VIGRE Nonlinear Dynamics Group Lectures

Week of June 12th, 2000 Notes prepared by Smita Sihag

   These lectures follow the general outline presented in Strogatz,

NonLinear Dynamics and Chaos

1 Failure of Perturbation Theory in Solving the Duffing Equation (Andrew Rudman)

1.1 Weakly Nonlinear Oscillators

Small perturbations of the linear oscillator ẋ + x = 0 are given by εh(x, ẋ) in equations of the form

ẍ + x + εh(x, ẋ) = 0, 0 ≤ ε ≪ 1  (1.1)

As a result, (1.1) describes systems termed weakly nonlinear oscillators. One example is the Duffing Equation:

ẍ + x + εx^{3} = 0  (1.2)

1.2 Regular Perturbation Theory Method

Let x(t, ε) be a solution where

x(t, ε) = x_{0}(t) + εx_{1}(t) + ε^{2}x_{2}(t) + ...  (1.3)

Here, we assume that the solution is a power series in ε, and hope to ignore terms of order ε^{2} and higher.

1.2.1 Example 1

Consider the weakly damped linear oscillator

ẍ + 2εẋ + x = 0  (1.4)

with initial conditions x(0) = 0 and ẋ(0) = 1. The exact solution of this second order differential equation is

x(t, ε) = (1 − ε)^{−1/2}e^{−εt} sin(1 − ε^{2})^{1/2}t  (1.5)

To solve (1.4) using Perturbation Theory, we want to plug (1.3) into (1.4). This yields

\frac{d^{2}}{dt^{2}} (x_{0} + εx_{1} + ...) + 2ε\frac{d}{dt}(x_{0} + εx_{1} + ...) + (x_{0} + εx_{1} + ...) = 0

Next, we group the relevant terms according to powers of ε like so

(ẍ_{0} + x_{0}) + ε(ẍ_{1} + 2ẋ_{0} + x_{1}) + O(ε^{2}) = 0

Since ε can take on any value much less than 1, the coefficients of each power of ε go to zero individually as follows

ẍ_{0} + x_{0} = 0  (1.6)
ẍ_{1} + 2ẋ_{0} + x_{1} = 0  (1.7)

(1.6) and (1.7) are just ordinary differential equations. To find x_{0} and x_{1} exactly, we now must consider the initial conditions. The first condition x(0) = 0 implies that both x_{0}(0) = 0 and x_{1}(0) = 0, while the second condition ẋ(0) = 1 implies that ẋ_{0}(0) = 1 and ẋ_{1}(0) = 0. Solving for x_{0} and x_{1}, we get

x_{0}(t) = sin t  (1.8)
x_{1}(t) = −t sin t  (1.9)

The final solution is therefore

x(t, ε) = sin t − εt sin t + O(ε^{2})  (1.10)

Notice that there is a secular term in the solution.

Definition 1.1. A secular term is a term that grows without bound as time t goes to infinity, i.e. t sin t.

The existence of secular terms eventually cause Perturbation Theory to break down. Refer to Mathematica hand-outs to see graphs comparing the Exact Solution and the Perturbation Solution for various time-scales. Figure 1.2 shows that both solutions match up almost perfectly from t = 0 to t = 10. However, for t > 30 the Perturbation Solution appears to be oscillating heavily, thus becoming more and more unstable while the Exact Solution seems to be stabilizing around the t-axis, as expected for a damped oscillator.

1.3 Does Perturbation Work for the Duffing Equation?

Let’s try to apply the above method to the Duffing Equation (1.2). Again, plugging (1.3) into (1.2) gives

\frac{d^{2}}{dt^{2}} (x_{0} + εx_{1} + ...) + (x_{0} + εx_{1} + ...) + ε(x_{0} + εx_{1} + ...)^{3} = 0

Now, we group the terms while ignoring all ε^{2} and higher order terms to get

(ẍ_{0} + x_{0}) + ε(ẍ_{1} + x_{1} + x_{0}^{3}) + O(ε^{2}) = 0

Letting each coefficient go to zero yields

ẍ_{0} + x_{0} = 0  (1.11)
ẍ_{1} + x_{1} + x_{0}^{3} = 0  (1.12)

The initial conditions here are the same as before: x(0) = 0 and ẋ = 0. Therefore, as before, the solution to (1.11) is

x_{0}(t) = sin t  (1.13)

We then use (1.13) to solve (1.12) like so

ẍ_{1} + x_{1} = − sin^{3} t  (1.14)

Looking at (1.14), we expect a secular term to emerge as the solution of x_{1}(t). This secular term causes Perturbation Theory to fail in solving the Duffing Equation. Again, refer to the Mathematica hand-outs to observe the huge discrepancy between the Exact Solution and the Perturbation Theory Solution. Figure 2.3 especially depicts just how different the two solutions are from t = 0 to t = 30.

1.4 Two-Timing

All of the solutions above clearly exhibit two time scales–one fast, and one slow. The method of two-timing takes this fact into consideration and offers much better approximations than Regular Perturbation Theory. First, let τ = t and T = εt denote fast time and slow time respectively. Treating these two times as independent variables, the general solution expands as follows

x(t, ε) = x_{0}(τ, T) + εx_{1}(τ, T) + O(ε^{2})  (1.15)

Secondly, the chain rule is needed to transform the time derivatives in terms of both τ and T:

ẋ = \frac{dx}{dt} = \frac{∂x}{∂τ} + \frac{∂x}{∂T} \frac{∂T}{∂t} = \frac{∂x}{∂τ} + ε\frac{∂x}{∂T}  (1.16)

This is two-timing in a nutshell.

2 The Torus Phase Space and The Uncoupled System (Kareen Rozen)

2.1 Intro to Coupled Oscillators

For systems of the following form:

\begin{matrix}Θ̇_{1} = f_{1}(Θ_{1}, Θ_{2}) \\ Θ̇_{2} = f_{2}(Θ_{1}, Θ_{2}) \end{matrix}  (2.1)

the natural phase space is a torus. Note that f_{1} and f_{2} are periodic in both Θ_{1} and Θ_{2}. A model of coupled oscillators is given by

Θ̇_{1} = ω_{1} + K_{1} ∗ sin(Θ_{2} − Θ_{1})

Θ̇_{2} = ω_{2} + K_{2} ∗ sin(Θ_{1} − Θ_{2}) (2.2)

Here Θ_{1}, Θ_{2} are the phases, while ω_{1}, ω_{2} > 0 are the natural frequencies, and K_{1}, K_{2} ≥ 0 are the coupling constants. One good way of thinking about (2.2) is by using the two runners analogy. If two people are running around a circular track, Θ_{1}(t), Θ_{2}(t) represent their respective positions while ω_{1}, ω_{2} are related to their desired running speeds. If they are not friends, the system is uncoupled, so the faster runner will periodically lap the slower runner. If they are friends, the system is coupled, so they need to agree on one running speed in order to achieve phase-locking.

2.2 The Torus Phase Space

Setting up a coordinate system with longitude and latitude coordinates Θ_{1} and Θ_{2} allows us to depict the flow in torus phase space. But, accurately drawing the trajectories onto the curvature of the torus is too difficult. As a result, a much simpler way to draw such phase portraits involves a square with periodic boundary conditions of 0 ≤ Θ_{1}, Θ_{2} ≤ 2π. One easy rule applies: trajectories that run off one edge of the square directly continue around to the other edge.

2.3 The Uncoupled System

This is the trivial case where K_{1}, K_{2} = 0, thus decoupling the system. Equation (2.2) therefore becomes

\begin{matrix}Θ̇_{1} = ω_{1} \\ Θ̇_{2} = ω_{2} \end{matrix}  (2.3)

The trajectories on the square phase diagram are just straight lines with slope

\frac{dΘ_{2}}{dΘ_{1}} = \frac{ω_{2}}{ω_{1}}

If the slope is rational: then ω_{1}/ω_{2} = p/q for some integers p, q with no common factors. All trajectories are closed orbits on the torus. For the values p = 3 and q = 2, the trajectory yields a trefoil knot when plotted on the torus. If the slope is irrational: then the flow is quasiperiodic. In other words, trajectories neither close nor self-intersect.

3 The Coupled System and A Brief Overview of Chaos (Martin Anderson)

3.1 The Coupled System

Θ̇_{1} = ω_{1} + K_{1} ∗ sin(Θ_{2} − Θ_{1})

Θ̇_{2} = ω_{2} + K_{2} ∗ sin(Θ_{1} − Θ_{2}) (3.1)

In the coupled case, K_{1}, K_{2} > 0. Let the phase difference ϕ = Θ_{1} − Θ_{2}. So

ϕ̇ = Θ̇_{2} − Θ̇_{1} = ω_{2} − ω_{1} + (K_{2} + K_{1}) sin(ϕ)  (3.2)

Suppose there exists two fixed points:

sin ϕ∗ = \frac{ω_{1} − ω_{2}}{K_{1} + K_{2}}  (3.3)

Then the compromise frequency is given by

ω∗ = Θ̇_{1} = Θ̇_{2} = ω_{2} + K_{2} sin ϕ∗ = \frac{K_{1}ω_{2} + K_{2}ω_{1}}{K_{1} + K_{2}}  (3.4)

Basically, this is simply the weighted average of the two frequencies. Refer to Strogatz, p.278 for details on the phase portrait.

3.2 Intro to Chaos

The Lorenz equations are

\begin{matrix}ẋ = σ(y − x) \\ ẏ = rx − y − xz \\ ż = xy − bz \end{matrix}  (3.5)

In this 3-dimensional system, originally designed to model weather patterns, σ, r, b > 0 are parameters. In particular, there are some special properties of this system worth examining. First is the symmetry. If (x, y) goes to (−x, −y), the system remains unchanged. Second is the fact that the volume in phase space shrinks exponentially fast. Refer to Strogatz p.313 for an active proof of this idea. But, in short, contraction of volume essentially implies that all fixed points in the system will be attracting.

3.2.1 Defining Chaos

Although there is no universally agreed upon definition of chaos, there are three basic elements to the concept.

Definition 3.1. A chaotic system is deterministic, and exhibits aperiodic longterm behavior as well as sensitive dependence on initial conditions.