#### 5.1 Example 1 (Stable limit cycle)

The following system of ode’s gives solutions that have limit cycle. \begin{align*} x'(t) &= -b y + a x (1-x^2-y^2)\\ y'(t) &= a x + b y (1-x^2-y^2)\\ \end{align*}

When $$a$$ and $$b$$ is not $$1$$, then the solution shows more iterations until it reaches the limit cycle, which is a circle of radius $$1$$.

The following animations are for diﬀerent initial conditions (one inside the limit cycle and one outside) and for diﬀerent values of $$a,b$$.

As $$a$$ becomes closer to zero, it takes many more cycles to appraoch the limit cycle.

 $$a=0.15,b=2$$ and initial conditions inside $$x(0)=0.15,y(0)=0.2$$ Pause Play Restart Step forward Step back $$a=0.15,b=2$$ and initial conditions outside $$x(0)=2,y(0)=1.7$$ Pause Play Restart Step forward Step back $$a=0.05,b=2$$ and initial conditions inside $$x(0)=0.15,y(0)=0.2$$ Pause Play Restart Step forward Step back $$a=0.05,b=2$$ and initial conditions outside $$x(0)=2,y(0)=1.7$$ Pause Play Restart Step forward Step back $$a=1,b=1$$ and initial conditions inside $$x(0)=0.15,y(0)=0.2$$ Pause Play Restart Step forward Step back $$a=1,b=1$$ and initial conditions outside $$x(0)=2,y(0)=1.7$$ Pause Play Restart Step forward Step back