If I am not wrong, if such systems exist, they are examples of non-Shilnikov chaos. I will mention systems with no equilibrium [1] or with only stable equilibria [2] as other examples of non-Shilnikov chaos.
[1] S. Jafari, J.C Sprott, S.M.R.H Golpayegani, Elementary quadratic chaotic flows with no equilibria, Phys. Lett. A 377, 699 (2013).
[2] M. Molaie, S. Jafari, J.C. Sprott, R.H.G.S. Mohammad, Simple chaotic flows with one stable equilibrium, International Journal of Bifurcation and Chaos 23, 13501888 (2013).
I have heard about the lyapunov exponent which is positive for chaotic systems. Will go through the eigen value also. Thanks for new information.
I believe that, for topological reasons (index theory, degree) it is impossible for any function which defines a flow (chaotic or not) to have all the eigenvalues with real parts of the same sign for all its equilibrium points (at least they are in finite number and they are all hyperbolic). Although not a flow, not even invertible, the famous logistic map x(t+1) = 4*x(t)*(1-x(t)) on [0,1] has 2 equilibria, one x=0 with f'(0)>1 and another with f' < -1. One can relate this with a 3D flow and probably obtain a counterexample to your claim. I am sorry I cannot be more precise at this point!
Sorry, I forgot to mention that it is clear that all equilibria must have ate least one eigenvalue with positive real part, and also all periodic orbits must have some multiplier > 1 or < -1. Bye!
Dear Jose,
I didn’t mean maps. My question is about flows. You mentioned that “it is impossible to have a all the eigenvalues with real parts of the same sign for all its equilibrium”. However in “M. Molaie, S. Jafari, J.C. Sprott, R.H.G.S. Mohammad, Simple chaotic flows with one stable equilibrium, International Journal of Bifurcation and Chaos 23, 13501888 (2013).” You can find many chaotic flows with only one equilibrium in them that equilibrium has eigenvalues with real parts of the negative sign.
I will take a look at your paper, Sajad. To have an asymptotically stable equilibrium point goes against the definition of chaos which I learned (a long time ago, let me say, when Yorke and Li published their "period 3 implies chaos").
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