Have a look at :: Conference Paper Assertion Checking Using Dynamic Inference

Chapter Monotone Dynamical Systems

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The assertion is wrong. Set X(t)=X0+t/(t+1), t0=0. (X0=X(0))

Then X0+1 is an accumulation point, because it is even a limit point. But X*=X0+1 is not a fixed point, nor is it part of a limit cycle.

K. Kassner: The X(t) you posted does not satisfy the semigroup property, and so is not a trajectory of a dynamical system.

If the trajectory is dense (as in the chaotic case) then you don't necessarily get a periodic point. You can get subtrajectories with every possible limit.

Tyler Meadows Could you elaborate? I am a physicist, not a mathematician. The X(t) I gave is the trajectory of a particle in one dimension satisfying the equation of motion \ddot(X) = -2 \dot(X)^{3/2}. While this type of equations is rare in physics, it would definitely qualify as a dynamical system with a two-dimensional phase space.

Of course, one can make my example autonomous, by setting X1=X, X2=t. Then it reads \dot(X1) = 1/(X2+1)^2, \dot(X2)=1. Hmm, but I see that then it does not have an accumulation point anymore. So it is not a counterexample... Interesting.

Hello, thanks for these answers. I should have added that the dynamical system is autonomous and 1 or 2-dimensional. In this case, this assumption certainly is part of the Poincaré-Bendixson's theorem proof.

Gabriel

K. Kassner: In the definition of a dynamical system, trajectories must satisfy the semigroup property. What this means is that if $\Phi_t(x_0)$ is a trajectory of a dynamical system, then for any t>0 and s>0 $\Phi_{t+s}(x_0) = \Phi_s(\Phi_t(x_0))$. This is a natural property to want for a dynamical system, as it says that if the system evolves for t+s seconds, that's the same as evolving for t seconds and then evolving for s seconds.

To add to your original response. The trajectory in this case would refer to the trajectory through two-dimensional phase space. In which case, the initial velocity is also needed to fully describe a general solution. In doing this, you can check that the semigroup property is satisfied by the two-dimensional system, although I suspect the calculations get a bit hairy.

Gabriel Thomas With the restriction of the system being one- or two-dimensional (and continuous), I suppose the assertion is correct and should be provable, the only attractors being fixed points and limit cycles. All that needs to be proven is that an accumulation point must belong to an attractor. That should almost follow from the definition of "attractor".

In more than two dimensions, I still think that the assertion is wrong. To obtain a proof for its refutation in a large set of Hamiltonian systems, one might use Poincaré's recurrence theorem. This states that if the system is ergodic (or a subset of nonzero measure is) any trajectory (or one of the subset) will come arbitrarily close to its initial point, given sufficient time. Which means that the initial point is an accumulation point. But not all these ergodic trajectories are fixed points or limit cycles. And there are many (quas)ergodic systems.