Hamilton College Dynamics Final Paper

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AN EXPLORATION OF JOHN BANKS’ ”ON DEVANEY’S DEFINITION OF CHAOS” MIKE HOWARD

1. Introduction In this paper we will simplify the criteria required for a dynamical system to be classified as chaotic. Module 7 presented Devaney’s definition of chaos which requires that a dynamical system meet three properties in order to be considered a chaotic dynamical system. Devaney’s Definition of Chaos Let X be a metric space. A continuous map f : X → X is said to be chaotic on X if •f is transitive, • the periodic points of f are dense in X, •f has sensitive dependence on initial conditions. We will show that only two of the three criteria listed above are required in order for f to be chaotic on X. Using John Banks’ paper On Devaney’s Definition of Chaos, as a model, we will prove the following, Theorem If f : X → X is transitive and has dense periodic points then f has sensitive dependence on initial conditions. Thus, Banks claims that a system need only be transitive and have dense periodic points in order to be classified a chaotic dynamical system. Before proving this claim, it is interesting to highlight a topological aspect of Banks’ argument. Banks observes that dense periodic points and transitivity are two topological conditions while sensitive dependence is not topological. It is intriguing to note that two topological properties of a system, when placed together, imply a third, non-topological property of a dynamical system. 2. Helpful Notes In order to prove Banks’ claim, we will employ the following: • Let d(xo , x1 ) denote the distance between two points, xo , x1 ∈ X. 1

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• An open ball of radius r on the metric space X is defined as follows, Br (p) ≥ {x ∈ X|d(x, p) < r}. This ball about a point x is analogous to a higher dimensional neighborhood about a point x. • Let b ∈ X be a periodic point of period m. Let the distance between a point x ∈ X and the orbit of b, {b, f (b), ..., f m−1 (b)} = O(b), be min({d(b, x), d(f (b), x) . . . d(f m−1 (b), x)}). • Recall the triangle inequality, which states that, for x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z). 3. Proving Banks’ Claim Suppose that f : X → X is transitive and has dense periodic points. We will prove that f has sensitive dependence on initial conditions with sensitivity constant δ. Let x ∈ X and let 0 < . Let N be an -neighborhood of x and let Bδ (x) denote a ball of radius δ centered at x. Since the periodic points of f are dense, there exists a periodic point, p, in the intersection U = N ∊ Bδ (x). Let n denote the period of p. We will now show that there exists a periodic point q ∈ X whose orbit, O(q) is of distance greater than or equal to 4δ from x. Proof. Since periodic points of f are dense in X, f has infinitely many periodic points in X. Note that the period of all of these periodic points is finite. Thus, we may choose a point, q, of some finite period, m ∈ N, such that for all i ∈ [0, m − 1], d(x, f i (q)) > 4δ. Set V =

n T

f −i (Bδ (f i (q))). Note that V is open since it is the finite intersection of open

i=0 f i (q)

sets. Since ∈ Bδ (f i (q)), f −i (f i (q)) = q ∈ f −i (Bδ (f i (q))) for all i. Hence q ∈ V and V is nonempty. Similarly, note that U is open since it is a finite intersection of open sets. Also, recall that U is nonempty since p ∈ U . Since f is transitive, there exists a k ∈ N such that f k (U ) ∊ W 6= {0}. Hence, there exists y ∈ U such that f k (y) ∈ V . Now, let j be the integer part of nk + 1. Thus, there exist p, q ∈ Z with q 6= 0, such that, p p k k k n + 1 = j + q . Observe that since q ≼ 0, n + 1 ≼ j. It follows that, 1 ≼ j − n . Multiplying both sides by n > 0 yields, n ≼ nj − k. Since p is a periodic point of peroid n and j ∈ Z, f nj (p) = p. By the triangle inequality, d(x, f nj−k (q)) ≤d(x, f nj (p)) + d(f nj (p), f nj−k (q)) ≤d(x, f nj (p)) + d(f nj (p), f nj (y)) + d(f nj (y), f nj−k (q)) =d(x, p) + d(p, f nj (y)) + d(f nj (y), f nj−k (q)). Consequently, (1)

d(p, f nj (y)) ≼ d(x, f nj−k (q)) − d(f nj (y), f nj−k (q)) − d(x, p).

AN EXPLORATION OF JOHN BANKS’ ”ON DEVANEY’S DEFINITION OF CHAOS”

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Note that we have already determined some of these distances. Recall that since p ∈ U , p ∈ Bδ (x). Thus d(x, p) < δ. Further, recall that O(q) is of distance at least 4δ from x, thus d(x, f nj−k (q)) > 4δ. We now claim that d(f nj (y), f nj−k (q)) < δ. Proof. Observe that f nj (y) = f nj−k (f k (y)). Since f k (y) ∈ V , f nj (y) ∈ f nj−k (V ) = n T f nj−k ( f −i (Bδ (f i (q)))). i=0

We will now show that for any two sets A ∩ B, f (A ∩ B) ⊆ f (A) ∩ f (B) ⊆ f (A), f (B). Let z ∈ f (A ∩ B), then there exists an w ∈ A ∩ B such that z = f (w). So w ∈ A and w ∈ B. Thus z = f (w) ∈ f (A) and z = f (w) ∈ f (B). Hence z ∈ f (A) ∩ f (B). Further, note that f (A) ∩ f (B) ⊆ f (A), f (B) Since 1 ≤ nj − k ≤ n, by expanding V and distributing f using the above fact f (A ∩ B) ⊆ f (A) ∩ f (B) ⊆ f (A), f (B), it follows that, n T f nj−k ( f −i (Bδ (f i (q)))) ⊆ f nj−k (Bδ (q))∩f nj−k (f −1 Bδ (f (q)))∩· · ·∩f nj−k (f −(nj−k) Bδ (f nj−k (q)))∩ i=0

· · · ∩ f nj−k (f −n Bδ (f n (q))) ⊆ f nj−k (f −(nj−k) Bδ (f nj−k (q))) = Bδ (f nj−k (q))). Hence f nj (y) ∈ Bδ (f nj−k (q))). So d(f nj (y), f nj−k (q)) < δ as desired.

Substituting the known distances into (1) yields, (2)

d(p, f nj (y)) > 4δ − δ − δ = 2δ.

Equivalently, since f nj (p) = p, (2) becomes d(f nj (p), f nj (y)) > 2δ. By the triangle inequality, 2δ < d(f nj (p), f nj (y)) ≤ d(f nj (p), f nj (x)) + d(f nj (x), f nj (y)). There exist three possibilites: • f nj (x) is at least δ away from both points, f nj (p) and f nj (y). Then there exists a point p arbitrarily close to x such that d(f nj (x), f nj (p)) > δ. Or, • f nj (x) is at least δ away from only f nj (p). Then there exists a point p arbitrarily close to x such that d(f nj (x), f nj (p)) > δ. Or, • f nj (x) is at least δ away from only f nj (y). Then there exists a point y arbitrarily close to x such that d(f nj (x), f nj (y)) > δ. In all three cases, f has sensitive dependence on initial conditions with sensitivity constant δ. 4. Conclusion Banks demonstrated that dense periodic points and transitivity necessarily imply sensitive dependence on initial conditions. Thus a dynamical system need only be shown to be transitive and have dense periodic points in order to be classified as a chaotic dynamical system using Devaney’s definition. This is exciting because it significantly reduces the number of criteria that one needs to check to classify a dynamical system as chaotic. The following quote from Banks’ paper highlights another interesting aspect of chaos, ”[Sensitive dependence on initial conditions] captures the idea that in chaotic systems minute errors in experimental readings eventually lead to large scale divergence. Sensitive

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dependence on initial conditions is thus widely understood as being the central idea in chaos.” It is fascinating that, when combined, the two topological, ’less central’ properties of Devaney’s definition do in fact imply that a system is chaotic and also imply the third, nontopological, condition that people view as the defining characteristic of chaotic dynamical system.

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