A system that does not explicitly contain dependence on the independent variable (say t) is called autonomous. An example is
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Definition 1
An equilibrium solution or critical point of the autonomous system (1.21) is a constant vector such that .
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, , or
We rewrite this as a system, so that , . The autonomous system is The critical points are , which yields . 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 x_{1}-x_{2} plane where: given , we plot trajectories on an plane, or given the system
we plot trajectories in the x_{1}-x_{2} plane. In these cases, the independent variable t is a parameter.A phase vector diagram shows tangent vectors to the trajectories in the x_{1}-x_{2} plane. Below is a figure showing a phase vector diagram for the example
The method is to first put a grid on the x_{1}-x_{2} plane. At each point on the grid, evaluate the tangent vector [ x_{1}+ x_{2}, -4x_{1}+x_{2}]. For example, at (1,1), the tangent vector is . One choice is to plot vectors of the same length, i.e., ,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.
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 True, Ticks None, Frame True, AspectRatio 1]
We can add ScaleFunction (1&) inside the square brackets to scale the vectors to unit length.
Basic properties of phase plane diagrams for
, , x_{1}(0)=a, x_{2}(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 , passes through (a,b) at t=t_{0}, and another trajectory passes through (a,b) at t=t_{1}, 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,
, , there can be different tangent
vectors
going through (x_{1}, x_{2}) for different
t and there is the possibility of cross-overs in the phase trajectory
plot. An example is which has the solution . This
solution exhibits the same value of x for different values of t.
For instance x=c for and , but and 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).