1. Concepts of Phase Plane Analysis

These two differential equations can be solved graphically and plot with ? being horizontal axis and being vertical axis.

Example:

For the system

ode_option = odeset(Maxstep,0.01);
[t1, x1]=ode45(@odefun,[0,10],[2,2],ode_option);
[t2, x2]=ode45(@odefun,[0,10],[-3,1],ode_option);
[t3, x3]=ode45(@odefun,[0,10],[0.5,0.5],ode_option);
?
hold on;
scatter(x1(:,1),x1(:,2),2,t1,filled);
scatter(x2(:,1),x2(:,2),2,t2,filled);
scatter(x3(:,1),x3(:,2),2,t3,filled);
?
colorbar;
function dxdt = odefun(~,x)
dxdt=zeros(2,1);
dxdt(1) = x(2);
dxdt(2) = -0.6*x(2)-3*x(1)-x(1)^2;
end


2. Symmetry in Phase Plane Portraits

The above three equations represent symmetry about ?, symmetry about ?, symmetry about origin.

3. How to Construct

Analytical Method

Method of Isoclines

4. Phase Plane Analysis of Linear System

Stable, Unstable, Saddle, Stable focus, Unstable focus, Center point

5. Limit Circle

1. Stable limit circleAll trajectories in the vicinity of limit circle converge to it.
2. Unstable limit circleAll trajectories in the vicinity of limit circle diverge from it.
3. Some converge to it while some diverge from it.

6. Existence of Limit Circle

N is the number of nodes, centers and foci enclosed by a limit circle.

S is the number of saddle points enclosed by a limit circle.

1. PoincareIf a limit circle exists in the second-order autonomous system, then ?the proof is mathematically involved and the inference is that from this theorem, a limit circle must enclose at least one equilibrium point.
2. Poincare-BendixsonIf a trajectory of the second order autonomous system remains in a finite region ? , then one of the following is true
1. the trajectory goes to an equilibrium point
2. the trajectory tends to an asymptotically stable limit circle
3. the trajectory is itself a limit circle.

1. BendixsonFor the nonlinear system, no limit circle can exist in a region ? of the phase plane in which ? does not vanish and does not change sign.

7. Exercise

1. Draw Phase Plane for the system

ode_option = odeset(Maxstep,0.01);
[t1, x1]=ode45(@odefun,[0,10],[0,0],ode_option);
[t2, x2]=ode45(@odefun,[0,10],[-1,1],ode_option);
[t3, x3]=ode45(@odefun,[0,10],[0.1,0.1],ode_option);
?
hold on;
scatter(x1(:,1),x1(:,2),2,t1,filled);
scatter(x2(:,1),x2(:,2),2,t2,filled);
scatter(x3(:,1),x3(:,2),2,t3,filled);
?
colorbar;
function dxdt = odefun(~,x)
dxdt=zeros(2,1);
dxdt(1) = x(2);
dxdt(2) = -10*x(2)-100*x(1)+100;
end


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