Hartman-Grobman and Stable Manifold Theorem

Home , Stable manifold, Stable manifold theorem

THE HARTMAN-GROBMAN AND STABLE MANIFOLD THEOREM

SLOBODAN N. SIMIC´

This is a summary of some basic results in dynamical systems we discussed in class. Recall that there are two kinds of dynamical systems: discrete and continuous. A discrete dynamical system is map f : M → M of some space M.A continuous dynamical system (or flow) is a collection of maps φt : M → M, with t ∈ R, satisfying φ0(x) = x and φs+t(x) = φs(φt(x)), for all x ∈ M and s, t ∈ R. We call M the phase space. It is usually either a subset of a Euclidean space, a metric space or a smooth manifold (think of surfaces in 3-space). Similarly, f or φt are usually nice maps, e.g., continuous or differentiable. We will focus on smooth (i.e., differentiable any number of times) discrete dynamical systems. Let f : M → M be one such system. Definition 1. The positive or forward orbit of p ∈ M is the set 2 O+(p) = {p, f(p), f (p),...}. If f is invertible, we define the (full) orbit of p by k O(p) = {f (p): k ∈ Z}. We can interpret a point p ∈ M as the state of some physical system at time t = 0 and f k(p) as its state at time t = k. The phase space M is the set of all possible states of the system. The goal of the theory of dynamical systems is to understand the long-term behavior of “most” orbits. That is, what happens to f k(p), as k → ∞, for most initial conditions p ∈ M? The first step towards this goal is to understand orbits which have the simplest possible behavior, namely fixed and periodic ones. Definition 2. A point p ∈ M is called fixed if f(p) = p. It is called periodic if f k(p) = p, for some k ∈ N. The smallest such k is called the period of p. Thus if p is periodic with period k, then O(p) = {p, f(p), . . . , f k−1(p)}. Example 1. Let M be the unit circle S1, thought of as the set of all complex numbers with absolute value one, and let f : S1 → S1 be the rotation by α radians, for some real α. That is, f(z) = e2παiz. It is possible to show that there is a dichotomy: if α is a rational number, then every orbit is periodic. If α is irrational, then every orbit is dense in S1. This means that for every z ∈ S1, O(z) is a dense subset of S1, i.e., for each w ∈ S1 and ε > 0 there exists k ∈ Z such that the distance between w and f k(z) is less then ε.

Date: February 14, 2007. 1 2 S. N. SIMIC´

As is usually the case, linear systems are easier to understand than the nonlinear ones. A dynamical system f defined on some open set in Rn is called linear if f(x) = Ax, for some n × n matrix A. Example 2. Consider 2 0 A = : 2 → 2. 0 1/2 R R The eigenvalues of A are 2 and 1/2; the corresponding eigenspaces are the x-axis and the y-axis, respectively. If v lies on the x-axis, then Akv → (0, 0), as k → −∞ (or Amv → ∞, as m → ∞), and if v is on the y-axis, then Akv → (0, 0), as k → ∞. Accordingly, we call the x-axis the unstable space of A and denote it by Eu; the y-axis is called the stable space of A and is denoted by Es. If v does not lie on any of the axes, then the positive orbit of v converges to Eu but is repelled from the origin in the horizontal direction. What if a nonlinear map f is a small perturbation of A? That is, what if f(x, y) = A(x, y)T + higher order terms? Is the long-term behavior of f in any way similar to the long-term behavior of A? We will soon see that the answer is yes, thanks to the fact that A is what is called a hyperbolic matrix. But first we have to define what we mean by the term “similar” or qualitatively the same. Definition 3. Dynamical systems f : M → M and g : N → N are called topologically conjugate if there exists a homeomorphism h : M → N such that f = h−1 ◦ g ◦ h. In this case, h is called a topological conjugacy between f and g. Recall that a homeomorphism is an invertible map that is continuous together with its inverse. We can think of a homeomorphism as a continuous change of coordinates. If h : M → N is a homeomorphism, then from the point of view of topology, M and N are the same, or homeomorphic. If a homeomorphism is differentiable together with its inverse, it is called a diffeomorphism. Example 3. Any two closed intervals in R are diffeomorphic. Any open interval in R is diffeomorphic to R. The torus (the surface of a doughnut) is diffeomorphic to the surface of a coffee cup (hence topologists can’t distinguish between the two). The unit circle is not homeomorphic to the interval [0, 2π) (or any interval for that matter). The unit sphere is not homeomorphic to the torus. Suppose that f and g are topologically conjugate via a conjugacy h, as above, and let p ∈ M be arbitrary. Then it is not hard to check that h(f k(p)) = gk(h(p)). Therefore, h takes the f-orbit of p to the g-orbit of h(p). If p is fixed or periodic, then so is h(p) (and it has the same period as p). In other words, after a change of coordinates given by h, orbits of f become orbits of g, and every dynamical property of f translates into a property of g: f and g are dynamically the same. Intuitively the orbits of f and g have the same “fate”. THE HARTMAN-GROBMAN AND STABLE MANIFOLD THEOREM 3

Example 4. Let M = N = R and assume f(x) = x/3 and g(y) = y/2. It is easy to see that the origin is a fixed point of both f and g and all orbits converge to it under forward iteration of f and g. Therefore, at least intuitively, f and g should be topologically conjugate. Let us look for a topological conjugacy of the form h(x) = xα, for some α > 0. To find α, we need to consider the equation f(x) = h−1(g(h(x))), which is equivalent to x xα 1/α = . 3 2 We obtain 3 = 21/α, which yields α = log 2/ log 3. Observe that h is only continuous; it is not differentiable at 0, since α < 1. In fact, there exists no differentiable conjugacy between f and g. (Suppose that there were one and differentiate h ◦ f = g ◦ h at zero.) Definition 4. We call p a hyperbolic fixed point of a dynamical system f if f(p) = p and the derivative Df(p) is a hyperbolic matrix. That is, no eigenvalue of Df(p) has absolute value one. The linear map A = Df(p) is called the linearization of f at p. The matrix in Example 2 is hyperbolic. Observe that it is allowed for all eigenvalues to be less then one in absolute value or to be greater than one in absolute value. In the former case we have the following result. Theorem 1. Suppose p is a fixed point of a f and every eigenvalue λ of Df(p) satisfies |λ| < 1. Then p is asymptotically stable, i.e., there exists a neighborhood U of p such that for every q ∈ U, f k(q) → p, as k → ∞. Proof. Since the norm of A = Df(p) equals max |λ|, where λ runs over all eigenvalues of A, it follows that kAk < 1. Choose µ such that kAk < µ < 1. Recall that we are assuming f is smooth. In particular, Df(x) is a continuous function of x, so there exists an open ball U around p such that for all q ∈ U, kDf(q)k ≤ µ. Let q ∈ U be arbitrary. By the Mean Value Theorem, |f(q) − f(p)| ≤ max kDf(x)k |q − p| , x∈[p,q] where [p, q] is the straight line segment from p to q. Since maxx∈[p,q] kDf(x)k ≤ µ and f(p) = p, it follows that |f(q) − f(p)| ≤ µ |q − p| . In particular, f(q) ∈ U. By induction we obtain

k k k f (q) − f (p) ≤ µ |q − p| , which converges to zero, as k → ∞, since 0 < µ < 1. We can now state 4 S. N. SIMIC´

Hartman-Grobman Theorem. Suppose p is a hyperbolic ﬁxed point of a smooth map f : Rn → Rn. Then there exists a neighborhood U of p and a homeomorphism h : U → Rn such that −1 f U = h ◦ A ◦ h, where A = Df(p). In other words, in a neighborhood of p, f is topologically conjugate to its linearization.

Remark. Here f U denotes the restriction of f to U. In general, there exists no diﬀerentiable conjugacy between f U and A. The theorem says that near a hyperbolic ﬁxed point, the dynamical behavior of a nonlinear system is qualitatively the same as the behavior of its linearization near the origin.

Example 5. Consider the map f : R2 → R2 defined by 1 1 f(x, y) = (2x + x2 − xy, y − xy). 2 2 Then f(0, 0) = (0, 0) and Df(0, 0) = A from Example 2, so 0 = (0, 0) is a hyperbolic fixed point of f. By the Hartman-Grobman theorem, in a neighborhood of the origin, f is topologically conjugate to A via a homeomorphism h. Define W u(0) = h−1(Eu) and W s(0) = h−1(Es), where as before, Eu and Es are the unstable and stable space for A. Since h−1 takes orbits of A to orbits of f, it follows that for every p ∈ W u(0), f k(p) → 0, as k → −∞ and for every q ∈ W s(0), f k(q) → 0, ask → +∞. It is not hard to see that both W u(0) and W s(0) are invariant with respect to f; that is, f(W σ(0)) ⊂ W σ(0), where σ ∈ {u, s}. Since h−1 is only known to be continuous, all we can say about W u(0) and W s(0) is that they are continuous curves passing through the origin. Let us try to compute W u(0) explicitly. We will try to express W u(0) as the graph of some real valued function u defined on a neighborhood J of zero. That is, we want W u(0) = {(x, u(x)) : x ∈ J}. We now want to use invariance of W u(0) under f to compute u(x). Let us write f(p) = u u (f1(p), f2(p)). Then f(W (0)) ⊂ W (0) amounts to (check this!)

f2(x, u(x)) = u(f1(x, u(x))). This is equivalent to 1 1 u(x) − xu(x) = u 2x − x2 − xu(x) . 2 2 We now have to solve this functional equation for u. These types of equations are in general very hard to solve. However, this particular equation has a solution u(x) = x2. Therefore, W u(0) is an arc of the parabola y = x2. This means that the unstable manifold is smooth. This is not a coincidence. See the Stable Manifold Theorem below. In general, for a ﬁxed point p of f we can always deﬁne: THE HARTMAN-GROBMAN AND STABLE MANIFOLD THEOREM 5

Definition 5. The stable set of p is W s(p) = {q : lim f k(q) = p}. k→+∞ The unstable set of p is W u(p) = {q : lim f k(q) = p}. k→−∞ In other words, the stable set of p is the set of all points whose positive orbits converge to p. The unstable set of p is the set of all points whose negative (or backward) orbits converge to p. In a similar way, we can define the stable and unstable manifolds of a periodic point. These sets are clearly very important for the understanding of the dynamics of f. A priori, these sets may be very irregular, but we saw in the previous example that for a hyperbolic fixed point, they are at least continuous curves. In fact, more can be said: Stable Manifold Theorem. The stable and unstable set of a hyperbolic fixed (or periodic) point p of a map f are smooth manifolds and are accordingly called stable and unstable manifold. They intersect at a nonzero angle p and are invariant with respect to f The tangent space to the stable manifold at p equals the stable space of the linearization Df(p). Similarly for the unstable manifold. Don’t worry if you don’t know what a smooth manifold is. Basically, it is a space which locally looks like an open subset of some Euclidean space (think of a torus or a sphere, which can both be covered by charts that look like open subsets of R2). The statement “The tangent space to the stable manifold at p equals the stable space of the linearization Df(p).” means that we think of the phase space of the linearization A as being tangent to the phase space of f at p. (To fully understand this, one needs to the concept of a differentiable manifold, which you can learn about in a course such as Math 213, which I may teach in the Fall 2007). Now try to imagine the following picture. Somewhere far away from a hyperbolic fixed point p, W s(p) and W u(p) cross again, at some point q. Such a point is called a (transverse) homoclinic point. Since q ∈ W s(p), we have f k(q) → p, as k → +∞. But q also lies on W u(p), so f k(q) → p, as k → −∞. In other words, in both positive and negative time, the orbit of q converges to p. This means that W s(p) and W u(p) actually cross at a whole sequence of u points qk → p and that W (p) accumulates on itself. Try to draw this picture! It is clearly very complicated and is an indication of chaotic behavior. In fact, there is a result (called the Birkhoff-Smale theorem) that says that the existence of a transverse homoclinic point implies chaos. To be continued...

Department of Mathematics, San Jose´ State University, San Jose,´ CA 95192-0103 E-mail address: [email protected]