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Why is it not possible to have a period-2 orbit(or greater) for a 2D autonomous system whereas the same is possible for a non-autonomous system?

I can explain the existence of period-2 orbits in non-autonomous systems by saying that the forcing function changes the phase and hence the point is not essentially the same point when the orbit returns back to the original point, but ,I am not able to come-up with a justification for a autonomous system.Can anyone? enter image description here

Sunny Marella
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2 Answers2

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Consider the system $(x,y)\to((\cos a) x-(\sin a) y,(\sin a) x+(\cos a) y)$. If $a=2\pi/k$ for some integer $k$, this autonomous 2D system has period-$k$ orbits.

For example, $(x,y)\to(-x,-y)$ has period-2 orbits and $(x,y)\to(-y,x)$ has period-4 orbits.

Edit: The edited version of the question seems to ask about trajectories of dynamical systems intersecting themselves. This is an entirely different problem, and the mention of "period-2 orbit" may seem unfortunate in this context. Thus, consider a dynamical system $x'(t)=f(x(t),y(t))$, $y'(t)=g(x(t),y(t))$. The configuration of the picture is impossible since the point $(x^*,y^*)$ at the intersection would be such that $(f(x^*,y^*),g(x^*,y^*))$ takes two different values at the same time. In fact, if $(x(t+s),y(t+s))=(x(t),y(t))$ for some given $t$ and $s\gt0$, one has $(x(t+s+u),y(t+s+u))=(x(t+u),y(t+u))$ for every $u\geqslant0$, that is, the trajectory is a cycle.

Did
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  • I meant system of continuous differential equations with their solution curves represented in 2D phase space. I remember reading somewhere that trajectories can never cross in a autonomous system(except at an equilibrium point or a fixed point in discrete system case). Am I clear with the question now?I edited the original question with a diagram. – Sunny Marella Apr 04 '13 at 09:48
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No, the intersection like you shown in the picture can never happen. The reason is that if your differential equations have a Lipschitz continuous r.h.s, the solution to the ODE with a given initial condition are unique. Let us say the point of intersection is $(x_0,y_0)$ at time $t_0$. Now if you integrate your ODE's with initial condition $(x_0,y_0)$, your picture shows that it could go in two different directions, which means the solution is NOT unique. Hence the contradiction. Note that at the point of intersection ,there are 2 different directions on the curve along which the time is increasing. So how will the ODE know which one to take, since ODE solution at time $t_0+\delta$ depends solely on $(x_0,y_0)$.

nonlinearism
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  • Yes,I understand now that the Lipschitz condition $$|f(t,u)-f(t,v)| \leq L*|u-v| $$ and the Lipschitz Lemma that the partial derivative $$ \frac{\partial f}{\partial y} \leq K $$ are sufficient conditions to check uniqueness of solution. But,my question is: 'Is there anyway that the solution curve intersects itself(like in diag.) in a non-autonomous Diff system?' I mean,won't "phase change" be a suitable enough condition to say that when the solution curve intersects itself(like in diag.), it is actually unique in the sense that the phase has changed from the original. Is there a fn. example? – Sunny Marella Apr 08 '13 at 06:12
  • More curious with counter-arguments, is it possible that the domain D for the Lipschitz condition is different for non-autonomous systems? I'm not aware so much whether domains are defined differently for non-autonomous systems. – Sunny Marella Apr 08 '13 at 06:57
  • i.e could you give an example where the Lipschitz condition does not hold,like when there is a periodic forcing function? Hope I have made my query clear. – Sunny Marella Apr 08 '13 at 13:05
  • Any non-autonomous system can be written as autonomous by taking time as the (n+1)th coordinate. So 'phase change' is equivalent to saying that the (n+1)th coordinate is different, hence the trajectories are not crossing themselves, but only look like crossing because you are taking a projection on a lower-dimensional manifold. – nonlinearism Apr 08 '13 at 14:32