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- 1. Tutorials and ReviewsInternational Journal of Bifurcation and Chaos, Vol. 13, No. 11 (2003) 3147–3233 c World Scientiﬁc Publishing Company TOWARD A THEORY OF CHAOS A. SENGUPTA Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India osegu@iitk.ac.in Received February 23, 2001; Revised September 19, 2002 This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multi- valued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000], and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this uniﬁed mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical con- vergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain eﬃciently a multiplicity of states that otherwise would remain unavailable to it. Keywords: Chaos; complexity; ill-posed problems; graphical convergence; topology; multifunc- tions.Prologue of study of so-called “strongly ” nonlinear system.1. Generally speaking, the analysis of chaos is ex- . . . Linearity means that the rule that determines whattremely diﬃcult. While a general deﬁnition for chaos a piece of a system is going to do next is not inﬂu-applicable to most cases of interest is still lacking, enced by what it is doing now. More precisely thismathematicians agree that for the special case of iter- is intended in a diﬀerential or incremental sense: Foration of transformations there are three common char- a linear spring, the increase of its tension is propor-acteristics of chaos: tional to the increment whereby it is stretched, with the ratio of these increments exactly independent of1. Sensitive dependence on initial conditions, how much it has already been stretched. Such a spring2. Mixing,3. Dense periodic points. can be stretched arbitrarily far . . . . Accordingly no real spring is linear. The mathematics of linear objects is [Peitgen, Jurgens & Saupe, 1992] particularly felicitous. As it happens, linear objects en-2. The study of chaos is a part of a larger program joy an identical, simple geometry. The simplicity of 3147
- 2. 3148 A. Senguptathis geometry always allows a relatively easy mental 5. One of the most striking aspects of physicsimage to capture the essence of a problem, with the is the simplicity of its laws. Maxwell’s equations,technicality, growing with the number of parts, basi- Schroedinger’s equations, and Hamilton mechanicscally a detail. The historical prejudice against nonlinear can each be expressed in a few lines. . . . Everythingproblems is that no so simple nor universal geometry is simple and neat except, of course, the world. Everyusually exists. place we look outside the physics classroom we see a Mitchell Feigenbaum’s Foreword (pp. 1–7) world of amazing complexity. . . . So why, if the laws in [Peitgen et al., 1992] are so simple, is the world so complicated? To us com- plexity means that we have structure with variations.3. The objective of this symposium is to explore the Thus a living organism is complicated because it hasimpact of the emerging science of chaos on various dis- many diﬀerent working parts, each formed by varia-ciplines and the broader implications for science and tions in the working out of the same genetic coding.society. The characteristic of chaos is its universality Chaos is also found very frequently. In a chaotic worldand ubiquity. At this meeting, for example, we have it is hard to predict which variation will arise in a givenscholars representing mathematics, physics, biology, place and time. A complex world is interesting becausegeophysics and geophysiology, astronomy, medicine, it is highly structured. A chaotic world is interestingpsychology, meteorology, engineering, computer sci- because we do not know what is coming next. Ourence, economics and social sciences. 1 Having so many world is both complex and chaotic. Nature can pro-disciplines meeting together, of course, involves the duce complex structures even in simple situations andrisk that we might not always speak the same lan- obey simple laws even in complex situations.guage, even if all of us have come to talk about [Goldenfeld & Kadanoﬀ, 1999]“chaos”. 6. Where chaos begins, classical science stops. For as Opening address of Heitor Gurgulino de Souza, long as the world has had physicists inquiring into Rector United Nations University, Tokyo the laws of nature, it has suﬀered a special ignorance [de Souza, 1997] about disorder in the atmosphere, in the turbulent sea,4. The predominant approach (of how the diﬀerent in the ﬂuctuations in the wildlife populations, in theﬁelds of science relate to one other ) is reductionist: oscillations of the heart and the brain. But in the 1970sQuestions in physical chemistry can be understood a few scientists began to ﬁnd a way through disor-in terms of atomic physics, cell biology in terms of der. They were mathematicians, physicists, biologists,how biomolecules work . . . . We have the best of rea- chemists . . . (and) the insights that emerged led di-sons for taking this reductionist approach: it works. rectly into the natural world: the shapes of clouds,But shortfalls in reductionism are increasingly appar- the paths of lightning, the microscopic intertwiningent (and) there is something to be gained from sup- of blood vessels, the galactic clustering of stars. . . .plementing the predominantly reductionist approach Chaos breaks across the lines that separate scientiﬁcwith an integrative agenda. This special section on disciplines, (and) has become a shorthand name for acomplex systems is an initial scan (where) we have fast growing movement that is reshaping the fabric oftaken a “complex system” to be one whose properties the scientiﬁc establishment.are not fully explained by an understanding of its com- [Gleick, 1987]ponent parts. Each Viewpoint author 2 was invited todeﬁne “complex” as it applied to his or her discipline. 7. order (→) complexity (→) chaos. [Gallagher & Appenzeller, 1999] [Waldrop, 1992]1 A partial listing of papers is as follows: Chaos and Politics: Application of Nonlinear Dynamics to Socio-Political issues;Chaos in Society: Reﬂections on the Impact of Chaos Theory on Sociology; Chaos in Neural Networks; The Impact of Chaoson Mathematics; The Impact of Chaos on Physics; The Impact of Chaos on Economic Theory; The Impact of Chaos onEngineering; The impact of Chaos on Biology; Dynamical Disease: And The Impact of Nonlinear Dynamics and Chaos onCardiology and Medicine.2 The eight Viewpoint articles are titled: Simple Lessons from Complexity; Complexity in Chemistry; Complexity in Biolog-ical Signaling Systems; Complexity and the Nervous System; Complexity, Pattern, and Evolutionary Trade-Oﬀs in AnimalAggregation; Complexity in Natural Landform Patterns; Complexity and Climate, and Complexity and the Economy.
- 3. Toward a Theory of Chaos 31498. Our conclusions based on these examples seem sim- essary that we have a mathematically clear physi-ple: At present chaos is a philosophical term, not a cal understanding of these notions that are suppos-rigorous mathematical term. It may be a subjective edly reshaping our view of nature. This paper is annotion illustrating the present day limitations of the attempt to contribute to this goal. To make thishuman intellect or it may describe an intrinsic prop- account essentially self-contained we include here,erty of nature such as the “randomness” of the se- as far as this is practical, the basics of the back-quence of prime numbers. Moreover, chaos may be ground material needed to understand the paper inundecidable in the sense of Godel in that no matter the form of Tutorials and an extended Appendix.what deﬁnition is given for chaos, there is some ex- The paradigm of chaos of the kneading of theample of chaos which cannot be proven to be chaotic dough is considered to provide an intuitive basisfrom the deﬁnition. of the mathematics of chaos [Peitgen et al., 1992], [Brown & Chua, 1996] and one of our fundamental objectives here is to re- count the mathematical framework of this process9. My personal feeling is that the deﬁnition of a “frac- in terms of the theory of ill-posed problems arisingtal” should be regarded in the same way as the biolo- from non-injectivity [Sengupta, 1997], maximal ill-gist regards the deﬁnition of “life”. There is no hard posedness, and graphical convergence of functionsand fast deﬁnition, but just a list of properties char- [Sengupta & Ray, 2000]. A natural mathematicalacteristic of a living thing . . . . Most living things have formulation of the kneading of the dough in themost of the characteristics on the list, though there form of stretch-cut-and-paste and stretch-cut-and-are living objects that are exceptions to each of them. fold operations is in the ill-posed problem arisingIn the same way, it seems best to regard a fractal as from the increasing non-injectivity of the functiona set that has properties such as those listed below, f modeling the kneading operation.rather than to look for a precise deﬁnition which willcertainly exclude some interesting cases. [Falconer, 1990] Begin Tutorial 1: Functions and10. Dynamical systems are often said to exhibit chaos Multifunctionswithout a precise deﬁnition of what this means. A relation, or correspondence, between two sets X [Robinson, 1999] and Y , written M: X –→ Y , is basically a rule that → associates subsets of X to subsets of Y ; this is often1. Introduction expressed as (A, B) ∈ M where A ⊂ X and B ⊂ YThe purpose of this paper is to present an uniﬁed, and (A, B) is an ordered pair of sets. The domainself-contained mathematical structure and physical defunderstanding of the nature of chaos in a discrete D(M) = {A ⊂ X : (∃Z ∈ M)(πX (Z) = A)}dynamical system and to suggest a plausible expla- and rangenation of why natural systems tend to be chaotic. defThe somewhat extensive quotations with which we R(M) = {B ⊂ Y : (∃Z ∈ M)(πY (Z) = B)}begin above, bear testimony to both the increas- of M are respectively the sets of X which underingly signiﬁcant — and perhaps all-pervasive — M corresponds to sets in Y ; here πX and (πY )role of nonlinearity in the world today as also our are the projections of Z on X and Y , respectively.imperfect state of understanding of its manifesta- Equivalently, (D(M) = {x ∈ X : M(x) = ∅}) andtions. The list of papers at both the UN Confer- (R(M) = x∈D(M) M(x)). The inverse M− of Mence [de Souza, 1997] and in Science [Gallagher & is the relationAppenzeller, 1999] is noteworthy if only to justify M− = {(B, A) : (A, B) ∈ M}the observation of Gleick [1987] that “chaos seemsto be everywhere”. Even as everybody appears to so that M− assigns A to B iﬀ M assigns B to A.be ﬁnding chaos and complexity in all likely and In general, a relation may assign many elements inunlikely places, and possibly because of it, it is nec- its range to a single element from its domain; of
- 4. 3150 A. Sengupta ¨ ¡ £ ¤¢ ¤ © £ $ # ! ¥ ¦¢ § ¦ ¥ § (a) (a) (a) (b) (b) (b) 3 4 ( ) 6 9 @ 1 0 % 2 7 8 5 (c) (c) (c) (d) (d) (d)Fig. 1. Functional and non-functional relations between two sets X and Y : while f and g are functional relations, M is not.(a) f and g are both injective and surjective (i.e. they are bijective), (b) g is bijective but f is only injective and f −1 ({y2 }) := ∅,(c) f is not 1:1, g is not onto, while (d) M is not a function but is a multifunction.especial signiﬁcance are functional relations f 3 that linear homogeneous diﬀerential equation with con-can assign only a unique element in R(f ) to any stant coeﬃcients of order n 1 has n linearlyelement in D(f ). Figure 1 illustrates the distinc- independent solutions so that the operator D n oftion between arbitrary and functional relations M D n (y) = 0 has a n-dimensional null space. Inversesand f . This diﬀerence between functions (or maps) of non-injective, and in general non-bijective, func-and multifunctions is basic to our development and tions will be denoted by f − . If f is not injectiveshould be fully understood. Functions can again be thenclassiﬁed as injections (or 1:1) and surjections (or def A ⊂ f − f (A) = sat(A)onto). f : X → Y is said to be injective (or one-to-one) if x1 = x2 ⇒ f (x1 ) = f (x2 ) for all x1 , x2 ∈ X, where sat(A) is the saturation of A ⊆ X induced bywhile it is surjective (or onto) if Y = f (X). f is f ; if f is not surjective thenbijective if it is both 1:1 and onto. f f − (B) := B f (X) ⊆ B. Associated with a function f : X → Y is its in-verse f −1 : Y ⊇ R(f ) → X that exists on R(f ) iﬀ If A = sat(A), then A is said to be saturated, andf is injective. Thus when f is bijective, f −1 (y) := B ⊆ R(f ) whenever f f − (B) = B. Thus for non-{x ∈ X: y = f (x)} exists for every y ∈ Y ; infact f is injective f , f − f is not an identity on X just asbijective iﬀ f −1 ({y}) is a singleton for each y ∈ Y . f f − is not 1Y if f is not surjective. However theNon-injective functions are not at all rare; if any- set of relationsthing, they are very common even for linear mapsand it would be perhaps safe to conjecture that f f − f = f, f −f f − = f − (1)they are overwhelmingly predominant in the non- that is always true will be of basic signiﬁcance inlinear world of nature. Thus for example, the simple this work. Following are some equivalent statements3 We do not distinguish between a relation and its graph although technically they are diﬀerent objects. Thus although afunctional relation, strictly speaking, is the triple (X, f, Y ) written traditionally as f : X → Y , we use it synonymously withthe graph f itself. Parenthetically, the word functional in this paper is not necessarily employed for a scalar-valued function,but is used in a wider sense to distinguish between a function and an arbitrary relation (that is a multifunction). Formally,whereas an arbitrary relation from X to Y is a subset of X × Y , a functional relation must satisfy an additional restrictionthat requires y1 = y2 whenever (x, y1 ) ∈ f and (x, y2 ) ∈ f . In this subset notation, (x, y) ∈ f ⇔ y = f (x).
- 5. Toward a Theory of Chaos 3151on the injectivity and surjectivity of functions f : set of X under ∼, denoted by X/ ∼:= {[x]: x ∈ X},X →Y. has the equivalence classes [x] as its elements; thus(Injec) f is 1:1 ⇔ there is a function f L : Y → X [x] plays a dual role either as subsets of X or as ele-called the left inverse of f , such that f L f = 1X ⇔ ments of X/ ∼. The rule x → [x] deﬁnes a surjectiveA = f − f (A) for all subsets A of X ⇔ f ( Ai ) = function Q: X → X/ ∼ known as the quotient map. f (Ai ). Example 1.1. Let(Surjec) f is onto ⇔ there is a function f R : Y → Xcalled the right inverse of f , such that f f R = 1Y ⇔ S 1 = {(x, y) ∈ R2 ) : x2 + y 2 = 1}B = f f − (B) for all subsets B of Y . be the unit circle in R2 . Consider X = [0, 1] as a As we are primarily concerned with non- subspace of R, deﬁne a mapinjectivity of functions, saturated sets generated byequivalence classes of f will play a signiﬁcant role q : X → S 1, s → (cos 2πs, sin 2πs), s ∈ X ,in our discussions. A relation E on a set X is said from R to R2 , and let ∼ be the equivalence relationto be an equivalence relation if it is 4 on X(ER1) Reﬂexive: (∀x ∈ X)(xEx). s ∼ t ⇔ (s = t) ∨ (s = 0, t = 1) ∨ (s = 1, t = 0) .(ER2) Symmetric: (∀x, y ∈ X)(xEy ⇒ yEx).(ER3) Transitive: (∀x, y, z ∈ X)(xEy ∧ yEz ⇒ If we bend X around till its ends touch, the resulting xEz). circle represents the quotient set Y = X/ ∼ whoseEquivalence relations group together unequal ele- points are equivalent under ∼ as followsments x1 = x2 of a set as equivalent according to [0] = {0, 1} = [1], [s] = {s} for all s ∈ (0, 1) .the requirements of the relation. This is expressedas x1 ∼ x2 (mod E) and will be represented here by Thus q is bijective for s ∈ (0, 1) but two-to-one forthe shorthand notation x1 ∼E x2 , or even simply the special values s = 0 and 1, so that for s, t ∈ X,as x1 ∼ x2 if the speciﬁcation of E is not essential. s ∼ t ⇔ q(s) = q(t) .Thus for a non-injective map if f (x1 ) = f (x2 ) forx1 = x2 , then x1 and x2 can be considered to be This yields a bijection h: X/ ∼ → S 1 such thatequivalent to each other since they map onto thesame point under f ; thus x1 ∼f x2 ⇔ f (x1 ) = q =h◦Qf (x2 ) deﬁnes the equivalence relation ∼ f induced deﬁnes the quotient map Q: X → X/ ∼ by h([s]) =by the map f . Given an equivalence relation ∼ on q(s) for all s ∈ [0, 1]. The situation is illustrated bya set X and an element x ∈ X the subset the commutative diagram of Fig. 2 that appears as def [x] = {y ∈ X : y ∼ x} an integral component in a diﬀerent and more gen-is called the equivalence class of x; thus x ∼ y ⇔ eral context in Sec. 2. It is to be noted that com-[x] = [y]. In particular, equivalence classes gener- mutativity of the diagram implies that if a givenated by f : X → Y , [x]f = {xα ∈ X : f (xα ) = equivalence relation ∼ on X is completely deter-f (x)}, will be a cornerstone of our analysis of chaos mined by q that associates the partitioning equiva-generated by the iterates of non-injective maps, and lence classes in X to unique points in S 1 , then ∼ isthe equivalence relation ∼f := {(x, y): f (x) = f (y)} identical to the equivalence relation that is inducedgenerated by f is uniquely deﬁned by the partition by Q on X. Note that a larger size of the equivalencethat f induces on X. Of course as x ∼ x, x ∈ [x]. classes can be obtained by considering X = R + forIt is a simple matter to see that any two equiva- which s ∼ t ⇔ |s − t| ∈ Z+ .lence classes are either disjoint or equal so that theequivalence classes generated by an equivalence re- End Tutorial 1lation on X form a disjoint cover of X. The quotient4 An alternate useful way of expressing these properties for a relation R on X are(ER1) R is reﬂexive iﬀ 1X ⊆ X(ER2) R is symmetric iﬀ R = R−1(ER3) R is transitive iﬀ R ◦ R ⊆ R,with R an equivalence relation only if R ◦ R = R.
- 6. 3152 A. Sengupta ¡ α∈D M(Aα ) and M α∈D Aα ⊆ α∈D M(Aα ) where D is an index set. The following illustrates the diﬀerence between the two inverses of M. Let X be a set that is partitioned into two disjoint M- ¢ invariant subsets X1 and X2 . If x ∈ X1 (or x ∈ X2 ) then M(x) represents that part of X1 (or of X2 ) that is realized immediately after one application §¥¡ ¦ ¤ £ © ¨ of M, while M− (x) denotes the possible precursors of x in X1 (or of X2 ) and M+ (B) is that subset of X whose image lies in B for any subset B ⊂ X. Fig. 2. The quotient map Q. In this paper the multifunctions that we shall be explicitly concerned with arise as the inverses of non-injective maps.One of the central concepts that we consider and The second major component of our theory isemploy in this work is the inverse f − of a nonlin- the graphical convergence of a net of functions toear, non-injective, function f ; here the equivalence a multifunction. In Tutorial 2 below, we replace forclasses [x]f = f − f (x) of x ∈ X are the saturated the sake of simplicity and without loss of generality,subsets of X that partition X. While a detailed the net (which is basically a sequence where the in-treatment of this question in the form of the non- dex set is not necessarily the positive integers; thuslinear ill-posed problem and its solution is given in every sequence is a net but the family 5 indexed, forSec. 2 [Sengupta, 1997], it is suﬃcient to point out example, by Z, the set of all integers, is a net andhere from Figs. 1(c) and 1(d), that the inverse of a not a sequence) with a sequence and provide thenon-injective function is not a function but a mul- necessary background and motivation for the con-tifunction while the inverse of a multifunction is a cept of graphical convergence.non-injective function. Hence one has the generalresult that f is a non-injective function ⇔ f − is a multifunction . Begin Tutorial 2: Convergence of (2) f is a multifunction Functions ⇔ f − is a non-injective function This Tutorial reviews the inadequacy of the usual notions of convergence of functions either to limitThe inverse of a multifunction M: X –→ Y is a gen- → functions or to distributions and suggests the mo-eralization of the corresponding notion for a func- tivation and need for introduction of the notiontion f : X → Y such that of graphical convergence of functions to multifunc- def tions. Here, we follow closely the exposition of M− (y) = {x ∈ X : y ∈ M(x)} Korevaar [1968], and use the notation (f k )∞ to de- k=1leads to note real or complex valued functions on a bounded or unbounded interval J. M− (B) = {x ∈ X : M(x) B = ∅} A sequence of piecewise continuous functionsfor any B ⊆ Y , while a more restricted inverse (fk )∞ is said to converge to the function f , nota- k=1that we shall not be concerned with is given as tion fk → f , on a bounded or unbounded intervalM+ (B) = {x ∈ X : M(x) ⊆ B}. Obviously, J6M+ (B) ⊆ M− (B). A multifunction is injective if (1) Pointwise ifx1 = x2 ⇒ M(x1 ) M(x2 ) = ∅, and commonlywith functions, it is true that M α∈D Aα = fk (x) → f (x) for all x ∈ J ,5 A function χ: D → X will be called a family in X indexed by D when reference to the domain D is of interest, and a netwhen it is required to focus attention on its values in X.6 Observe that it is not being claimed that f belongs to the same class as (fk ). This is the single most important cornerstoneon which this paper is based: the need to “complete” spaces that are topologically “incomplete”. The classical high-schoolexample of the related problem of having to enlarge, or extend, spaces that are not big enough is the solution space of algebraicequations with real coeﬃcients like x2 + 1 = 0.
- 7. Toward a Theory of Chaos 3153i.e. Given any arbitrary real number ε 0 there It is to be observed that apart from point-exists a K ∈ N that may depend on x, such that wise and uniform convergences, all the other modes|fk (x) − f (x)| ε for all k ≥ K. listed above represent some sort of an averaged con-(2) Uniformly if tribution of the entire interval J and are therefore not of much use when pointwise behavior of the sup |f (x) − fk (x)| → 0 as k → ∞ , limit f is necessary. Thus while limits in the mean x∈J are not unique, oscillating functions are tamed byi.e. Given any arbitrary real number ε 0 there m-integral convergence for adequately large valuesexists a K ∈ N, such that supx∈J |fk (x) − f (x)| ε of m, and convergence relative to test functions,for all k ≥ K. as we see below, can be essentially reduced to m-(3) In the mean of order p ≥ 1 if |f (x) − f k (x)|p is integral convergence. On the contrary, our graphicalintegrable over J for each k convergence — which may be considered as a point- wise biconvergence with respect to both the direct |f (x) − fk (x)|p → 0 as k → ∞ . and inverse images of f just as usual pointwise con- J vergence is with respect to its direct image onlyFor p = 1, this is the simple case of convergence in — allows a sequence (in fact, a net) of functions tothe mean. converge to an arbitrary relation, unhindered by ex-(4) In the mean m-integrally if it is possible to select ternal inﬂuences such as the eﬀects of integrationsindeﬁnite integrals and test functions. To see how this can indeed mat- x x1 ter, consider the following (−m) fk (x) = πk (x) + dx1 dx2 Example 1.2. Let fk (x) = sin kx, k = 1, 2, . . . and c c xm−1 let J be any bounded interval of the real line. Then ··· dxm fk (xm ) 1-integrally we have c x (−1) 1 1and fk (x) = − cos kx = − + sin kx1 dx1 , k k 0 x x1 f (−m) (x) = π(x) + dx1 dx2 which obviously converges to 0 uniformly (and c c therefore in the mean) as k → ∞. And herein lies xm−1 the point: even though we cannot conclude about ··· dxm f (xm ) the exact nature of sin kx as k increases indeﬁ- c nitely (except that its oscillations become more andsuch that for some arbitrary real p ≥ 1, more pronounced), we may very deﬁnitely state that (−m) p limk→∞(cos kx)/k = 0 uniformly. Hence from |f (−m) − fk | →0 as k → ∞. J x (−1) fk (x) → 0 = 0 + lim sin kx1 dx1where the polynomials πk (x) and π(x) are of degree 0 k→∞ m, and c is a constant to be chosen appropriately. it follows that(5) Relative to test functions ϕ if f ϕ and f k ϕ are lim sin kx = 0 (3)integrable over J and k→∞ ∞ 1-integrally. (fk − f )ϕ → 0 for every ϕ ∈ C0 (J) as k → ∞ , Continuing with the same sequence of func- J tions, we now examine its test-functional conver- ∞where C0 (J) is the class of inﬁnitely diﬀerentiable 1 gence with respect to ϕ ∈ C0 (−∞, ∞) that vanishescontinuous functions that vanish throughout some for all x ∈ (α, β). Integrating by parts, /neighborhood of each of the end points of J. For ∞ βan unbounded J, a function is said to vanish in fk ϕ = ϕ(x1 ) sin kx1 dx1some neighborhood of +∞ if it vanishes on some −∞ αray (r, ∞). 1 While pointwise convergence does not imply = − [ϕ(x1 ) cos kx1 ]β α kany other type of convergence, uniform conver- βgence on a bounded interval implies all the other 1 − ϕ (x1 ) cos kx1 dx1convergences. k α
- 8. 3154 A. Sengupta ©¦ £¦ ¨ § ¦ ( ) A¥£7 B@ 98 % 6 C ¤ § £¦ A¥£7 B@ 98 @ ¢ £¡ ¢ # ! $ 4 0 2 31 0 6 5 ¥ ¤ ¤ (a) (b) (c)Fig. 3. Incompleteness of function spaces. (a) demonstrates the classic example of non-completeness of the space of real-valued continuous functions leading to the complete spaces Ln [a, b] whose elements are equivalence classes of functions with bf ∼ g iﬀ the Lebesgue integral a |f − g|n = 0. (b) and (c) illustrate distributional convergence of the functions fk (x) ofEq. (5) to the Dirac delta δ(x) leading to the complete space of generalized functions. In comparison, note that the spaceof continuous functions in the uniform metric C[a, b] is complete which suggests the importance of topologies in determiningconvergence properties of spaces.The ﬁrst integrated term is 0 due to the condi- converges in the mean to f (−m) ϕ(m) so thattions on ϕ while the second also vanishes because β β 1ϕ ∈ C0 (−∞, ∞). Hence (−m) (m) fk ϕ = (−1)m fk ϕ α α ∞ β fk ϕ → 0 = lim ϕ(x1 ) sin ksdx1 β β −∞ α k→∞ → (−1)m f (−m) ϕ(m) = f ϕ. α αfor all ϕ, and leading to the conclusion that In fact the converse also holds leading to the following Equivalences between m-convergence in lim sin kx = 0 (4) k→∞ the mean and convergence with respect to test- functions [Korevaar, 1968].test-functionally. Type 1 Equivalence. If f and (fk ) are functions This example illustrates the fact that if on J that are integrable on every interior subinter-Supp(ϕ) = [α, β] ⊆ J,7 integrating by parts suf- val, then the following are equivalent statements.ﬁciently large number of times so as to wipe outthe pathological behavior of (fk ) gives (a) For every interior subinterval I of J there is an integer mI ≥ 0, and hence a smallest in- β fk ϕ = fk ϕ teger m ≥ 0, such that certain indeﬁnite inte- (−m) J α grals fk of the functions fk converge in the β β mean on I to an indeﬁnite integral f (−m) ; thus (−1) (−m) m = fk ϕ = · · · = (−1)m fk ϕ (−m) − f (−m) | → 0. α α I |fk ∞ (b) J (fk − f )ϕ → 0 for every ϕ ∈ C0 (J). (−m) x x xwhere fk (x) = πk (x) + c dx1 c 1 dx2 · · · c m−1 A signiﬁcant generalization of this Equivalence isdxm fk (xm ) is an m-times arbitrary indeﬁnite in- β (−m) obtained by dropping the restriction that the limittegral of fk . If now it is true that α fk → object f be a function. The need for this gener- β (−m) (m) α f (−m) , then it must also be true that fk ϕ alization arises because metric function spaces are7 ∞ By deﬁnition, the support (or supporting interval) of ϕ(x) ∈ C0 [α, β] is [α, β] if ϕ and all its derivatives vanish for x ≤ αand x ≥ β.
- 9. Toward a Theory of Chaos 3155known not to be complete: Consider the sequence can be associated with the arbitrary indeﬁniteof functions [Fig. 3(a)] integrals 0, if a≤x≤0 a≤x≤0 0, 1 1 fk (x) = kx, if 0≤x≤ (5) def (−1) Θk (x) = δk (x) = kx, 0 x k k 1 1 1, if ≤x≤b 1, ≤x≤b k kwhich is not Cauchy in the uniform metricρ(fj , fk ) = supa≤x≤b |fj (x) − fk (x)| but is Cauchy of Fig. 3(c), which, as noted above, converge b in the mean to the unit step function Θ(x);in the mean ρ(fj , fk ) = a |fj (x) − fk (x)|dx, or ∞ β β (−1)even pointwise. However in either case, (f k ) cannot hence −∞ δk ϕ ≡ α δk ϕ = − α δk ϕ → βconverge in the respective metrics to a continuous − 0 ϕ (x)dx = ϕ(0). But there can be no func-function and the limit is a discontinuous unit step β tional relation δ(x) for which α δ(x)ϕ(x)dx = ϕ(0)function for all ϕ ∈ C0 1 [α, β], so that unlike in the case in 0, if a ≤ x ≤ 0 Type 1 Equivalence, the limit in the mean Θ(x) Θ(x) = (−1) 1, if 0 x ≤ b of the indeﬁnite integrals δk (x) cannot be ex- pressed as the indeﬁnite integral δ (−1) (x) of somewith graph ([a, 0], 0) ((0, b], 1), which is also in- function δ(x) on any interval containing the ori-tegrable on [a, b]. Thus even if the limit of the se- gin. This leads to the second more general type ofquence of continuous functions is not continuous, equivalence.both the limit and the members of the sequenceare integrable functions. This Riemann integration Type 2 Equivalence. If (fk ) are functions on Jis not suﬃciently general, however, and this type that are integrable on every interior subinterval,of integrability needs to be replaced by a much then the following are equivalent statements.weaker condition resulting in the larger class ofthe Lebesgue integrable complete space of functions (a) For every interior subinterval I of J there is anL[a, b].8 integer mI ≥ 0, and hence a smallest integer The functions in Fig. 3(b), m ≥ 0, such that certain indeﬁnite integrals (−m) k, if 0 x 1 fk of the functions fk converge in the mean k on I to an integrable function Θ which, unlike δk (x) = 1 in Type 1 Equivalence, need not itself be an 0, x ∈ [a, b] − 0, , k indeﬁnite integral of some function f .8 Both Riemann and Lebesgue integrals can be formulated in terms of the so-called step functions s(x), which are piecewiseconstant functions with values (σi )I on a ﬁnite number of bounded subintervals (Ji )I i=1 i=1 (which may reduce to a point or defImay not contain one or both of the end points) of a bounded or unbounded interval J, with integral J s(x)dx = i=1 σi |Ji |.While the Riemann integral of a bounded function f (x) on a bounded interval J is deﬁned with respect to sequencesof step functions (sj )∞ and (tj )∞ satisfying sj (x) ≤ f (x) ≤ tj (x) on J with J (sj − tj ) → 0 as j → ∞ as j=1 j=1R J f (x)dx = lim J sj (x)dx = lim J tj (x)dx, the less restrictive Lebesgue integral is deﬁned for arbitrary functions fover bounded or unbounded intervals J in terms of Cauchy sequences of step functions J |si − sk | → 0, i, k → ∞, convergingto f (x) as sj (x) → f (x) pointwise almost everywhere on J ,to be def f (x)dx = lim sj (x)dx . J j→∞ JThat the Lebesgue integral is more general (and therefore is the proper candidate for completion of function spaces) isillustrated by the example of the function deﬁned over [0, 1] to be 0 on the rationals and 1 on the irrationals for which anapplication of the deﬁnitions verify that while the Riemann integral is undeﬁned, the Lebesgue integral exists and has value1. The Riemann integral of a bounded function over a bounded interval exists and is equal to its Lebesgue integral. Becauseit involves a larger family of functions, all integrals in integral convergences are to be understood in the Lebesgue sense.
- 10. 3156 A. Sengupta(b) ck (ϕ) = ∞ fk ϕ → c(ϕ) for every ϕ ∈ C0 (J). system evolves to a state of maximal ill-posedness. J The analysis is based on the non-injectivity, and (−m)Since we are now given that I fk (x)dx → hence ill-posedness, of the map; this may be viewed (−m) (m) as a mathematical formulation of the stretch-and- I Ψ(x)dx, it must also be true that fk ϕ con-verges in the mean to Ψϕ(m) whence fold and stretch-cut-and-paste kneading operations of the dough that are well-established artifacts in (−m) (m) the theory of chaos and the concept of maximal ill- fk ϕ = (−1)m fk ϕ J I posedness helps in obtaining a physical understand- ing of the nature of chaos. We do this through the → (−1)m Ψϕ(m) = (−1)m f (−m) ϕ(m) . fundamental concept of the graphical convergence of I I a sequence (generally a net) of functions [SenguptaThe natural question that arises at this stage is Ray, 2000] that is allowed to converge graphically,then: What is the nature of the relation (not func- when the conditions are right, to a set-valued maption any more) Ψ(x)? For this it is now stipulated, or multifunction. Since ill-posed problems naturallydespite the non-equality in the equation above, that lead to multifunctional inverses through functionalas in the mean m-integral convergence of (f k ) to a generalized inverses [Sengupta, 1997], it is naturalfunction f , to seek solutions of ill-posed problems in multifunc- x (−1) def tional space Multi(X, Y ) rather than in spaces of Θ(x) := lim δk (x) = δ(x )dx (6) functions Map(X, Y ); here Multi(X, Y ) is an ex- k→∞ −∞ tension of Map(X, Y ) that is generally larger thandeﬁnes the non-functional relation (“generalized the smallest dense extension Multi | (X, Y ).function”) δ(x) integrally as a solution of the inte- Feedback and iteration are natural processes bygral equation (6) of the ﬁrst kind; hence formally 9 which nature evolves itself. Thus almost every pro- dΘ cess of evolution is a self-correction process by which δ(x) = (7) dx the system proceeds from the present to the future through a controlled mechanism of input and eval-End Tutorial 2 uation of the past. Evolution laws are inherently nonlinear and complex; here complexity is to be un- derstood as the natural manifestation of the non-The above tells us that the “delta function” is not linear laws that govern the evolution of the system.a function but its indeﬁnite integral is the piecewise This paper presents a mathematical descriptioncontinuous function Θ obtained as the mean (or of complexity based on [Sengupta, 1997] and [Sen-pointwise) limit of a sequence of non-diﬀerentiable gupta Ray, 2000] and is organized as follows.functions with the integral of dΘk (x)/dx being pre- In Sec. 1, we follow [Sengupta, 1997] to give anserved for all k ∈ Z+ . What then is the delta overview of ill-posed problems and their solution(and not its integral)? The answer to this ques- that forms the foundation of our approach. Sec-tion is contained in our multifunctional extension tions 2 to 4 apply these ideas by deﬁning a chaoticMulti(X, Y ) of the function space Map(X, Y ) con- dynamical system as a maximally ill-posed problem;sidered in Sec. 3. Our treatment of ill-posed prob- by doing this we are able to overcome the limi-lems is used to obtain an understanding and inter- tations of the three Devaney characterizations ofpretation of the numerical results of the discretized chaos [Devaney, 1989] that apply to the speciﬁc casespectral approximation in neutron transport the- of iteration of transformations in a metric space,ory [Sengupta, 1988, 1995]. The main conclusions and the resulting graphical convergence of func-are the following: In a one-dimensional discrete sys- tions to multifunctions is the basic tool of our ap-tem that is governed by the iterates of a nonlin- proach. Section 5 analyzes graphical convergence inear map, the dynamics is chaotic if and only if the Multi(X) for the discretized spectral approximation9 The observant reader cannot have failed to notice how mathematical ingenuity successfully transferred the “troubles” of ∞(δk )k=1 to the suﬃciently diﬀerentiable benevolent receptor ϕ so as to be able to work backward, via the resultant trouble free (−m)(δk )∞ , to the ﬁnal object δ. This necessarily hides the true character of δ to allow only a view of its integral manifestation k=1on functions. This unfortunately is not general enough in the strongly nonlinear physical situations responsible for chaos, andis the main reason for constructing the multifunctional extension of function spaces that we use.
- 11. Toward a Theory of Chaos 3157of neutron transport theory, which suggests a nat- Example 2.1. As a non-trivial example of an in-ural link between ill-posed problems and spectral verse problem, consider the heat equationtheory of nonlinear operators. This seems to oﬀeran answer to the question of why a natural sys- ∂θ(x, t) ∂ 2 θ(x, t) = c2tem should increase its complexity, and eventually ∂t ∂x2tend toward chaoticity, by becoming increasingly for the temperature distribution θ(x, t) of a one-nonlinear. dimensional homogeneous rod of length L satisfy- ing the initial condition θ(x, 0) = θ 0 (x), 0 ≤ x ≤ L,2. Ill-Posed Problem and Its and boundary conditions θ(0, t) = 0 = θ(L, t), 0 ≤ Solution t ≤ T , having the Fourier sine-series solutionThis section based on [Sengupta, 1997] presents ∞ nπ 2a formulation and solution of ill-posed problems θ(x, t) = An sin x e−λn t (8) Larising out of the non-injectivity of a function f : n=1X → Y between topological spaces X and Y . A where λn = (cπ/a)n andworkable knowledge of this approach is necessary asour theory of chaos leading to the characterization a 2 nπof chaotic systems as being a maximally ill-posed An = θ0 (x ) sin x dx L 0 Lstate of a dynamical system is a direct application ofthese ideas and can be taken to constitute a math- are the Fourier expansion coeﬃcients. While the di-ematical representation of the familiar stretch-cut- rect problem evaluates θ(x, t) from the diﬀerentialand paste and stretch-and-fold paradigms of chaos. equation and initial temperature distribution θ 0 (x),The problem of ﬁnding an x ∈ X for a given y ∈ Y the inverse problem calculates θ0 (x) from the inte-from the functional relation f (x) = y is an inverse gral equationproblem that is ill-posed (or, the equation f (x) = y 2 ais ill-posed) if any one or more of the following con- θT (x) = k(x, x )θ0 (x )dx , 0 ≤ x ≤ L, L 0ditions are satisﬁed. when this ﬁnal temperature θT is known, and(IP1) f is not injective. This non-uniqueness prob-lem of the solution for a given y is the single most ∞ nπ nπ 2signiﬁcant criterion of ill-posedness used in this k(x, x ) = sin x sin x e−λn T L Lwork. n=1(IP2) f is not surjective. For a y ∈ Y , this is the is the kernel of the integral equation. In terms ofexistence problem of the given equation. the ﬁnal temperature the distribution becomes(IP3) When f is bijective, the inverse f −1 is not ∞continuous, which means that small changes in y nπ 2 θT (x) = Bn sin x e−λn (t−T ) (9)may lead to large changes in x. L n=1 A problem f (x) = y for which a solution exists, with Fourier coeﬃcientsis unique, and small changes in data y that lead 2 a nπto only small changes in the solution x is said to Bn = θT (x ) sin x dx . L 0 Lbe well-posed or properly posed. This means thatf (x) = y is well-posed if f is bijective and the In L2 [0, a], Eqs. (8) and (9) at t = T and t = 0inverse f −1 : Y → X is continuous; otherwise the yield respectivelyequation is ill-posed or improperly posed. It is to ∞be noted that the three criteria are not, in general, L 2 2 θT (x) 2 = A2 e−2λn T ≤ e−2λ1 T θ0 n 2 (10)independent of each other. Thus if f represents a 2 n=1bijective, bounded linear operator between Banach ∞spaces X and Y , then the inverse mapping theo- 2 L 2 θ0 = Bn e2λn T . 2 (11)rem guarantees that the inverse f −1 is continuous. 2 n=1Hence ill-posedness depends not only on the alge-braic structures of X, Y , f but also on the topolo- The last two equations diﬀer from each other ingies of X and Y . the signiﬁcant respect that whereas Eq. (10) shows
- 12. 3158 A. Senguptathat the direct problem is well-posed according to (b) For a linear operator A: Rn → Rm , m n, sat-(IP3), Eq. (11) means that in the absence of similar isfying (1) and (2), the problem Ax = y reduces Abounds the inverse problem is ill-posed. 10 to echelon form with rank r less than min{m, n}, when the given equations are consistent. The solu- tion however, produces a generalized inverse leadingExample 2.2. Consider the Volterra integral equa- to a set-valued inverse A− of A for which the inversetion of the ﬁrst kind images of y ∈ R(A) are multivalued because of the x non-trivial null space of A introduced by assump- y(x) = r(x )dx = Kr tion (1). Speciﬁcally, a null-space of dimension n−r a n is generated by the free variables {x j }j=r+1 which are arbitrary: this is illposedness of type (1). In ad-where y, r ∈ C[a, b] and K: C[0, 1] → C[0, 1] is dition, m − r rows of the row reduced echelon formthe corresponding integral operator. Since the dif- of A have all 0 entries that introduce restrictionsferential operator D = d/dx under the sup-norm m on m − r coordinates {yi }i=r+1 of y which are now r = sup0≤x≤1 |r(x)| is unbounded, the inverse r related to {yi }i=1 : this illustrates ill-posedness ofproblem r = Dy for a diﬀerentiable function y type (2). Inverse ill-posed problems therefore gen-on [a, b] is ill-posed, see Example 6.1. However, erate multivalued solutions through a generalizedy = Kr becomes well-posed if y is considered to be inverse of the mapping.in C 1 [0, 1] with norm y = sup0≤x≤1 |Dy|. This il- (c) The eigenvalue problemlustrates the importance of the topologies of X andY in determining the ill-posed nature of the prob- d2lem when this is due to (IP3). + λ2 y = 0 y(0) = 0 = y(1) dx2 Ill-posed problems in nonlinear mathematics oftype (IP1) arising from the non-injectivity of f has the following equivalence class of 0can be considered to be a generalization of non- d2uniqueness of solutions of linear equations as, for [0]D2 = {sin(πmx)}∞ , m=0 D2 = + λ2 ,example, in eigenvalue problems or in the solution of dx2a system of linear algebraic equations with a larger as its eigenfunctions corresponding to the eigenval-number of unknowns than the number of equations. ues λm = πm.In both cases, for a given y ∈ Y , the solution set of Ill-posed problems are primarily of interest tothe equation f (x) = y is given by us explicitly as non-injective maps f , that is under f − (y) = [x]f = {x ∈ X : f (x ) = f (x) = y} . the condition of (IP1). The two other conditions (IP2) and (IP3) are not as signiﬁcant and play onlyA signiﬁcant point of diﬀerence between linear and an implicit role in the theory. In its application tononlinear problems is that unlike the special im- iterative systems, the degree of non-injectivity of fportance of 0 in linear mathematics, there are no deﬁned as the number of its injective branches, in-preferred elements in nonlinear problems; this leads creases with iteration of the map. A necessary (butto a shift of emphasis from the null space of linear not suﬃcient) condition for chaos to occur is theproblems to equivalence classes for nonlinear equa- increasing non-injectivity of f that is expressed de-tions. To motivate the role of equivalence classes, scriptively in the chaos literature as stretch-and-foldlet us consider the null spaces in the following lin- or stretch-cut-and-paste operations. This increasingear problems. non-injectivity that we discuss in the following sec-(a) Let f : R2 → R be deﬁned by f (x, y) = x + y, tions, is what causes a dynamical system to tend(x, y) ∈ R2 . The null space of f is generated by the toward chaoticity. Ill-posedness arising from non-equation y = −x on the x–y plane, and the graph surjectivity of (injective) f in the form of regular-of f is the plane passing through the lines ρ = x ization [Tikhonov Arsenin, 1977] has receivedand ρ = y. For each ρ ∈ R the equivalence classes wide attention in the literature of ill-posed prob-f − (ρ) = {(x, y) ∈ R2 : x + y = ρ} are lines on the lems; this however is not of much signiﬁcance ingraph parallel to the null set. our work.10 Recall that for a linear operator continuity and boundedness are equivalent concepts.
- 13. Toward a Theory of Chaos 3159 %¨§ # ¡$ ¡ ¨§ # P ! 3 5) B 6 @ ¡ £ £ 6 GF 8@ ¡ £ © £ 3 5 I1 ) ¡ ¥£ © ¤ 8 HF 921ED 3 ) ¤ ¥£ 4210( 3 ) ¡ ©¨§ ¦ ¨§ ¦ ¡$ A C ¡¢ 8 95 6 75 (a) (b)Fig. 4. (a) Moore–Penrose generalized inverse. The decomposition of X and Y into the four fundamental subspaces of Acomprising the null space N (A), the column (or range) space R(A), the row space R(AT ) and N (AT ), the complement ofR(A) in Y , is a basic result in the theory of linear equations. The Moore–Penrose inverse takes advantage of the geometricorthogonality of the row space R(AT ) and N (A) in Rn and that of the column space and N (AT ) in Rm . (b) When X andY are not inner-product spaces, a non-injective inverse can be deﬁned by extending f to Y − R(f ) suitably as shown bythe dashed curve, where g(x) := r1 + ((r2 − r1 )/r1 )f (x) for all x ∈ D(f ) was taken to be a good deﬁnition of an extensionthat replicates f in Y − R(f ); here x1 ∼ x2 under both f and g, and y1 ∼ y2 under {f, g} just as b is equivalent tob in the Moore–Penrose case. Note that both {f, g} and {f − , g − } are both multifunctions on X and Y , respectively. Ourinverse G, introduced later in this section, is however injective with G(Y − R(f )) := 0. map a) is the noninjective map deﬁned in terms of the row and column spaces of A, row(A) = R(A T ),Begin Tutorial 3: Generalized col(A) = R(A), asInverseIn this Tutorial, we take a quick look at the equation def (a|row(A) )−1 (y), if y ∈ col(A)a(x) = y, where a: X → Y is a linear map that need GMP (y) = 0, if y ∈ N (AT ) .not be either one-one or onto. Speciﬁcally, we willtake X and Y to be the Euclidean spaces R n and (12)Rm so that a has a matrix representation A ∈ R m×nwhere Rm×n is the collection of m×n matrices with Note that the restriction a|row(A) of a to R(AT )real entries. The inverse A−1 exists and is unique iﬀ is bijective so that the inverse (a| row(A) )−1 is well-m = n and rank(A) = n; this is the situation de- deﬁned. The role of the transpose matrix appearspicted in Fig. 1(a). If A is neither one-one or onto, naturally, and the GMP of Eq. (12) is the uniquethen we need to consider the multifunction A − , a matrix that satisﬁes the conditionsfunctional choice of which is known as the general-ized inverse G of A. A good introductory text for AGMP A = A, GMP AGMP = GMP , (13)generalized inverses is [Campbell Mayer, 1979]. (GMP A)T = GMP A, (AGMP )T = AGMPFigure 4(a) introduces the following deﬁnition ofthe Moore–Penrose generalized inverse G MP . that follow immediately from the deﬁnition (12); hence GMP A and AGMP are orthogonal projec-Deﬁnition 2.1 (Moore–Penrose Inverse). If a: tions11 onto the subspaces R(AT ) = R(GMP ) andRn → Rm is a linear transformation with matrix R(A), respectively. Recall that the range spacerepresentation A ∈ Rm×n then the Moore–Penrose R(AT ) of AT is the same as the row space row(A)inverse GMP ∈ Rn×m of A (we will use the same of A, and R(A) is also known as the column spacenotation GMP : Rm → Rn for the inverse of the of A, col(A).11 A real matrix A is an orthogonal projector iﬀ A2 = A and A = AT .
- 14. 3160 A. SenguptaExample 2.3. For a: R5 → R4 , let rank is 4. This gives 9 1 18 2 1 −3 2 1 2 − − 275 275 275 55 3 −9 10 2 9 − 27 3 54 6 A= − 2 −6 4 2 4 275 275 275 55 2 −6 8 1 7 10 6 20 16 GMP = − − 143 143 143 143 238 57 476 59 By reducing the augmented matrix (A|y) to the − − 3575 3575 3575 715 row-reduced echelon form, it can be veriﬁed that 129 106 258 47 the null and range spaces of A are three- and two- − −dimensional, respectively. A basis for the null space 3575 3575 3575 715 (14)of AT and of the row and column space of A ob-tained from the echelon form are respectively as the Moore–Penrose inverse of A that readily ver- iﬁes all the four conditions of Eqs. (13). The basic point here is that, as in the case of a bijective map, 1 0 −3 0 GMP A and AGMP are identities on the row and col- −2 1 1 0 umn spaces of A that deﬁne its rank. For later use — 0 −1 0 1 0 1 , ; and 3 , 1 ; , . when we return to this example for a simpler inverse 1 0 2 0 2 − 4 G — given below are the orthonormal bases of the 0 1 1 3 −1 1 four fundamental subspaces with respect to which 2 4 GMP is a representation of the generalized inverse of A; these calculations were done by MATLAB. The basis forAccording to its deﬁnition Eq. (12), the Moore–Penrose inverse maps the middle two of the above (a) the column space of A consists of the ﬁrst twoset to (0, 0, 0, 0, 0)T , and the A-image of the ﬁrst columns of the eigenvectors of AAT :two (which are respectively (19, 70, 38, 51) T and T(70, 275, 140, 205)T lying, as they must, in the span 1633 363 3317 363 − ,− , ,of the last two), to the span of (1, −3, 2, 1, 2) T and 2585 892 6387 892(3, −9, 10, 2, 9)T because a restricted to this sub- T 929 709 346 709space of R5 is bijective. Hence − , , ,− 1435 1319 6299 1319 1 0 (b) the null space of AT consists of the last two −3 0 −2 columns of the eigenvectors of AAT : 1 0 1 0 −1 T 3185 293 3185 1777 GMP A 3 A 1 − ,− 1 , , 0 2 −4 8306 2493 4153 3547 1 3 0 1 T 323 533 323 1037 , , , 2 4 1732 731 866 1911 1 0 0 0 (c) the row space of A consists of the ﬁrst two −3 0 0 0 columns of the eigenvectors of AT A: 0 1 0 0 421 44 569 659 1036 = 3 1 . , ,− ,− , 13823 14895 918 2526 1401 2 −4 0 0 661 412 59 1523 303 1 3 , , ,− ,− 0 0 690 1775 2960 10221 3974 2 4 (d) the null space of A consists of the last threeThe second matrix on the left is invertible as its columns of AT A:
- 15. Toward a Theory of Chaos 3161 571 369 149 291 389 (T3) Arbitrary unions of members of U belong − ,− , ,− ,− to U. 15469 776 25344 350 1365 281 956 875 1279 409 Example 2.4 − , , ,− , 1313 1489 1706 2847 1473 (1) The smallest topology possible on a set X is 292 876 203 621 1157 its indiscrete topology when the only open sets ,− , , , 1579 1579 342 4814 2152 are ∅ and X; the largest is the discrete topologyThe matrices Q1 and Q2 with these eigenvectors where every subset of X is open (and hence also(xi ) satisfying xi = 1 and (xi , xj ) = 0 for i = j closed).as their columns are orthogonal matrices with the (2) In a metric space (X, d), let Bε (x, d) = {y ∈ X:simple inverse criterion Q−1 = QT . d(x, y) ε} be an open ball at x. Any subset U of X such that for each x ∈ U there is a d- ball Bε (x, d) ⊆ U in U , is said to be an openEnd Tutorial 3 set of (X, d). The collection of all these sets is the topology induced by d. The topological space (X, U) is then said to be associated withThe basic issue in the solution of the inverse ill- (induced by) (X, d).posed problem is its reduction to an well-posed one (3) If ∼ is an equivalence relation on a set X, thewhen restricted to suitable subspaces of the do- set of all saturated sets [x]∼ = {y ∈ X: y ∼ x}main and range of A. Considerations of geometry is a topology on X; this topology is called theleading to their decomposition into orthogonal sub- topology of saturated sets.spaces is only an additional feature that is not cen- We argue in Sec. 4.2 that this constitutestral to the problem: recall from Eq. (1) that any the deﬁning topology of a chaotic system.function f must necessarily satisfy the more general (4) For any subset A of the set X, the A-inclusionset-theoretic relations f f −f = f and f − f f − = f − topology on X consists of ∅ and every supersetof Eq. (13) for the multiinverse f − of f : X → Y . of A, while the A-exclusion topology on X con-The second distinguishing feature of the MP-inverse sists of all subsets of X − A. Thus A is openis that it is deﬁned, by a suitable extension, on all in the inclusion topology and closed in the ex-of Y and not just on f (X) which is perhaps more clusion, and in general every open set of one isnatural. The availability of orthogonality in inner- closed in the other.product spaces allows this extension to be made The special cases of the a-inclusion and a-in an almost normal fashion. As we shall see be- exclusion topologies for A = {a} are deﬁned inlow the additional geometric restriction of Eq. (13) a similar fashion.is not essential to the solution process, and in- (5) The coﬁnite and cocountable topologies in whichfact, only results in a less canonical form of the the open sets of an inﬁnite (resp. uncount-inverse. able) set X are respectively the complements of ﬁnite and countable subsets, are examples of topologies with some unusual properties that are covered in Appendix A.1. If X is itself ﬁnite (respectively, countable), then its coﬁniteBegin Tutorial 4: Topological Spaces (respectively, cocountable) topology is the dis-This Tutorial is meant to familiarize the reader with crete topology consisting of all its subsets. It isthe basic principles of a topological space. A topo- therefore useful to adopt the convention, unlesslogical space (X, U) is a set X with a class 12 U of stated to the contrary, that coﬁnite and co-distinguished subsets, called open sets of X, that countable spaces are respectively inﬁnite andsatisfy uncountable.(T1) The empty set ∅ and the whole X belong to U In the space (X, U), a neighborhood of a point(T2) Finite intersections of members of U belong x ∈ X is a nonempty subset N of X that con-to U tains an open set U containing x; thus N ⊆ X is a12 In this sense, a class is a set of sets.

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