Conley Theory: A combinatorial approach to dynamics

Transcription

1 Conley Theory: A combinatorial approach to dynamics Konstantin Mischaikow November 21, 2010 Contents 1 Why? Physics Biology Data Data & Planetary Motion Modeling via Data Elementary Definitions Motivating examples Definitions Invariant sets Reality Check Alpha and Omega limit sets Another Reality Check Equivalences of dynamical systems Attractor-Repeller Decompositions Lattice of Invariant Sets Attractors and repellers Attracting neighborhoods Based on notes with W. Kalies and R. Vandervorst mischaik@math.rutgers.edu 1

2 3.2.2 Attractor within an attractor Attractor-Repeller pairs Lyapunov functions Lyapunov functions for attractor-repeller pairs Attracting blocks Lattice structures for attractor-repeller decompositions Lattices of attractors and repellers Lattices of Lyapunov functions Filtrations Birkhoff s Representation Theorem Morse Decompositions Attractor lattices and Morse decompositions Constructing a p-morse decomposition from a lattice of attractors A Dynamics version of Birkhoff s Representation Theorem Lyapunov Functions for Morse Decompositions Directed Graphs Definitions Digraph Dynamics Combinatorialization Time Discretization Space Discretization Grids and covering grids Multivalued mappings for grids and covering grids Digraph Representation of Dynamics Convergence 93 9 Reduced Dynamics Continuity Index Pairs Index Pairs from Digraph Dynamics

3 10 Conley Index Shift Equivalence Conley index Extracting Dynamics Parameterized Dynamics 113 3

4 Forward. These are lecture notes for a course given at Jagiellonian University and are based in part on a book being prepared jointly with W. Kalies and R. Vandervorst. These notes are incomplete in several ways. First, because they are only intended to re-enforce the material presented in class there is no proper acknowledgement or credit given to original sources. Similarly, there is no attempt at completeness, either mathematically or grammatically. Second, many proofs are not included due to time constraints in the lectures. Third, there has been no attempt to correct errors and typos. 4

5 1 Why? 1.1 Physics Nicholaus Copernicus ( ) Heliocentric model Tycho Brahe ( ) Recognized that classical measurements were inconsistent. Careful systematic measurements - goal to prove an earth centric model Johannes Kepler ( ) Continued Brahe s measurements and had access to Brahe s data. Assigned to study Mars - most elliptical of orbit of planets. Three laws 1. The orbits of the planets are ellipses, with the sun at one focus of the ellipse. 2. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. 3. The ratio of the squares of the revolutionary periods P i for two planets is equal to the ratio of the cubes of half of their major axes R i : P1 2 P2 2 = R3 1 R 3 2 Remark: These are descriptive laws derived from data as opposed to results obtained from a theoretical framework. Galileo Galilei ( ) Newton s First Law Isaac Newton ( ) Principia Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. 5

6 F = ma For every action there is an equal and opposite reaction. Remarks: Notice that it took approximately 100 years from collecting quality data to a mathematical model that provided a description of the phenomenon. Mass m is a quantity that can be measured very precisely. In applications to celestial mechanics approximations are made, i.e. two-body problems, point masses. The precession of the perihelion of Mercury does not agree with the predictions of Newtonian celestial mechanics. (one of the earliest experimental supports for general relativity) 1.2 Biology A parasitoid is an insect whose females lay their eggs in or on the bodies of a host insect. Typically, hosts that have been parasitized give rise to the next generation of parasitoids. Only hosts that have not been parasitized give rise to the next generation of hosts. We want to make a population model for the host/parasitoids interaction. Let H denote the population of hosts and P denote the population of parasitoids. Let f(h, P ) denote the fraction of hosts not parasitized. Then f(h, P )H = number of hosts not parasitized [1 f(h, P )] H = number of hosts that are parasitized Assume 1: The only control on the growth of the hosts population are the parasitoids. (this means that we are interested in the situation in which the parasites keep the host population from getting so large that they compete with themselves for resources). H n+1 = kf(h n, P n )H n 6

7 Observe that k is the reproductive rate for the non parasitized hosts. If k < 1, then the hosts will die out whether or not there are any parasitoids. Thus from now on we assume k > 1. Assume 2: The average number of eggs laid in a single host that grows to be an adult parasitoid is constant. P n+1 = c [1 f(h n, P n )] H n What model shall we choose for f? It clearly depends on how the hosts and parasitoids interact. Assume 3: The host and parasitoids meet at random and this probability is independent of whether the host is already infected. If we use the concept from chemistry of mass action kinetics, then the probability of meeting should be proportional to the product of the number of hosts and parasitoids. Thus the average number of meetings per host is ν = ahp H = ap i.e. it is proportional to the number of parasitoids. This is an average. Some hosts will have more encounters and some less. Let p(i) = the probability that a host meets i parasitoids Random, independent encounters are modeled using a Poisson distribution p(i) = νi e ν Assume 4: Assume that if the host and parasitoid meet then the host is parasitized. f(h, P ) = p(0) = ν0 e ν = e ν = e ap 0! Thus the model is This is the Nicholson-Bailey model. Remarks: i! H n+1 = kh n e apn (1) P n+1 = ch n [ 1 e ap n ] 7

8 Has been used to fit data for approximately two dozen generations of populations of the greenhouse whitefly Trialeurodes vaporairorum and the parasitoid Encarsia formosa grown under laboratory conditions. Almost all solutions become unbounded as time goes to infinity. Fitting values to parameters k, c, a is not obvious. Summary: Typical issues associated with mathematical models of biology (or many multiscale) processes: 1. Understanding the dynamics is often essential 2. Nonlinearities are often based on heuristics rather than first principles 3. Multiple, poorly measured parameters 4. Limited resolution of measurement 1.3 Data Data & Planetary Motion Aerospace web page: High overhead, more than 20,000 kilometers above Earth, GPS satellites race by at speeds approaching 3800 meters per second. The movements of these spacecraft are generally described by the laws of planetary motion developed by Johannes Kepler almost 400 years agobut they are by no means certain or simple. Each satellite must contend with diverse forces that constantly nudge and pull it from its desired orbit. Yet in spite of this, the positions of GPS satellites must be known at all times with exceptional accuracy. Modeling these orbits is a complex affair... There is no mention of Newton! Remark: As a mathematician and scientist I do not want to give up on Newton - theory that explains why an observable phenomenon occurs. 8

9 But this takes time and what we have is lots of data and what we will have shortly is phenomenal amounts of data (Petabyte data sets). Applied science, engineering, medicine, social sciences are developing techniques based on data not theory. My opinion is that because these techniques are being used it is essential to understand them in a rigorous way, i.e. we need to have mathematical theories that can incorporate these techniques. 1.5 Modeling via Data Anthony R. Ives, Árni Einarsson, Vincent A. A. Jansen and Arnthor Gardarsson, High-amplitude fluctuations and alternative dynamical states of midges in Lake Myvatn, Nature, vol. 452 (7183) pp Lake Myvatn is a shallow, naturally eutrophic lake in northern Iceland. Midges, Tanytarsus gracilentus, are the dominant herbivore/detritivore in Myvatn. Two non-overlapping generations per year (first in May, second in late July early August) Statistical evidence suggests that fluctuations in midge populations are driven by consumerresource interactions, with midges being the consumers and algae/detritus the resources, as opposed to predatorprey interactions with midges being the prey Population levels of midges have been collected since Solid line derived from measured data. 9

10 Proposed network of interactions: 7 x 1 midge population 7 7 x 2 algae population x 3 detritus population It is assumed that if all the algae are consumed then the algal populations can recover through the input of small subsidies from outside the midgealgaedetritus system. These subsidies represent small influxes of algae and detritus into the muddy midge habitat from hard-bottom areas where midges are few. There is no direct measurement of this input, but much of the algae 10

11 and detritus in the lake occurs in areas inaccessible to midge larvae, and the hydrological mixing of the shallow lake makes influxes of small amounts of this material into the midge habitat a certainty. Deterministic model ( ) q x 1 (t) x 1 (t + 1) = r 1 x 1 (t) 1 + (2) x 2 (t) + px 3 (t) x 2 (t) x 2 (t + 1) = r x 2 (t) x 2 (t) x 2 (t) + px 3 (t) x 1(t + 1) + c (3) where x 3 (t + 1) = dx 3 (t) + x 2 (t) r 1 is the intrinsic population growth rate for midges r 2 is the intrinsic population growth rate for algae q density dependence parameter p quality of detritus for midges in relation to algae c is influx rate of algae from environment d retention rate of detritus in environment px 3 (t) x 2 (t) + px 3 (t) x 1(t + 1) + c (4) This model is built on the interaction network information, heuristic nonlinearities, and an attempt to minimize the number of parameters. Question: How can we justify this model? Consider different models. Gompertz log-linear model. Let u i = ln x i u i (t + 1) = 3 b ij u j (t) j=1 Lotka-Volterra model ( ) 3 x i (t + 1) = r i x i (t) exp 1 + b ij x j (t) j=1 11

12 and compare global dynamics of these three models. First set of results: There exist parameter values of IEJG model which exhibit multiple basins of attraction (we will give precise definitions of this later). This is not true for other models. (a) r 1 = 3.873, r 2 = , c = , d = , p = , q = (b) σ 1 = , σ 2 = σ 3 =

13 Remark: The lake Myvatn ecosystem is more complicated that that which has been modelled. This is accounted for by introducing stochastic fluctuations. IEJG model: ( ) q x 1 (t) x 1 (t + 1) = r 1 x 1 (t) 1 + e ɛ 1(t) x 2 (t) + px 3 (t) [ ] x 2 (t) x 2 (t + 1) = r x 2 (t) x 2 (t) x 2 (t) + px 3 (t) x 1(t + 1) + c x 3 (t + 1) = [ dx 3 (t) + x 2 (t) where the ɛ i are normal random variables. Gompertz log-linear model. ( 3 ) u i (t + 1) = b ij u j (t) Lotka-Volterra model x i (t + 1) = r i x i (t) exp px 3 (t) x 2 (t) + px 3 (t) x 1(t + 1) + c j=1 ( 1 + e ɛ i(t) ) 3 b ij x j (t) j=1 e ɛ i(t) e ɛ 2(t) ] e ɛ 3(t) Remark: Observe that the c is extremely small. However, dynamics is very sensitive to value of c. A striking biological conclusion from the model is the sensitivity of the amplitude of midge fluctuations to very small amounts of resource input, c; the resource input sets the lower boundary of midge abundance and hence the severity of population crashes. Thus, even though resource input might be six orders of magnitude less than the abundance of resources in the lake in most years, this vanishingly small source of resources is nevertheless critical in setting the depth of the midge population nadir and the subsequent rate of recovery. This sensitivity to resource subsidies might explain changes in midge dynamics that have apparently occurred over the last decades. Although Myvatn has supported a (5) (6) (7) 13

14 local charr (salmonid) fishery for centuries, this fishery collapsed in the 1980s, coincident with particularly severe midge population crashes. Over the same period, waterbird reproduction in Myvatn was also greatly reduced during the crash years. These changes might have been caused by dredging in one of the two basins in the lake that started in 1967 to extract diatomite from the sediment. Hydrological studies indicate that dredging produces depressions that act as effective traps of organic particles, hence reducing algae and detritus inputs to the midge habitat. Our model predicts that even a slight reduction in subsidies can markedly increase the magnitude of midge fluctuations. Such slight environmental changes can then have seriously negative con- sequences for fish and bird populations. Conclusion: Being able to derive quantitative conclusions from these data driven, heuristically generated models is important. 14

15 2 Elementary Definitions 2.1 Motivating examples Example 2.1 The logistic equation is ẋ = dx dt = rx(κ x) (8) where r is the reproduction rate and κ is the carrying capacity of the environment. The unknown function x(t) describes a population density. Given an initial population x 0, this equation can be solved explicitly by x(t) = x 0 κe rκt κ x 0 + x 0 e rκt. Dynamical systems approach to differential equations is to consider the collection of all solutions as a function of both time t and initial value x. In the case of the logistic equation we obtain ϕ(t, x) = xκe rκt. (9) κ x + xerκt Definition 2.2 Let X be a metric space. Let D R X be an open set with the property that D x := D (R {x}) = (τ x, τ + x ) where τ x [, 0) and τ + x (0, ]. A continuous function ϕ : D X is a local flow if (i) ϕ(0, x) = x for all x X and (ii) ϕ(t, ϕ(s, x)) = ϕ(t + s, x) for all s D x and t D ϕ(s,x). If D x = R for all x X, then ϕ is a (global) flow. The following exercise shows that we can reparameterize a vector field so that the local flow becomes a global flow. 15

16 Exercise 2.3 Suppose that F : R n R n is a locally Lipschitz vector field. Show that the differential equation ẋ = F (x) 1 + F (x) 2 generates a global flow on R n whose orbits are the same as the orbits of ẋ = F (x). Example 2.4 The logistic differential equation (8) models a population in which the change of the population is continuous. However, in certain populations (recall the midges of Lake Myvatn) births happen only at certain fixed times and hence are modeled by a discrete time system. In this context, one can consider the dynamics of iterating the logistic map given by x n+1 = f(x n ) = rx n (1 x n ). (10) Orbits of this system are sequences {x n } n Z such that x n+1 = f(x n ) For the logistic map with r = 2.5 both { {x n = 0.6} n Z and x n = { } f n (0.6) for n < for n 0 n Z (11) are orbits through x 0 = 0.6 where f 1 (x) := (1 1 (8x)/5)/2. Clearly we do not have uniqueness in backward time. This is caused by the fact that f is not a monomorphism. Example 2.5 The partial differential equation u t = u xx u, u(t, 0) = u(t, π) = 0, x [0, π] (12) gives rise to the same phenomenon: we can solve the initial value problem u 0 (x) = u(0, x) forward in time but not necessarily backward in time. 16

17 2.2 Definitions Let (X, d) be a metric space X with metric d, and let T denote the space of time variables which is either Z or R. The restriction T + denotes either Z + = {0, 1, 2,...} or R + = [0, ). Definition 2.6 A dynamical system on X is a continuous map ϕ : T + X X that satisfies the following two properties: (i) ϕ(0, x) = x for all x X, and (ii) ϕ(t, ϕ(s, x)) = ϕ(t + s, x) for all s, t T + and all x X. Remark 2.7 Most references refer to this type of system as a semi-dynamical system. For them a dynamical system satisfies properties (i) and (ii) for all t T. Observe that in this case we have a group action of T on X. As will become clear I want to de-emphasize the group action. Though ϕ is only defined for nonnegative times understanding the preimages of points with respect to time is often essential. This leads to the following extension. Definition 2.8 Let ϕ : T + X X be a dynamical system. The backward extension ϕ : T X X is defined by ϕ( t, x) := {y X ϕ(t, y) = x}. Proposition 2.9 Let U X and let s T. (i) If t 0, then ϕ(t, ϕ(s, U)) ϕ(t + s, U). Moreover, if ϕ(s, x) for all x U and t + s 0, then (ii) If t 0, then Moreover, if s 0, then ϕ(t, ϕ(s, U)) = ϕ(t + s, U). ϕ(t, ϕ(s, U)) ϕ(t + s, U). ϕ(t, ϕ(s, U)) = ϕ(t + s, U). Proof is left as an exercise, the only comment is that if ϕ(s, x) =, then ϕ(t, ϕ(s, x)) =. 17

18 2.3 Invariant sets The starting point for the analysis of dynamical systems is the time evolution of a single point. Definition 2.10 Let ϕ : T + X X be a dynamical system. The forward orbit or trajectory through x X is the set {ϕ(t, x) t T + }, denoted by γ + x. A (complete) orbit or trajectory through x X is the image {γ x (t) X t T} of a function γ x : T X such that γ x (0) = x and γ x (t + s) = ϕ(s, γ x (t)) for all t T and s T +. A backward orbit or trajectory through x X is the image of the restriction of a complete orbit to t T and is denoted by γ x where T = {t T t 0}. Remark 2.11 For simplicity of notation, γ x will be used both to denote the function γ x : T X which defines a trajectory through x and its image {γ x (t) X t T} which is the trajectory. Note that γ x need not be unique nor exist for all x. The qualitative study of dynamical systems involves the analysis of the structure of its orbits. One of the fundamental ideas in the theory of dynamical systems is that it is often useful to consider collections of orbits rather than individual orbits. Definition 2.12 Given a dynamical system ϕ : T + X X, a set S X is an invariant set if ϕ(t, S) = S for all t T +. A set S X is called strongly invariant if ϕ(t, S) = S for all t T. Notation: Given a dynamical system ϕ : T + X X the collection of invariant sets and strongly invariant sets in X are denoted by Invset(X, ϕ) and SInvset(X, ϕ), respectively. Clearly, SInvset(X, ϕ) Invset(X, ϕ). The set of forward invariant sets is denoted by Invset + (X, ϕ) := { S X ϕ(t, S) S, t T +} and the set of backward invariant sets is denoted by Invset (X, ϕ) := { S X ϕ(t, S) S, t T }. 18

19 Definition 2.13 Let ϕ : T + X X be a dynamical system. The forward image of a set U X is defined by Γ + (U) := t 0 ϕ(t, U) = ϕ ( [0, ), U ). The backward image of a set U X is defined by Γ (U) := t 0 ϕ(t, U) = ϕ ( (, 0], U ). The complete image of U is Γ(U) = Γ + (U) Γ (U). Γ + τ (U) := ϕ ( τ, Γ + (U) ) and Γ τ(u) := ϕ ( τ, Γ + (U) ). For τ 0, define Remark 2.14 Observe that for any dynamical system, the empty set is strongly invariant since ϕ( t, ) := {x X ϕ(t, x) } = Proposition 2.15 If the dynamical system ϕ : T + X X is invertible then an invariant set is strongly invariant. Proof is left as an exercise. Definition 2.16 Let ϕ : T + X X be a dynamical system. An element x X is an equilibrium point or fixed point if ϕ(t, x) = x for all t T +. For the flow ϕ of the logistic equation ẋ = rx(κ x) φ(t, 0) = 0 and φ(t, κ) = κ for all t R, and thus {0} and {κ} are strongly invariant sets. In contrast, if one lets ψ : Z + R R denote the dynamical system generated by the logistic map f(x) = rx(1 x) for r > 1, then ψ(n, 0) = 0 and ψ ( ) n, r 1 r = r 1 for all n 0. Thus, 0 and r r 1 are invariant sets. However, 1 ψ( 1, 0) and hence 0 is not a strongly r invariant set. 19

20 Definition 2.17 Let ϕ : T + X X be a dynamical system. A point x X is called a periodic point with period τ if there exists τ > 0 such that ϕ(τ, x) = x. If ϕ(t, x) x for all t (0, τ), then τ is the minimal period. The orbit {ϕ(t, x) 0 t τ } through a periodic point x is called a periodic orbit. Returning to the logistic equation ẋ = rx(κ x). From the phase portrait we see that κ is an attracting fixed point; that is, any initial condition chosen sufficiently close to κ limits to κ in forward time. 0 is a repelling fixed point in that any initial condition chosen sufficiently close to 0 moves a uniform distance away from 0 in forward time. There is one other bounded orbit (0, κ). Taking a more global point of view note that for each x (0, κ), lim ϕ(x, t) = 0 and lim ϕ(x, t) = κ. t t The biological implication of this is that given an arbitrarily small but positive population, asymptotically the population will tend towards the carrying capacity. Thus, understanding the limits of orbits with respect to time is of considerable importance. Definition 2.18 Let ϕ : T + X X be a dynamical system. A heteroclinic orbit is a complete orbit γ x such that lim γ x(t) = y and lim γ x (t) = z. t t where y and z are distinct equilibria of ϕ. Using the language developed so far, the structure of the dynamics of the logistic equation is as follows. The invariant set composed of bounded orbits consists of two equilibria, 0 and κ, and a unique heteroclinic orbit from 0 to κ. 20

21 Example 2.19 (Doubling Map) Consider the dynamical system generated by f : S 1 S 1 defined by f(x) = x mod 1 Write x base 2, i.e. x = 0.x 1 x 2 x 3... where x i {0, 1}. Then is the shift operator. f(x) = 0.x 2 x 3 x 4... For every period τ N there exist 2 τ periodic points. Given any two periodic points we can construct a heteroclinic orbit from one to the other. There is an orbit which is dense in S 1. Given any two distinct initial conditions x, y S 1, there exists t > 0 such that ϕ(t, x) ϕ(t, y) > 1/4. The point of this example is that set of bounded orbits can be extremely complicated (this is an example of chaos) 2.4 Reality Check Gauss fitting of logisitic map. We are back to fundamental question of qualitative vs. quantitative or heuristic vs. predictive. In this setting the notion of an invariant set (equilibrium point) is more of Platonic ideal than a physical reality. One of the things we want to develop is a language which allows us to replace these platonic ideals by experimentally measurable quantities. 2.5 Alpha and Omega limit sets Of fundamental interest is what happens in future time. The following concept captures the asymptotic future. A point x X need not have a limit under ϕ as t. For example points y = ϕ(t, x) approaching a periodic orbit. In order to describe the limiting behavior of a point x a logical step is to consider the possible limiting behaviors for arbitrary time sequences t n. 21

22 Definition 2.20 Let ϕ : T + X X be a dynamical system. A point y is called a omega limit point of a set U X under ϕ if there exist times t n and points x n U such that lim n ϕ(t n, x n ) = y. The set of all omega limit points y is called the omega limit set of U and is denoted by ω(u, ϕ). Exercise 2.21 Construct an example to show that in general ω(u, ϕ) x U ω(x, ϕ). Proposition 2.22 Let U X. Then ω(u) = t 0 cl ( ϕ([t, ), U) ) = ω ( Γ + (U) ). (13) The omega limit set ω(u) is closed, forward invariant, and contained in cl(γ + (U)). If U X is forward invariant, then ω(u) = t 0 cl ( ϕ(t, U) ), (14) and ω(u) cl(u) (equality when U is invariant). Proof: First prove ω(u) = t 0 cl ( ϕ([t, ), U) ) Let y ω(u), then y cl ( ϕ([t, ), U)) ), t T +, and thus ω(u) t 0 cl ( ϕ ( [t, ), U )). In the other direction, if y t 0 cl ( ϕ ( [t, ), U )), then y cl ( ϕ([t, ), U)) ), t T +. Choose an increasing sequence of t n T + and x n U such that d(ϕ(t n, x n ), y) < 1/n. Since d(ϕ(t n, x n ), y) 0 as t n, it follows that y ω(u) which proves the other inclusion. 22

23 Now prove ω(u) = t 0 cl ( ϕ(t, U) ). Since U X is forward invariant, the semi-group property implies, ϕ(t + s, U) = ϕ ( t, ϕ(s, U) ) ϕ(t, U) s, t 0. Therefore ϕ([t, ), U) = ϕ(t, U). The second part of the first statement ω(u) = ω ( Γ + (U) ) follows from the semi-group property, i.e. ϕ([t, ), U) = ϕ(t, Γ + (U)), Equation (14) and the forward invariance of Γ + (U). Closedness follows immediately from the definition. As for the forward invariance of ω(u) we argue as follows. We first show forward invariance of ω(u) when U is forward invariant. For any t T + we have that ( ϕ(t, ω(u)) = ϕ t, cl ( ϕ(s, U) )) s 0 cl (ϕ ( t, ϕ(s, U) )) s 0 = cl (ϕ ( s, ϕ(t, U) )) s 0 ω(u). For general U, use the fact that Γ + (U) is forward invariant. Therefore, ϕ(t, ω(u)) = ϕ ( t, ω(γ + (U)) ) ω(γ + (U)) = ω(u), which proves forward invariance for general U. The omega limit set is obviously contained in cl(γ + (U)). Proposition 2.23 Suppose Γ + τ (U) is precompact for some τ 0, then (i) ω(u) is compact and invariant; 23

24 (ii) U, implies ω(u) ; (iii) U connected and T = R, implies that ω(u) is connected; (iv) for all x U, d(ϕ(t, x), ω(u)) 0, as t. Proof: (Compactness) For t τ, cl ( ϕ([t, ), U)) ) cl(γ + τ (U)) is compact, and thus ω(u) cl ( ϕ([t, ), U)) ) cl(γ + τ (U)) t τ is compact. Since the latter is an intersection of nested non-empty compact sets it is non-empty (this also implies Property (ii)). (Invariance) We need to show that ϕ(t, ω(u)) = ω(u), t T +. Let y ω(u). By definition there exists a sequence {(x n, t n )}, t n, such that d(ϕ(t n, x n ), y) 0, as n. Given t > 0, assume that t n > t, then the sequence {ϕ(t n t, x n )} Γ + (U) is well-defined, and since Γ + (U) is precompact, there exists a subsequence converging to some z ω(u) (a limit point has to be in ω(u) by definition). By continuity ϕ(t, z) = y, for all y ω(u). Similarly, if z ω(u) then there exists a sequence {(x n, t n )}, t n, such that d(ϕ(t n, x n ), z) 0, as n. Let y = ϕ(t, z). By continuity d(ϕ(t n + t, x n ), y) 0, as n. Thus, y ω(u). Therefore ω(u) is invariant. When T = R and U is connected, then ϕ ( t, Γ + (U) ) is connected. Using the precompactness of Γ + (U) we derive that cl ( ϕ ( t, Γ + (U) )) is a nested sequence of compact and connected sets. Therefore, t 0 cl( ϕ ( t, Γ + (U) )) is connected, which proves Property (iii). If d(ϕ(t, x), ω(u)) 0 as t, then d(ϕ(t n, x), ω(u) δ > 0, for some sequence t n. Since Γ + (U) is precompact, the sequence {ϕ(t n, x)} has a limit point y, with d(y, ω(u)) > 0, which is a contradiction and therefore proves Property (iv). 24

25 Remark 2.24 If X is compact then Γ + τ (U) is precompact for all U X and for all τ 0. Proposition 2.25 Let U, V X, then the omega limit sets satisfy the following list of properties: (i) if V U, then ω(v ) ω(u); (ii) ω(u V ) = ω(u) ω(v ) and ω(u V ) ω(u) ω(v ); (iii) if V ω(u), then ω(v ) ω(u); (iv) ω(u) = ω(cl(u)), i.e. cl(ω(u)) = ω(cl(u)); (v) ω(u) = ω ( ϕ(t, U) ) for all t T. (vi) if there exists a backward orbit γ x U, then x ω(u); Describing the limiting behavior of ϕ as t is more involved since ϕ lacks backward uniqueness as well as continuity for t 0 in general and backward images may be empty. Definition 2.26 Let ϕ : T + X X be a dynamical system. A point y is called a alpha limit point of a set U X under ϕ if there exist times t n and points x n U, y n ϕ(t n, x n ) such that lim n y n = y. The set of all alpha limit points y is called the alpha limit set of U and is denoted by α(u, ϕ). As for omega limit sets we have a similar characterization for alpha limit sets. Proposition 2.27 Let U X. Then α(u) = t 0 cl ( ϕ((, t], U) ) = α ( Γ (U) ). (15) The alpha limit set α(u) is closed, forward invariant, and contained in cl(γ (U)). If U X is backward invariant, then α(u) = t 0 cl ( ϕ(t, U) ), (16) and α(u) cl(u). 25

26 The proof is left as an exercise. Definition 2.28 If γ x is an orbit of ϕ, then the alpha limit set of γ x is α o (γx ) := α ( ) x, ϕ cl(γx) = cl ( γ x ((, t]) ). (17) We emphasize that α o (γ x ) is defined only for (complete) backward orbits γ x. Proposition 2.29 Let γ x be a complete orbit. If γ x is precompact, then α o (γ x ) is non-empty, compact and invariant. t 0 Proposition 2.30 If S is an invariant set, then (i) ω(s) = cl(s) and in particular when S is closed, S = ω(s); (ii) S precompact, implies that cl(s) = ω(s) is a compact invariant set. If S is a strongly invariant, then (iii) α(s) = cl(s) and in particular when S is closed, S = α(s) = ω(s). Proof: Direct consequence of Equations (14) and (16). From Equation (14) it follows that cl(s) = ω(s). By Proposition 2.23, ω(s) and thus cl(s) is a compact invariant set. Lemma 2.31 Let S be forward invariant. Then, cl(s) is a closed forward invariant set. Proof: For all t 0, that ϕ(t, cl(s)) cl(ϕ(t, S)) cl(s), which proves that cl(s) is forward invariant. Proposition 2.32 If S is an invariant set, then (i) ω(s) = cl(s) and in particular when S is closed, S = ω(s); (ii) S precompact, implies that cl(s) = ω(s) is a compact invariant set. If S is a strongly invariant, then (iii) α(s) = cl(s) and in particular when S is closed, S = α(s) = ω(s). Proof: Direct consequence of Equations (14) and (16). From Equation (14) it follows that cl(s) = ω(s). By Proposition 2.23, ω(s) and thus cl(s) is a compact invariant set. 26

27 2.6 Another Reality Check Omega limit sets for the logistic map. Discuss the period doubling bifurcation. Feigenbaum Number. Let r n be the value of r where the n-th period doubling bifurcation occurs. δ = lim n r n r n 1 r n+1 r n = This is a universal number (for a 1-d map under weak hypothesis on derivatives of f one always gets the same number). It implies that there are bifurcations on all scales. Describe the global dynamics during the period doubling bifurcations in terms of graphs. Remark: We need to have a language in which we can talk about these decompositions when we only have a finite degree of precision in our measurements both in phase space and parameter space. Conley provides such a language with the concept of a Morse decomposition 2.7 Equivalences of dynamical systems Clearly the dynamics of the logistic maps changes as we change parameter values. Some changes are not important (exact value of stable fixed point) while other changes are important (period doubling bifurcation). We want to make this precise. For simplicity I will restrict my attention for the moment to discrete dynamical systems. Definition 2.33 Two discrete dynamical systems ϕ : Z + X X and ψ : Z + Y Y are conjugate if there exists a homeomorphism h : X Y such that h ( ϕ(t, x) ) = ψ ( t, h(x) ), (18) for all t Z + and all x X. The system ϕ is semiconjugate to ψ if h is continuous and surjective. Observe that if f(x) = ϕ(1, x) and g(x) = ψ(1, x), then the above relation reduces to h f = g h. Conjugacy and semiconjugacy is captured in the 27

28 following commutative diagram: Z + X id h Z + Y ϕ X h ψ Y Observe that under conjugacy invariant sets of one dynamical system are mapped to invariant sets of the other dynamical system. Consider a parameterized family of dynamical systems, that is a continuous map ϕ: Z + X Λ X where Λ is a connected metric space. Let be defined by ϕ λ (t, x) = ϕ(t, x, λ). ϕ λ : Z + X X Definition 2.34 An element λ 0 Λ is a bifurcation point if for any open neighborhood U Λ of λ 0, there exists λ 1 Λ such that ϕ λ0 is not conjugate to ϕ λ1. Example 2.35 The points in parameter space at which the period doubling occurs are bifurcation point for the logistic map. Fact: There exist parameterized families of dynamical systems where the set of bifurcation points contain Cantor sets of positive measure. This has profound implications for modeling! 28

29 3 Attractor-Repeller Decompositions 3.1 Lattice of Invariant Sets Definition 3.1 A partially ordered set or poset (P, ) is a set P with a binary relation, called a partial order, which satisfies the following axioms: (i) (reflexivity) p p, for all p P, (ii) (anti-symmetry) p q, and q p, then p = q, (iii) (transitivity) if p q, and q r, then p r. A relation < which satisfies only the transitivity property and p p for all p P is called a strict partial order and is denoted by (P, <). Posets and strict poset are equivalent. A partial order defines a strict order via: p < q if and only if p q and p q. Similarly, a strict order < defines a partial order via: p q if and only if p < q, or p = q. The notations (P, ) and (P, <) are used interchangeably to emphasize order or strict order. Example 3.2 Invset(X, ϕ) is a partially ordered set with respect to inclusion. This is denote this by ( Invset(X, ϕ), ). Definition 3.3 Let U X, then the maximal invariant set in U is defined by Inv(U, ϕ) := S S Invset(U,ϕ) Other characterizations of the maximal invariant set include Inv(U, ϕ) = {x U γ x U } = sup {S Invset(X, ϕ) S U }. Definition 3.4 Let P and P be posets. A mapping f : P P is called order preserving if f(p) f(q) for all p q and order reversing if f(p) f(q) for all p q. 29

30 Let P(X) denote the power set of X, that is, the set of all subsets of X. Observe that for a given dynamical system ϕ, the maximal invariant set can be viewed as defining an order-preserving morphism Inv: P(X) Invset(X, ϕ) U Inv(U, ϕ). We can regard the infimum and supremum as binary operations on a poset P if for any two elements p, q P the infimum and supremum exist: p q = sup(p, q), p q = inf(p, q). (19) A poset for which infimum and supremum exist for all pairs p, q P is called a lattice, and introduces an algebraic structure to P. The operation is called vee or join and the operation is called wedge or meet. A lattice can also be defined independently as an agebraic structure. Definition 3.5 A lattice (L,, ) is a set L with the binary operations, : L L L satisfying the following axioms: (i) (idempotent) a a = a a = a for all a L, (ii) (commutative) a b = b a, and a b = b a for all a, b L, (iii) (associative) a (b c) = (a b) c and a (b c) = (a b) c for all a, b, c L, (iv) (absorption) a (a b) = a (a b) = a for all a, b L. A distributive lattice satisfies the additional axiom (v) (distributive) a (b c) = (a b) (a c) and a (b c) = (a b) (a c) for all a, b, c L. A lattice is bounded if there exist neutral elements 0 and 1 with property that (vi) 0 a = 0, 0 a = a, for all a L, and 1 a = a, 1 a = 1, for all a L. A set K L is a sublattice if a b K and a b K for all a, b K. A sublattice that contains 0 and 1 is called a (0, 1)-sublattice. 30

31 Proposition 3.6 The set ( Invset(X, ϕ),, ) with S S = S S, S S = Inv(S S, ϕ), (20) is a bounded distributive lattice. neutral elements. The sets and S = Inv(X, ϕ) are the Remark: S, S Invset(X, ϕ) does not imply that S S Invset(X, ϕ). Lemma 3.7 For any pair S, S Invset (X, ϕ), or S, S Invset + (X, ϕ) Inv(S S ) = Inv(S) Inv(S ) = Inv(S) Inv(S ); Inv(S S ) = Inv ( Inv(S) Inv(S ) ) = Inv(S) Inv(S ). Proof: Let S, S Invset (X, ϕ), or S, S Invset + (X, ϕ), then Inv(S) Inv(S ) Inv(S S ). As for the reversed inclusion we argue as follows. Let x Inv(S S ) and γ x Inv(S S ) a complete orbit. Since S and S are both forward or backward invariant sets it follows that for all y γ x we have that γ y ± S when y S, and γ y ± S, when y S. This implies that y S, or y S for all y γ x, and thus γ x Inv(S), or γ x Inv(S ). Consequently, Inv(S S ) Inv(S) Inv(S ). Combining these two inclusions we obtain Inv(S) Inv(S ) = Inv(S S ). In terms of join-operation : Inv(S S ) = Inv(S) Inv(S ). As for intersections we have that Inv(S) Inv(S ) S S, and therefore Inv ( Inv(S) Inv(S ) ) Inv(S S ). On the other hand Inv(S S ) Inv(S), and Inv(S S ) Inv(S ), which implies Inv(S S ) Inv(S) Inv(S ) and Inv(S S ) Inv ( Inv(S) Inv(S ) ). Combining the inclusions yields Inv(S S ) = Inv ( Inv(S) Inv(S ) ), 31

32 and in terms of the meet-operation : which proves the lemma. Proof: Inv(S S ) = Inv(S) Inv(S ), That Invset(X, ϕ) is a bounded lattice is obvious. To prove distributivity we use the fact that Inv is a lattice homomorphism. For S, S, S Invset(X, ϕ), then sets S S and S S are forward invariant. Then by Lemma 3.7 (S S ) (S S ) = Inv(S S, ϕ) Inv(S S, ϕ) which completes the proof. = Inv((S S ) (S S ), ϕ) = Inv(S (S S ), ϕ) = Inv(S, ϕ) Inv(S S, ϕ) = S (S S ), 3.2 Attractors and repellers From now on we assume that X is a compact, or locally compact metric space. Definition 3.8 Let ϕ : T + X X be a dynamical system. A compact set N X is a trapping region if N is forward invariant and there exists a T > 0 such that ϕ(t, N) int(n). Not an asymptotic definition. Consider a parameterized family of dynamical systems ϕ : T + X Λ X For applications we probably want to choose T = 1. Let f : X Λ X be defined by f λ (x) = ϕ λ (1, x). Discuss robustness of trapping regions with respect to errors and perturbations in parameters! Assme N is a trapping region for f λ0. Then there exists ɛ > 0 such that B ɛ (f λ0 N) N 32

33 There exists δ > 0 such that if λ λ 0 < δ then f λ (x) f λ0 (x) < ɛ. In which case N remains a trapping region. Trapping regions are what are observable with respect to data! Definition 3.9 Let ϕ : T + X X be a dynamical system. A set A X is an attractor if there exists a trapping region N X such that A = Inv(N, ϕ). The set of attractors is denoted by Att(X, ϕ). The concept of a repeller is not a matter of time reversal and therefore requires a slight modification in its definition. Definition 3.10 Let ϕ : T + X X be a dynamical system. A compact set N X is a repelling region if N is backward invariant and there exists a T < 0 such that ϕ(t, N) int(n). Definition 3.11 A set R X is an repeller if there exists a repelling region N X such that R = Inv + (N, ϕ). The set of repellers is denoted by Rep(X, ϕ). Example 3.12 The empty set can serve as a trapping region or as a repelling region, and the corresponding attractor or repeller is. Remark 3.13 Let Y X be a compact forward invariant set. The restriction ϕ Y : T + Y Y, is a well-defined dynamical system. When considering this restricted system, interiors are taken relative to Y in determining trapping and repelling regions, and the corresponding attractors and repellers are denoted by Att(Y, ϕ Y ) and Rep(Y, ϕ Y ). By Proposition 2.23 we have ω(y, ϕ Y ) = Inv(Y, ϕ Y ), which is an attractor in Att(Y, ϕ Y ) since Y is a trapping region. Note that since Y is forward invariant, the attractor of ϕ Y are exactly the attractors of ϕ restricted to Y. Therefore we write Att(Y, ϕ) = Att(Y, ϕ Y ). 33

34 It is important to note that A Att(Y, ϕ) does not necessarily imply that A Att(X, ϕ). Consider the dynamical system ϕ : [0, ) R 2 R 2 defined by the differential equation ẋ = x ẏ = y. Observe that Y = [ 1, 1] {0} is a compact forward invariant set under ϕ and the point (0, 0) Y is an attractor for ϕ Y. However, {(0, 0)} Att(R 2, ϕ). Proposition 3.14 Let ϕ : T + X X be a dynamical system. (i) Attractors are compact, invariant sets. (ii) If N is a trapping region with corresponding attractor A = Inv(N, ϕ), then A = ω(n) and A int(n). If N, then A. Proof: Since attractors are defined as maximal invariant sets of a compact set, Proposition 2.32 implies that they are compact and invariant. Let N be a trapping region with corresponding attractor A = Inv(N, ϕ). If N =, by Proposition 2.23, ω(n) is nonempty and ω(n) = Inv(N, ϕ) = A. Since N is a trapping region, there exists T > 0 such that ϕ(t, N) int(n). By the invariance of A N we have A = ϕ(t, A) ϕ(t, N) int(n) Attracting neighborhoods An attractor is defined as the maximal invariant set inside a trapping region. However, the forward invariance required for a trapping region can sometimes be difficult to establish. It is often useful to have a weaker condition which guarantees that a region contains an attractor. Definition 3.15 Suppose ϕ : T + X X is a dynamical system. compact set N is an attracting neighborhood if ω(n) int(n). A The following proposition, which is of interest in its own right, is used to establish that an attracting neighborhood must contain an attractor. Proposition 3.16 Suppose ϕ : T + X X is a dynamical system. A compact set N is an attracting neighborhood if and only if there exists T > 0 such that ϕ(t, N) int(n) for all t T. 34

35 Proof: Suppose N is an attracting neighborhood so that ω(n) int(n). Fix x N. We claim that there exists δ x > 0 and T x > 0 such that ϕ(t, y) int(n) for all t T x and all y B δx (x). Suppose not. Then we can choose t n and y n x such that ϕ(t n, y n ) / int(n) and ϕ(t n, y n ) z ω(n) int(n). This is a contradiction since N \int(n) is closed. Therefore, by the compactness of N, we can choose T > 0 such that ϕ(t, N) int(n) for all t T. Now suppose there exists T > 0 such that ϕ(t, N) int(n) for all t T. Fix x ω(n). Since ω(n) is invariant by Lemma 2.23, there exists an orbit γ x : T ω(n). Then x = ϕ(t, γ x ( T )) int(n). Thus ω(n) int(n). Theorem 3.17 Let ϕ : T + X X be a dynamical system. If N is an attracting neighborhood, then A = ω(n) is an attractor. In particular, there exists a trapping region N N such that A = ω(n ). Moreover, if N, then A. Proof: Since A is compact by Proposition 2.23, there exists ɛ 0 > 0 such that cl(b ɛ (A)) int(n) for all 0 < ɛ < ɛ 0. Fix 0 < ɛ < ɛ 0. Proposition?? implies that ω(n) = ω(ω(n)) = ω(a) ω(cl(b ɛ (A))) ω(n), which further implies that ω(cl(b ɛ (A))) = A B ɛ (A). Thus cl(b ɛ (A)) is an attracting neighborhood. By Proposition 3.16 there exists T ɛ > 0 such that ϕ(t, cl(b ɛ (A))) B ɛ (A) for all t T ɛ. Define N ɛ = ϕ([0, T ɛ ], cl(b ɛ (A))). By definition N ɛ is forward invariant and compact, but N ɛ may not be a subset of N. We claim that ɛ can be chosen small enough so that N ɛ N. Suppose not and choose ɛ n 0 such that N ɛ n N. Then there exists x n B ɛn (A) and t n T ɛn such that x n x A and ϕ(t n, x n ) / N with ϕ(t n, x n ) z / int(n) since S\int(N) is closed. Passing to a subsequence either t n τ < or t n. In the former case ϕ(τ, x) = z, which contradicts the invariance of A. In the latter case z ω(n) int(n), a contradiction. Choose ɛ > 0 so that N ɛ N. By construction, ( ϕ(t ɛ, N ɛ) = ϕ T ɛ, ϕ ( t, cl(b ɛ (A)) )) = ϕ ( T ɛ, ϕ ( t, cl(b ɛ (A)) )) = t [0,T ɛ] t [0,T ɛ] so that N ɛ is a trapping region, and t [0,T ɛ] ϕ ( T ɛ + t, cl(b ɛ (A)) ) B ɛ (A) int(n ɛ), A = ω(a) ω(n ɛ) ω(n) = A. 35

36 By Proposition 2.23, we have A = Inv(N ɛ) so that A is an attractor and if N, then A. The following corollary gives the characterization that is often used as a definition of attractor in Conley theory. Corollary 3.18 Let ϕ : T + X X be a dynamical system. A set A is an attractor if and only if there exists a precompact neighborhood U of A such that A = ω(u). Corollary 3.19 Let ϕ : T + X X be a dynamical system. A set A is an attractor if and only if there exists ɛ 0 > 0 such that A = ω(b ɛ (A)) for every 0 < ɛ < ɛ 0. Proof: Fix an attracting neighborhood N of A, then by compactness there exists ɛ 0 > 0 such that B ɛ0 (A) N. Then A = ω(a) ω(b ɛ (A)) ω(n) = A. Remark 3.20 As the following exercise demonstrates the existence of a compact neighborhood N of an invariant set A such that for all x N, ω(x) A is not sufficient to imply that A is an attractor. Example 3.21 Consider the flow on R generated by the differential equation ẋ = x x 3. Let A = { 1, 0, 1} and N = [ 1.1, 0.9] [ 0.1, 0.1] [0.9, 1.1]. Observe that N is a compact neighborhood of A such that ω(x) A for all x N, but A is not an attractor Attractor within an attractor If Y is a compact forward invariant set for ϕ : T + X X, then Remark 3.13 points out that A Att(Y, ϕ Y ) = Att(Y, ϕ) does not imply that A Att(X, ϕ) in general. However, the result is true in the special case that Y Att(X, ϕ), as the following theorem indicates. Theorem 3.22 Let ϕ : T + X X be a dynamical system. Att(X, ϕ) and A Att(A, ϕ A ), then A Att(X, ϕ). If A The proof of Theorem 3.22 is based on an alternative characterization of an attractor as an invariant set A which has a neighborhood N with the property that all backward orbits outside of A must leave N. A consequence of this property is that a relative attractor inside an attractor must also be an attractor in the larger system. First we need a few technical results. 36

37 Definition 3.23 A compact set N X is an isolating neighborhood if S = Inv(N) int(n). Lemma 3.24 Let N be an isolating neighborhood. For T > 0 define N + T = {x N ϕ([0, T ], x) N }. Then there exists δ T > 0 such that B δt (S) N + T, i.e. ϕ([0, T ], B δ T (S)) N. Proof: By continuity and the invariance of S, for each x S there exists δ x > 0 such that ϕ([0, T ], B δx (x)) N. By compactness there exists finitely many such balls B δxi (x i ) such that S i B δxi (x i ). Since this union is open, there exists δ > 0 such that B δ (S) i B δxi (x i ) N + T. Lemma 3.25 Let N be an isolating neighborhood. Assume that for all x N \ Inv(N) there are no backward orbits γx : T N. Then for every ɛ > 0 there exists T > 0 such that there are no backward orbit segments γx : [ T, 0] N through x N \ B ɛ (Inv(N)). Proof: Define N T = { x N a backward orbit segment γ x : [ T, 0] N } for T > 0, and let ɛ T := sup {d(x, S)}. x N T First we show that ɛ T 0 as T. Suppose not. Then there exists a sequences T n and x n N with x n x N \ Inv(N) and γx n : [ T n, 0] N. Observe that this implies that there exists a backward orbit γx N, which is a contradiction. Finally choose T > 0 large enough so that ɛ T < ɛ. Proposition 3.26 Let ϕ : T + X X be a dynamical system. An invariant set A is an attractor if and only if there exists a compact neighborhood N of A such that there are no backward orbits γ x : T N through x N \ A. 37

38 Proof: The necessity of the condition is straightforward. Suppose A is an attractor and N is a trapping region so that A = ω(n). If there is a backward orbit γx N with x N \ A, then the complete orbit γ x lies in N, which implies x Inv(N) = A, a contradiction. Now suppose N is a neighborhood with the stated property. Since there are no backward orbits γx : T N with x N \ A and N is a neighborhood of A, we have A = Inv(N) int(n). By Lemma 3.24, there exists δ 1 > 0 such that B δ1 (A) N and ϕ([0, 1], B δ1 (A)) N. Fix ɛ < δ 1 and consider the neighborhood B ɛ (A). By Lemma 3.25, there exists T > 0 such that there are no backward orbit segments γx : [ T, 0] N for x N \ B ɛ (A). Now we construct an attracting neighborhood for A. By Lemma 3.24, there exists δ 2T > 0 such that the neighborhood U = B 2T (A) satisfies ϕ([0, 2T ], U) N. We show that U is an attracting neighborhood. Let V 0 = ϕ(t, U). For each x V 0 there exists a backward orbit segment γx : [ T, 0] N. The definition of T from Lemma 3.25 implies that that V 0 B ɛ (A). Moreover, by our choice of ɛ < δ 1 we have ϕ([0, 1], B ɛ (A)) N so that V 1 = ϕ([0, 1], V 0 ) N. Thus for each x V 1 there exists a backward orbit segment γx : [ T 1, 0] N which implies that γx ([ T, 0]) N so that V 1 B ɛ (A). We can repeat this argument inductively to prove that V k = ϕ([k 1, k], V 0 ) B ɛ (A) N, k > 0. Therefore ϕ([0, ), U) N. Finally, A U implies that A = ω(a) ω(u), and ϕ([0, ), U) N implies that ω(u) Inv(N) = A. Therefore A = ω(u), and A is an attractor by Theorem Proof of Theorem Let N be an attracting neighborhood for A in X. Choose a neighborhood N N of A in X such that N A is an attracting neighborhood for A in A. Suppose that γx is a backward orbit through x N \ A such that γx N. Then γx N so that Proposition 2.25 implies x ω(n) = A. Indeed the entire backward orbit γx is contained in ω(n) = A by a similar argument. Thus γx N A, and again applying Proposition 2.25 gives x ω(n A) = A, a contradiction. The criterion in Proposition 3.26 now yields that A is an attractor for ϕ in X. Corollary 3.27 Let ϕ : T + X X be a dynamical system. If N X is a trapping region and A Att(N, ϕ N ), then A Att(X, ϕ). 38

39 Proof: By definition there exists a trapping region N N for ϕ N, such that A = Inv(N, ϕ N ). Since N is a trapping region for ϕ we set A = Inv(N, ϕ) and since N N, we have that A A. Indeed A = Inv(N A, ϕ N ) = Inv(N A, ϕ A ). Moreover, since N and A are forward invariant, it follows that N A is forward invariant because Invset + (X, ϕ) is a lattice with respect to and. Therefore N A is also forward invariant for ϕ A. Now ϕ(t, N A) ϕ(t, N ) ϕ(t, A) int N (N ) A = int N (N ) int A (A) int N A (N A) = int A (N A), which shows that N A is a trapping region for A with respect to ϕ A. Thus, A Att(A, ϕ A ). From Theorem 3.22 it follows that A Att(X, ϕ). 3.3 Attractor-Repeller pairs Definition 3.28 Let ϕ : T + X X a dynamical system on a compact metric space X. Let A Att(X, ϕ). The dual repeller of A is defined by A := Inv + (X \ N, ϕ) where N X is a trapping region for A. A pair (A, A ) is called an attractorrepeller pair in X. Example 3.29 Let ϕ : T + X X a dynamical system on a compact metric space X. Let S = ω(x), then S Att(X, ϕ). If x X \ S, then ω(x) S. Thus, S =. One of the themes of Conley theory is that it is easier to identify regions than invariant sets. As the following result, which is a restatement of the definition, indicates our characterization of attractor-repeller pairs emphasizes this philosophy. 39

40 Proposition 3.30 Let X be a compact metric space and let N Y be a trapping region for ϕ: T + X X. Then ( Inv(N, ϕ), Inv + (X \ N, ϕ) ) is an attractor-repeller pair in X. We now show that dual repellers and dual attractors are well-defined by providing a characterization of these sets which is independent of the choice of trapping or repelling region. For A X define A := {x X ω(x) A = }. Lemma 3.31 If A is invariant, then A Invset + (X, ϕ). Proposition 3.32 Let ϕ : T + X X be a dynamical system on a compact metric space X. For an attractor A, the dual repeller A is well-defined, compact, and characterized by A = {x X ω(x) A = }. Moreover, if X is invariant, then A is strongly invariant. Proof: Let N be a trapping region for A with A = Inv(X \ N). We first show that A A. If x N, then since N is a trapping region, ω(x) ω(n) = A, and therefore A X \ N. Moreover, A is forward invariant by Lemma 3.31 so that A Inv + (X \ N) = A since Inv + (X \ N) is the maximal forward invariant set in X \ N. To see that A A we argue as follows. Since A X \ N, it holds that cl(a ) cl(x \ N), and cl(a ) is forward invariant by Lemma We now show that cl(a ) X \ N. Indeed, if x cl(a ) N, then since N is a trapping region, ϕ(t, x) int(n) for some T > 0. However, ϕ(t, x) cl(a ) cl(x \ N), which is a contradiction. Therefore A cl(a ) Inv + (X \ N) = A so that A = cl(a ) and hence is compact. If x A, then ω(x) ω(a ) cl(a ) = A X \N, which implies x A. Therefore A A. Combining these inclusions gives that A = A. The following proposition provides a correspondence between the attractorrepeller pairs in a forward invariant set X and the attractor-repeller pairs in S = Inv(X, ϕ). Therefore, to develop the theory of attractor-repeller pairs in a dynamical system, it is sufficient to consider systems defined on invariant sets, i.e. surjective systems. 40