Phase plane

A system that does not explicitly contain dependence on the independent variable (say t) is called autonomous. An example is  

$${dX\over dt} =f(X), X\in R^n.$$ (19)

Note that a non-autonomous system can be recast as an autonomous system at the expense of adding another dimension. An example is

$$X=\pmatrix{x_1\cr x_2},$$

$${dx_1(t)\over dt}=f_1(x_1,x_2,t),$$

$${dx_2(t)\over dt}=f_2(x_1,x_2,t).$$

This is not autonomous in ${\rm I\!R}^2$. Define x₃(t)=t. Then

$$X=\pmatrix{x_1\cr x_2\cr x_3},$$

$${dx_1(t)\over dt}=f_1(x_1,x_2,x_3),$$

$${dx_2(t)\over dt}=f_2(x_1,x_2,x_3),$$

$${dx_3(t)\over dt}=1,$$

which is autonomous in ${\rm I\!R}^3$.

Definition 1

 An equilibrium solution or critical point of the autonomous system (1.21) is a constant vector $\xi$ such that $f(\xi)=0$.

We are interested in stability properties of critical points; i.e., if we perturb the solution slightly from an equilibrium position, does it stay close or wander away?

Example 5

: Pendulum.   From Newton's law, we have the rate of change of angular momentum equal to the torque. That is, $ml^2 \ddot\theta=-mgl\sin\theta$, or

$$\ddot\theta+{g\over l}\sin\theta=0.$$

We rewrite this as a system,

$$x_1=\theta, \quad x_2=\dot\theta,$$

so that $\dot x_1=\dot\theta=x_2$, $\dot x_2=\ddot\theta= -(g/l)\sin x_1$. The autonomous system is

$${d\over dt}\pmatrix{x_1\cr x_2} =\pmatrix{x_2\cr -{g\over l} \sin x_1}.$$

The critical points are $x_2=0, -(g/l)\sin x_1=0$, which yields $x_1=n\pi, n=0, \pm 1,\pm 2,...$. Intuitively, when n = even, the solution is `stable'. When n=odd, the pendulum is inverted, and hence `unstable'. We will describe these notions further in the course.

For 2-D systems, it is convenient to study the phase plane, the x₁-x₂ plane where: given $\ddot x=f(x,\dot x)$, we plot trajectories on an $x-\dot x$ plane, or given the system

$$dX/dt=f(X),\quad X=\pmatrix{x_1\cr x_2},$$

we plot trajectories in the x₁-x₂ plane. In these cases, the independent variable t is a parameter.

A phase vector diagram shows tangent vectors to the trajectories in the x₁-x₂ plane. Below is a figure showing a phase vector diagram for the example

$$\dot x_1=x_1+x_2, \quad \dot x_2=-4x_1+x_2.$$

The method is to first put a grid on the x₁-x₂ plane. At each point on the grid, evaluate the tangent vector [ x₁+ x₂, -4x₁+x₂]. For example, at (1,1), the tangent vector is ${\bf t}=(2,-3)$. One choice is to plot vectors of the same length, i.e., ${\bf t}=(2,-3)/\sqrt{4+9}$,normalized to unit length. This then shows only the directions. Plotting the tangent vector without normalization results in the length of the vectors showing the magnitude of the speeds.

 

Figure 1.1: Phase vector diagram for $\dot x_1=x_1+x_2$, $\dot x_2=-4x_1+x_2$. 

On Matlab, try:

[x,y]=meshgrid(-3:0.475: 3, -3:0.475:3);

z1=x+y;

z2=-4*x+y;

quiver(x,y,z1,z2)

grid

On Mathematica, try:

Needs["Graphics `PlotField`"]

PlotVectorField[{ x+y, -4*x+y}, {x, -3,3 }, { y,-3,3}, Axes $\to$ True, Ticks $\to$ None, Frame $\to$ True, AspectRatio $\to$ 1]

We can add ScaleFunction $\to$(1&) inside the square brackets to scale the vectors to unit length.

Basic properties of phase plane diagrams for $\dot x_1=F(x_1,x_2)$, $\dot x_2=G(x_1,x_2)$, x₁(0)=a, x₂(0)=b.
1. Existence. There is a trajectory through each point (a,b) on the plane.
2. If F and G are smooth functions, we get uniqueness of the solution. Two trajectories through the same point are translations (in time) of each other. That is, if $x_1=\phi(t)$, $x_2=\psi(t)$passes through (a,b) at t=t₀, and another trajectory $x_1=\alpha(t), x_2=\beta(t)$ passes through (a,b) at t=t₁, then because the solution to each problem is unique, one solution is a translation of the other.
3. A trajectory may not cross itself. A trajectory can be a closed curve for periodic solutions.
If, on the other hand, the system is non-autonomous, $\dot x_1=F(x_1,x_2,t)$, $\dot x_2=G(x_1,x_2,t)$, there can be different tangent vectors $(F(x_1,x_2,t),\break G(x_1,x_2,t) )$ going through (x₁, x₂) for different t and there is the possibility of cross-overs in the phase trajectory plot. An example is $\dot x=\sin t$ which has the solution $x=-\cos t +c$. This solution exhibits the same value of x for different values of t. For instance x=c for $t=\pi/2$ and $t=3\pi/2$, but $x(t-\pi/2)$ and $x(t- 3\pi/2)$ are not translations of each other. This type of behavior means that you can have 2D chaos in non-autonomous systems, but not for autonomous systems. The property that trajectories cannot cross for 2D autonomous systems is a powerful tool for establishing the existence of periodic solutions (Poincaré-Bendixson theory).