39]. For planar dynamical systems the boundaries of stability regions are either sta- ble" trajectories of saddle equilibrium points or unstable limit cycles. They de ne the boundaries of all the domains of attraction. If the dimension of the state space n 3 then stable manifolds of trajectories Ws of the codimension 1 passing through saddle equilibrium points or saddle periodic trajec- tories are the most widespread boundary "components". Qualitative analysis of dynamical system is done before the investigation of stability regions. In particular, the analysis of stable and unstable invariant manifolds of critical elements like saddle equilibria and saddle closed orbits must be performed. First those saddle critical elements must be found whose unstable manifolds Wu have trajectories approaching to the stable equilibrium point or closed trajectory. 3.5.1 Computation of stability region boundaries Consider the algorithm of searching for the boundary of stability region formed by stable invariant manifold corresponding to saddle equilibrium with one positive eigenvalue 42]. The boundary of stability region Ws n;1 in the state space can be represented by a number of its cross-sections by two-dimensional planes P2 (see Fig.3.8). The intersection between P2 and Ws n;1 results in the curves @Sp : @SP = P2 Ws n;1 35
  37. 37. Figure 3.8: Algorithm computing stability region cross-sections These curves are the boundaries of the stability cross section. Trajectories passing through the points of the curve @SP belong to Ws n;1, and they approach the saddle point as t !1. If trajectory initial point is moved aside from the curve @SP (but still remains it in the plane P2), then the trajectory will miss the saddle point. One can introduce a frame of reference on the cross plane P2 - xk xi. Now it is necessary to de ne a measure of miss. Consider the hyperplane Ln;1 passing through the saddle point and tangent to the Ws n;1. Ln;1 is a linear n;1-dimensional hyperplane with a basis formed by stable eigenvectors of Jacoby matrix (computed at saddle point). The hypercylinder CR is de ned as a set of points being on the same distance R from the straight line passing through saddle point and parallel to unstable eigenvector . There exists a neighbourhood in the vicinity of a point on the curve @SP P2, all the trajectories started from the points belonging to the neighbourhood cross the surface of the cylinder CR. The maximum size of that neighbourhood is depends on the radius R of the cylinder. A distance from the intersection point (where trajectory crosses the surface of CR) to the hyperplane Ln;1 is used as a measure of miss, see Fig.3.8. The measure of miss H is de ned as H(x) = (x;xe)0 (3.41) where xe is a saddle point vector, and is a unit vector normal to Ln;1 hyperplane (it can be calculated as unstable eigenvector of transposed Jacoby matrix). 36
  38. 38. Scalar function r(x) evaluates the distance between current point on the trajectory x and the surface of the cylinder CR. r(x) = kx;xe ;H(x) k;R (3.42) where kxk is euclidean norm of the x vector. If r(x) changes its sign, it means that cylinder CR is crossed by the trajectory. The mapping of a point of a plane P2 onto the surface of the cylinder CR is determined by the condition r(x) = 0. For everypoint belonging to some neighbourhood of the curve @SP 2 P2, the mapping de nes a scalar value H(xR) which is denoted as HR(xk xi). The mapping HR(xk xi) is performed by numerical integration of dynamical system, the sign of r(x) is checked during integration. The problem of computing the curve@SP (i.e. thesection of stability region boundary Ws n;1 by the plane P2) is reduced to nding and continuation of a solution of the scalar equation HR(xk xi) = 0 (3.43) The equation (3.43) depends on two parameters xk and xi { the coordinates of the initial point of the mapping with respect to the frame of reference de ned on the P2 plane. To use algorithm e ciently, the radius of the cylinder R is to be chosen properly. The radius must be small enough for the tangent hyperplane Ln;1 to be a reasonable approximation of the invariant manifolds Ws n;1 inside the cylinder CR. At the same time, the radius must be large enough to minimize calculation time and to establish reasonable domain of algorithm convergence. Condition (3.43) de nes an approximate boundary of stability region. This boundary tends to the exact boundary of the section of invariant manifolds Ws n;1 when R !0. The solution of (3.43) is calculated using the continuation method. First the method converges to a solution of (3.43), and then a continuation of the curve @SP is performed : dxk dt = ;kHRHxk R ;Hxi R (Hxk R )2 + (Hxi R )2 dxi dt = Hxi R ;kHRHxk R (Hxk R )2 + (Hxi R )2 : (3.44) Required accuracy of computations is reached by means of proper selection of the co- e cient k and integration step t in (3.44). The partial derivatives Hxk R and Hxi R are obtained numerically. 3.5.2 Direct computation of stability region cross-sections If computer is powerful enough, then the direct computation of two-dimensional cross sections of stability region veryoften is more e cient than other methods. The algorithm includes the following stages : 1. Known attractors are written into memory. Periodical solutions and more complex attractors are represented by an arbitrary point belonging to the attractor. 37
  39. 39. Figure 3.9: Direct computation of stability region boundaries For each attractor some reasonably small region Si is ether determined or selected. The region Si must be a subset of attractor's stability region. The regions Si i = 1 2 :::, for example, can be estimated by means of Lyapunov function method 38], see g.3.9. To determine all the attractors in advance is desirable but not obligatory for this algorithm. 2. Maximum integration time and boundaries of an allowed state space region are speci ed. Maximum integration time should be big enough (not less that 100 characteristic periods of time for the system in question). 3. Two-dimensional secant plane is selected, and some frame of reference on that plane is de ned. 4. A grid on a secant plane is selected, the grid de nes initial points for numerical integration of the dynamical system. 5. System of equations is integrated starting from every grid point. Integration stops if a trajectory enters inside anyone of the regions of guaranteed stability Si i = 1 2 :::, or if maximum integration time is exceeded, or if the trajectory leaves allowed state-space region. If maximum integration time was exceeded,the process of automatic adding of new attractors may be trigged. Last trajectory point is registered as a new attractor and some small region of guaranteed stability is assigned to it. 38
  40. 40. 6. Grid points are classi ed according to the results of integration. Either they belong to i-th attractor stability region or situation is uncertain. As s result, a section of stability region may be determined with desired accuracy. The ability of the algorithm to look for new attractors may be handy for the analysis of the state space structure. The computation time for this algorithm depends on the number of grid points and on the size of the regions Si i = 1 2 :::. Though usually time-consuming, this method is simple, reliable and e cient. 3.6 Example: Global stability analysis in roll-coupling problem The roll coupling problem at fast roll rotation is usually described by the following 5th order system of equations (supposing that gravitational terms and velocity variation are negligible): _ = q ;(pcos ;rsin )tan + z _ = psin ;rcos + y _p = ;i1qr + l _q = i2rp + m _r = ;i3pq + n, (3.45) where p, q, r are angular velocity components in body-axis, , are angle of attack and sideslip, i1 = (Iz ;Iy)=Ix, i2 = (Iz ;Ix)=Iy, i3 = (Iy ;Ix)=Iz are non-dimensional inertia coe cients, l = L=Ix, m = M=Iy, n = N=Iz, y = Y=MV , z = Z=MV . The reduced normal and side forces z, y, pitch, roll and yaw moments m, l and n in equations (3.45) can be represented as a functions of state variables and control parameters both in linear or nonlinear forms: z = z( )+ zq( )q +z e ( ) e + y = y0 + y ( ) + yr( )r +y r ( ) r + m = m( ) +mq( )q + m e( ) e +4m( :::) l = l ( ) +lp( )p+lr( )r +l r ( ) r + l a ( ) a + 4l( :::) n = n ( ) + np( )p+nr( )r + n r( ) r + n a( ) a +4n( :::) (3.46) The equations (3.45) complemented by (3.46), form the closed nonlinear autonomous system of fth order with the state vector x = ( p q r)T. The nonlinearities in these equations can be divided into three groups: kinematic, inertia moments and aerodynam- ics terms. All these terms can lead to the cross coupling between the longitudinal and lateral modes of motion. 39
  41. 41. The main feature of the considered problem is that even in the case of linear represen- tation of aerodynamic coe cients the existence of multiple stable steady-state solutions (equilibrium and periodic) is possible. The bifurcational analysis of all the possible steady state solutions and their local and global stability analysis can show the genesis of stability loss and explain in many cases the very strange aircraft behaviour. The validity of the system (3.45), (3.46) is con ned in time. Therefore, in the cases of weakness or lack of stability of the steady states the conclusions resulting from the consideration of the asymptotic stability in Lyapunov sense, when t ! 1, can be wrong. The similar problem can arise for short-term control inputs. The prediction of the bifurcation analysis will be more consistent when the considered steady states have the su cient margin of asymptotic stability. In any case the numerical simulation of aircraft motion using the complete set of equations is necessary for nal veri cation of the bifurcational analysis results. To demonstrate the possibilities of bifurcation and global stability analysis, roll- coupling problem is studied for the hypothetical swept-wing ghter. Flight conditions are high altitudes and supersonic velocities. The simpli ed model considered in 42] is taken as an example for demonstration of stability region boundaries computation. Equations (3.45) can be reduced to 3rd order system provided the characteristic frequency of the longitudinal motion su ciently exceeds the characteristic frequency of the isolated motion in yaw. This may be valid at high supersonic ight. Angle of attack in such cases remains practically constant at the disturbed motion and q p . Approximate equations governing the variations of angular rates p, r and sideslip angle in this case have the following form: _p = a1 + a2 p ;a3 r + lc _r = (b1 ;b2 p2) + b3 r + b4 p _ = ;r cos 0 + p sin 0 + c1 (3.47) In the calculations the following values of the parameters were assumed: a1 = ;14s;2 a2 = ;1:4s;1 a3 = 0 b1 = 2s;2 b2 = 0:6 b3 = ;0:2s;1 b4 = 0 0 = 0:1 c1 = 0 During bifurcation analysis and stability regions computation the roll control lc is con- sidered as a parameter. At any value of the control parameter lc there are three equilibrium points. Fig.3.10 (left plot) shows the dependence of roll rate p on the control parameter lc for all three equilibrium points. Two branches with non-zero coordinates at lc = 0 are unstable at any value of the control parameter. These equilibria have one real positive eigenvalue and a complex pair with negative real part. The branch of solutions passing through the origin comprises the stable equilibria at lc < 3:55 and oscillatory unstable equilibria at lc 3:55. At Hopf bifurcation point (lc ' 3:55) the stable closed trajectories or limit cycles appear. Fig.3.10 (right plot) shows the projections of the closed trajectories onto (p r) plane for several values of lc. For each cycle the corresponding values of the 40
  42. 42. control parameter lc, time period of oscillations T and the value of the main characteristic multiplier of the Jacoby matrix of the Poincare point mapping at the xed point of the mapping are shown in g3.11. Time period of oscillations increases with the growth of the amplitude of the cycle. The increase is especially dramatic when the cycleapproaches saddle-type unstable equilibrium point. In Fig.3.10 one can see how the cycle changes its form and stretches towards saddle point. The change in the locations of the equilibrium points is insigni cant while the control parameter remains in the region where cycles exist. At lc 3:9 closed trajectory merges with saddle equilibrium point and disappear. At lc > 3:9 there are no cycles and all three equilibrium points are unstable. At step-wise control input with lc < 3:55 the motion parameters approach their equilibrium values aperiodically. At 3:55 < lc < 3:9 steady oscillations arise after some time of transient motion. If critical value lc ' 3:9 is exceeded, the parameters of the motion increase sharply and irreversibly after several oscillations. Every stable equilibrium or limit cycle has a certain region of attraction, which is bounded by two separating surfaces passing through saddle unstable equilibrium points. Fig.3.12 shows two sections of the stability region at lc = 0 de ned by thier boundaries computation. Cross plane P2 in both cases was chosen normal to the p axis. In the rst case (left plot) the plane passes through the stable solution located at the origin, and in the second case (right plot) the cross plane P2 passes through saddle equilibrium point. Change in the shape of the separating surfaces and corresponding deformation of attraction region of stable equilibrium point with the increase of the control parameter lc are demonstrated in Fig.3.13 (the cross plane is de ned by a condition r = 0 and passes through zero point). At lc = 1:5 the section of the region of attraction becomes double-linked due to spatial deformation and bending of the separating surface passing through saddle point to which stable equilibrium is approaching. Further growth of the control parameter (lc = 2:6) results in the appearance of another internal instability region at < 0, p < 0. At lc = 2:9 the internal instability regions merge leaving only narrow "capture" area. The periodic solutions of equations (3.47), shown in Fig.3.10, appear due to Hopf bifurcation at lc 3:55 and disappear at lc 3:9 because of the homoclinical bifurcation of saddle point unstable trajectory, when limit cycle merges with saddle point. The performed analysis reveals the sequence of bifurcations of steady-state solutions of equations (3.47) bringing about possible irreversible loss of stability at critical control input in roll. Excess of the critical value corresponding to the disappearance of stable oscillations, may result in unproportional rotation. The fth order system (3.45), which takes into account longitudinal dynamics, is considered for computation of stability region cross sections by direct method. The following linear aerodynamic coe cients were used in the numerical analysis:1 1Angles of attack and sideslip are in radians, rotation rates are in radians per second, control surface de ections are in degrees 41
  43. 43. Figure 3.10: Bifurcations of equilibrium and periodic solutions. 42
  44. 44. Figure 3.11: Bifurcations of equilibrium and periodic solutions. Closed orbit time period and multiplier. Figure 3.12: Stability region cross sections in the plane ( r) obtained by their bound- aries computation. 43
  45. 45. Figure 3.13: Stability region cross sections in the plane ( p) for di erent control pa- rameters lc, obtained by thier boundaries computation. zo = 0 yo = 0 i1 = 1:077 z = ;0:29 y = ;0:093 i2 = 1:008 z e = ;0:00059 y r = 0:00008 i3 = 0:804 l = ;28:74 mo = ;0:0763 n = 4:93 lp = ;0:607 m = ;16:84 np = 0:0203 lr = 0:305 m e = ;0:192 nr = ;0:149 l a = ;0:194 mq = ;0:277 n a = 0:0014 l r = 0:0929 m_ = ;0:0355 n r = ;0:0324 In this case two critical roll rates exist and are very close to Phillips' approximate values. As a result the equilibrium solutions are divided by this critical lines into di erent unconnected families, which are shown for di erent normal factor values in Fig.(3.14,3.15,3.16) at the bottom right plot. The dependencies of equilibrium roll rate on aileron de ection are presented for di erent elevator de ections, corresponding to normal factor parameters az = 1 0 ;1. Di erent line types indicate local stability of the computed equilibriums: solid lines correspond to stable solutions, dashed lines { to divergent solutions, dash-dotted lines { to oscillatory unstable solutions. There are di erent families of equilibrium curves, some of them are located between the critical rates p , p and in the outer regions. Some of them generate the stable autorotational rolling regimes at zero aileron de ection. The autorotational rolling so- lutions exist at all aileron de ections (see Fig.3.16), but at "pro-roll" aileron de ections (p > 0, a > 0) the autorotational equilibrium solutions become oscillatory unstable 44
  46. 46. (dash-doted lines), and after Hopf bifurcation point on each curve the family of stable closed orbits appear. A similar oscillatory instability of equilibrium solutions appears at subcritical equi- librium curves at large aileron de ections (see Fig.3.14). The Hopf bifurcation points in this case also give birth to the families of stable closed orbits (they are shown in Fig.3.14). 45
  47. 47. Figure 3.14: Stability region cross sections for level ight (az = 1) without rotation. Figure 3.15: Stability region cross sections for ight with zero normal factor (az = 0) without rotation. 46
  48. 48. Figure 3.16: Stability region cross sections for ight with negative normal factor (az = ;1) without rotation. The global stability analysis or investigation of asymptotic stability regions is very important in the case, when there are multiple locally stable equilibrium points and closed orbits. The reconstruction of these stability regions provides global information about state space of system in question and determines the critical disturbances of the state variables leading to the loss of stability. As already was noted, more convenient way of reconstruction of multidimensional stability region is the direct computation of its two-dimensional cross sections. The disturbances are only in two selected state variables (with other state variables xed). Cross-section is often selected so as to pass through the locally stable equilibrium point under consideration. At zero aileron and rudder de ections a = r = 0 and e = ;5:40 (az = 1:0) for supersonic ight regime (see Fig.3.14) there are ve equilibrium points, three of them - are locally stable (black circles) and two - are locally unstable (white cycles). Three di erent cross sections of the asymptotic stability regions of stable equilibrium points are presented in Fig.3.14. All the cross sections pass through the central stable equilibrium point without rotation. The rst cross section is placed in the plane of angle of attack and sideslip , the second one is placed in the plane of roll rate p and sideslip , and the third one is placed in the plane of yaw rate r and sideslip . The region of attraction for central equilibrium is marked with light points, the regions of attraction for critical equilibria are marked with more darker points. One can see, that more critical level of perturbation for central equilibrium is in yaw rate { rcr 0:4 1/s. Far from central equilibrium the boundary of stability region has thin structure with multiple folds. The size of asymptotic stability region of central equilibrium becomes signi cantly 47
  49. 49. less when aircraft is trimmed at zero or negative angle of attack (Figs.3.15,3.16). To illustrate this, similar cross sections of the stability regions for e = 4:40 (az = ;1:0) are presented in Fig.3.16. 48
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