11 Chaos in Continuous Dynamical Systems. 47

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11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 47

11 Chaos in Continuous Dynamical Systems.

Let’s consider a system of diﬀerential equations given by

˙x= f(x),

where x(t):R → R and f : R → R .

The linearization of the system near x0 is given by

˙x= Ax,

where A = Df(x0).

A point x0 ∈ R is an equilibrium point if f(x0) = 0, and an equilibrium point is called hyperbolic if none of the eigenvalues of Df(x0) have zero real part.

x2 − y2 − 1 Example. Let f(x) = . Then there are two equilibrium points,

2y ¡ ¡ 2x −2y at [1, 0] and [−1, 0] . The derivative matrix is Df(x) = , so 0 2 2 0 −2 0 Df(1, 0) = and Df(−1, 0) = . Both equilibrium points are 0 2 0 2 hyperbolic. In fact, (1, 0) is a source, while (−1, 0) is a saddle.

The Stable Manifold Theorem.

Let E be an open subset of R containing the origin, let f ∈ C1(E), and let φ(t, x) be the solution curve to the system, or ﬂow, through x, as guaranteed by the existance and uniqueness theorem. Suppose the system has a hyperbolic equilibrium point at the origin, so that f(0) = 0. If Df(0) has k eigenvalues with negative real part, then there exists a k-dimensional diﬀerentiable stable manifold

S such that φ ¢(S) ⊂ S for all t ≥ 0, and ¢

lim£φ (x0) = 0,

¤ ¢ for all x0 ∈ S.

Similarly, if Df(0) has n − k eigenvalues with positive real part, then there

exists a n − k dimensional diﬀerentiable unstable manifold U such that φ ¢(U) ⊂ U

for all t ≤ 0, and

£lim φ (x0) = 0,

¥ ¢

for all x0 ∈ U. 11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 48

If f ∈ C1(E), then the existence and uniqueness theorem guarantees there will be a continuous solution curve φ(t, x0) to the system of diﬀerential equations

through any initial condition x0. In general, the function φ ¢x) is called the flow of the system of differential equations. This flow will be the dynamical system that is the focus of our study. It satisfies the following two properties:

• φ0(x) = x for all x ∈ E.

¦¢¦ • φ ¢◦ φ (x) = φ + (x) for all s, t ∈ R and x ∈ E.

If we ﬁx an initial condition x0, then the ﬂow φ(t, x0):R → E is called the orbit through the point x0. We can identify this orbit as a curve or trajectory

Γ §0 = {x ∈ E|x = φ(t, x0), t ∈ R}.

A cycle or periodic orbit is any closed solution curve which is not an equilibrium point. The minimal value T for which φ(t + T,x0) = φ(t, x0) is called the period of the periodic orbit.

Example. Consider the following system:

x˙ = y y˙ = x + x2.

The system has two equilibrium points at (-1,0) and at the origin. (-1,0) is a center (and thus not hyperbolic), while the origin is a saddle (and thus hyperbolic). Any initial condition on the x-axis between -1 and 0 will lie on a periodic orbit circling around the center at (-1,0). The solution curves are deﬁned by 2 y2 − x2 − x3 = C. 3 The solution curve corresponding to C = 0 goes through the point (-3/2,0) and also the origin. In fact, both the unstable manifold and the stable manifold for the origin lie on this one solution curve. Such a solution curve is called a homoclinic orbit. 11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 49

11.1 Strange Attractors. The Lorenz System and the H´enon Map. A closed invariant set A ⊂ E is an attracting set of the system of diﬀerential equations if there is some neighborhood U of A such that for all x ∈ U,

• φ ¢(x) ∈ U for all t ≥ 0, and

• φ ¢(x) → A as t → ∞.

An attractor is an attracting set which contains a dense orbit.

Example. Consider the system

x˙ = −y + x(1 − x2 − y2) y˙ = x + y(1 − x2 − y2).

In polar coordinates the system is

r˙ = r(1 − r2) θ˙ = 1.

This system has the unit circle as an attractor. Orbits inside the unit circle spiral away from the origin towards the unit circle, and orbits outside of the unit circle spiral in, accumulating on the unit circle. The unit circle itself is a periodic orbit for the system. 11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 50

The Lorenz System.

Lorenz in 1963 introduced the following system in the development of exper- iments to predict the weather. The sytem is as follows:

x˙ = σ(x + y) y˙ = ρx − y − xz z˙ = −βz + xy

The system has two negative and one positive eigenvalues at the origin. For certain values of the parameters, the system exhibits sensitive dependence on initial conditions near the orgin, and has a very ”strange” attractor. The following is the projection in the x, z plane of a single orbit as it accumulates on the attractor. The parameter values are σ = 10, ρ = 28, and β = 8/3.

This sensitive dependence lead Lorenz to conclude that no matter how precise your data, you cannot predict the long-term behavior of the weather... the famous ”Butterﬂy Eﬀect”.

The actual attractor A consists of an inﬁnite number of branched surfaces which interweave and intersect, but a single orbit does not intersect itself, instead moving from one branch to another. The attractor A is called a strange attractor, meaning it has a countable number of periodic orbits of arbitrarily large period, and an uncountable number of non-periodic orbits, as well as a dense orbit.

There is an excellent applet on the web that shows two orbits with very close initial conditions at http://www.cmp.caltech.edu/∼mcc/Chaos Course/Lesson1/Demo8.html 11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 51

Another famous example of a strange attractor is the attractor exibited with certain parameter values of the H´enon Map, which is a discrete two-dimensional dynamical system given by:

x 1 − ax2 + y φ = . y bx

This map was proposed by Hénon in 1976 as a simplified model of the Poincaré map of the Lorenz system. A single orbit with parameter values a = 1.4 and b = 0.3 is shown below. The attractor of the Hénon Map consists of a string that loops and folds onto itself infinitely often.

A web applet that allows you to zoom in on the layers of the attractor is located at: http://www.sekine-lab.ei.tuat.ac.jp/∼kanamaru/Chaos/e/Henon/