In the first part of the presentation, we examine a laser diode system and a predator-prey system of equations. In both, we find a fixed point and use linear stability analysis to determine conditions on parameters that would support a generalized Hopf bifurcation. Then, we use multiple scale expansions in progressively slower time scales to construct the periodic orbit resulting from the bifurcations and use numerical techniques to verify our conclusions. This work is new and led to two papers.In the second part, we examine a 4-mode system that supports Double-Hopf bifurcations using the same process as above.
Here we show the variables N and Q, just to the left of the bifurcation point there is a stable solution, and just to the right, a periodic orbit has been created.
For the second parameter set, the population N approaches a stable fixed point before the bifurcation pointAfter the bifurcation a chaotic envelope has been created that contains the oscillating variable N. The third picture is zoomed in.The fourth is a plot of 3d space where we can see the radius changing chaotically inside the envelope.
For parameter set 1 in the laser diode system, before the bifurcation the variable goes to infinity, and afterwards a quasiperiodic orbit is created.Now to introduce the new work on Double Hopf bifurcations
The animation is of phyllotaxis – the placement of leaves around a plant stem. The only thing changing is the angle between each ball.
However, the fundamental and somewhat surprising result that diffusion could destabilize an otherwise stable equilibrium leading to nonuniform spatial patterns (referred to as prepattern) is not dependent on particular forms of R1 and R2.
In order to incorporate various realistic physical effects which may cause at least one of the physical variables to depend on the past history of the system, it is often necessary to introduce time-delays into the governing equations.
Substitution of the traveling wave variable and our particular reaction terms R1 and R2 into Turing’s reaction diffusion equation give the 4 mode systemNext we’ll do linear stability analysis
Now we need the eigenvalues of this matrix. They will come from a 4th order characteristic polynomial.
Presently we are interested in double Hopf bifurcations, so we want the four eigenvalues to be two pairs of purely imaginary complex conjugatesWe will use these parameter conditions to find stability boundaries of solutions.
We use epsilon as an indexer to keep terms together that are of the same order. At the end of the analysis we’ll set epsilon=1.This allows the nonlinear terms and the control parameters to occur at the same order.
First eqn solved for MiSecond for PiThird for QiPlug all into the last equation to get a composite equation in Ni
Coefficients of exponentials with identical arguments on either side are matched to determine N200, N201,…, N210.
Last two are the QP
Changing either of theses inequalities to an equality produces a critical line where a family of limit cycles bifurcates from the equilibrium solution
Changing the second inequality to an equality results in the third critical line L3, along which a secondary Hopf bifurcation takes place with frequency w2. The trajectories trace out a 2-D torus described by (4.4).
Changing the second inequality to an equality results in the fourth critical line L4, along which a secondary Hopf bifurcation takes place with frequency ω1
μ1 = -0.06 , μ2 = 0.9 Immediately after the line L3 a static bifurcation of the periodic orbit occurs
Hopf Bifurcations and nonlinear dynamics
ON THE DYNAMICAL CONSEQUENCES OF GENERALIZED AND DEGENERATEHOPF BIFURCATIONS: THIRD-ORDER EFFECTS Todd Blanton Smith PhD, Mathematics University of Central Florida Orlando, Florida Spring 2011 Advisor: Dr. Roy S. Choudhury
Presentation Outline• Intro to phase plane analysis and bifurcations• Demonstration of analytical technique for dividing parameter space into regions that support certain nonlinear dynamics. • Three-mode laser diode system and predator-prey system • Generalized Hopf bifurcations • Four-mode population system • Double Hopf Bifurcations
Phase plane analysis of nonlinear autonomous systems Consider a general nonlinear autonomous system Fixed points satisfy .Add a small perturbation A Taylor expansion gives behavior near the fixed point. 0
Bifurcations• A bifurcation is a qualitative change in the phase portrait of a system of ODEs. A static bifurcation: Eigenvalues are
Supercritical Hopf BifurcationsThey occur when the real part of a complexconjugate pair of eigenvalues movesthrough zero.
Subcritical Hopf Bifurcations Two unstable fixed points collapse onto a stable fixed point and cause it to loose stability
Generalized Hopf Bifurcations• A generalized Hopf bifurcation occurs in a system with three dimensions when there is one complex conjugate pair of eigenvalues and the other eigenvalue is zero.• In four dimensions there are three scenarios: (Double Hopf)
Laser Diode and Predator-Prey systemsFirst we find generalized Hopf bifurcations andnearby nonlinear dynamics in two systems:Laser Diode Predator Prey
Locating a Generalized Hopf Bifurcation In the Predator Prey model, our Jacobian matrix is The eigenvalues satisfy the characteristic equation where depend on the system parameters. We insist on the formwhich imposes the conditions and .
Analytical Construction of Periodic Orbits- Use multiple scale expansions to construct periodicorbits near the bifurcation.
Using these expansions and equating powers of yields equations at Where are differential operators and are source terms.These three equations can be solved simultaneously toform a composite operator
The Source Terms The first order sources are zero, so we can guess a functional form for the first order solution.
First-Order Solution where we impose and .This can be plugged into the source of the next ordercomposite equationNow eliminate source terms that satisfy thehomogeneous equations.
Set the coefficients of equal to 0 to getChange to polar coordinates for periodic orbits.
After the change of variables, setting equal the real andimaginary parts gives the polar normal form.The fixed points are
The post-generalized Hopf periodic orbit isThe evolution and stability of the orbit is determined bythe stability of the fixed point, so we evaluate theJacobian (eigenvalue) at the fixed point to get
Numerical confirmation of chaotic behavior Autocorrelation function Power Spectral Density Autocorrelation function PSD shows contributions Approaches zero in a finite From many small frequencies time
A stable quasi-periodic orbit is created after the bifurcation in the laser diode system.
Periodic and Quasiperiodic Wavetrains from Double Hopf Bifurcations in Predator-Prey Systems with General Nonlinearities• Morphogenesis- from Developmental Biology - The emergence of spatial form and pattern from a homogeneous state.
Turing studied reaction diffusion equations of the formThe reaction functions (or kinematic terms) R1 and R2 werepolynomials
The general two species predator-prey model rate of predation per predator Prey birth rate Carrying capacity rate of the prey’s contribution to the predator growthwhere N(t) and P(t) are the prey and predator populations, respectively
Here we investigate traveling spatial wave patterns in the form where is the traveling wave, or “spatial” variable, and v is the translation or wave speed.
The four-mode population system studied by Stefan Mancas:
Linear stability analysis The fixed points of the system are The functions F(N) and G(P) are kept general during the analysis but are subsequently chosen to correspond to three systems analyzed by Mancas .
The Jacobian matrixat the fixed point(N0, M0, P0, Q0) is
The eigenvalues λ of the Jacobian satisfy thecharacteristic equation where bi, i = 1,…, 4 are given by
Double Hopf bifurcation- two pairs of purely imaginary complex conjugates.Hence we impose the characteristic formThis determines that the conditions for a double Hopf bifurcation are
Analytical construction of periodic orbits We use the method of multiple scales to construct analytical approximations for the periodic orbits. is a small positive non-dimensional parameter that distinguishes different time scales. The time derivative becomesControl Parameters:
where the Li, i = 1, 2, 3, 4, are the differential operators
To find the solutions, we note that S1,i = 0 for i = 1, 2, 3, 4and so we choose a first order solution
Suppressing secular source terms (solutions of thehomogeneous equations for i = 1) gives the requirement Now we assume a second order solution of the form
This second order ansatz is plugged into the composite operator with i = 2.Evaluating the third order source term ᴦwith the first 3 and second order solutions and setting first order harmonics equal to zero produces the normal form. Note that c1 through c6 are complex. We can more easily find periodic orbits by switching to polar coordinates.
Plugging the polar coordinates into the normal form and separating real and imaginary parts of the resulting equations gives the normal form in polar coordinates.
Periodic solutions are found by setting the tworadial equations equal to zero and solving for p1(T2)and p2(T2). There are four solutions: the initial equilibriumsolution, the Hopf bifurcation solution with frequencyω1, the Hopf bifurcation solution with frequency ω2, andthe quasiperiodic solution with frequencies ω1 and ω2 .
The stability conditions for each of these solutions can bedetermined with the Jacobian of the radial equations. Evaluate this on the equilibrium solution and set theeigenvalues negative to see the stability conditions: and
Choosing μ1 = -0.24 and μ2 = 0.79between L1 and L2 results in the stableequilibrium solution shown in Figure 2
The second order deviation values μ1 = -0.4 and μ2 = 0.6 placethe sample point immediately after the line L1 where a Hopfbifurcation occurs.The values μ1 = -0.11 and μ2 = 0.9place the sample point immediatelyafter L2.
Evaluating the Jacobian on the first Hopfbifurcation solution yields the stability conditions: When a20 < 0, the above conditions also allow thesecond Hopf bifurcation solution to exist.
Evaluating the Jacobian on the second Hopfbifurcation solution yields the stability conditions: When b02 < 0, the above conditions also allow thefirst Hopf bifurcation solution to exist.
μ1 = -0.06 , μ2 = 0.9Quasiperiodic motion after line 3 Two frequency peaks indicate quasiperiodic motion.
The critical line L5, where a quasiperiodicsolution loses stability and may bifurcate into amotion on a 3-D torus, is given here:
Immediately after the lines L4 and L5 thesolutions fly off to infinity in finite time. The solutions in the regions after L4 and L5 are unstable
Conclusion• In this dissertation we have demonstrated an analytical technique for determining regions of parameter space that yield various nonlinear dynamics due to generalized and double-Hopf bifurcations. • Thank You: Dr Choudhury Dr Schober Dr Rollins Dr Chatterjee Dr Moore