If you mean the real part of the eigenvaues, I would say that it is not posible to have a bounded system, chaotic or non chaotic. Now, you could find that the transient towards infitiy might present chaotic characteristics, like trajectories very sensitive to the initial conditions.
Actually, thinking again, what control the final divergence of dynamic systems are the Lyapunov exponents. If by eingenvalues you mean the eigenvalues of the linearized system around the fixed points, then there are chaotic systems with positive (linear)eigenvalues, e.g. the Lorenz system.
Dear Professor Clausse,
Thank you. I corrected the question. I meant real parts.
As you said, I mean the eigenvalues of the linearized system around the fixed points, but I believe that Lorenz system is not an example of that. It has 3 fixed-points and in none of them ALL the eigenvalues are positive (real parts). I want a system in which any fixed-point has only positive (real part) eigenvalues.
Regards,
Sajad
I see. Well, in discrete maps there is at least one: the logistic map. It has only one eigenvalue and when it is chaotic the real part is positive.
No I believe you cannot. If you have all positive eigenvalues then all trajectories will move away from the fixed point and you cannot have a positive invariant set like in the Lorenz attractor. Now you could take a system with an equilibrium that has all positive eigenvalues when you linearize and attach it to another system which has a chaotic attractor. Then you can artificially build a beast that has one equilibrium that has all positive eigenvalues and the system itself exhibits chaos since it has a chaotic attractor, but I'm not sure this is what you mean.
I am not sure about the impossibility of bounding a system whose LINEAR eigenvalues has positive real parts. The latter will certainly repel the trajectories in the region close to the equilibrium points, but once the state variables are sufficiently far, non-linear effects can prevent the system from diverge. For example, the Van der Pol oscillator has two eigenvalues with positive real parts, but the system evolves to a limit cycle. The issue here is whether or not the mixing of trajectories characteristic of chaos is possible in such cases.
An intermediate issue here would be if it is possible to have only one positive Lyapunov exponent if all the real part of the linear eigenvalues are positive.
Dear Alejandro,
It seems to me that you are completely right. Thank you for the Van der Pol example.
Actually, I have read of systems of coupled Van der Pol oscillators that evolve to chaotic attractors. If that is so, those will be examples of chaotic systems having all real parts of the linear eigenvalues positive.
I would like to add that, in general, the sign of eigenvalues may not be equal the sign of Lyapunov exponents (see Nemytskii-Vinograd counterexample - http://www.math.spbu.ru/user/nk/PDF/Lyapunov-exponent-Sign-inversion-Perron-effects-Chaos.pdf)
Such systems exist, but they are rare. I have found such an example in a 3-D system of ODEs with quadratic nonlinearities, and a publication describing it is in preparation.
You can try to construct an example of such a system with a quadratic nonlinearities, so that all solutions have been limited (a system with dissipative). For this, I advise you to read the article of Lorenz, where he proved the limitations of solutions. But if all eigenvalues of the real part is positive, then the equilibrium will repel trajectory. These trajectories may move, for example, to limit cycle, but there is not a chaotic system.
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