If I have a 3D differential system of 3 variables and considerable number of parameters, and I on PURPOSE would like this system to exhibit chaos, how would I go about ensuring that the system has positive Lyapunov exponents analytically (rather than using the hit-and-try method)? My first thought is to ensure that the divergence of the system is less than zero AND use the Routh Hurwitz criterion on the solution of eigenvalues to make sure that the system is stable. But,all this still doesn't give analytical proof that the Lyapunov exponents are positive (unless I run it in a program and check).
Dear Marella,
I think your question is an open problem in chaos. There are some points that come to my mind:
1. I am sure that you cannot find a general method based on the eigenvalues of the equilibria; because we can have chaotic systems with unstable equilibria (Lorenz system and most of the simple systems in [1]), stable equilibria [2], and even without equilibria [3].
2. In a chaotic system, we have basin of attraction for the strange attractor(s), and may have basin of attraction for possible stable equilibria, basin for limit cycle and torus and regions which cause unstable answers. The worst part is that sometimes these basins have not determined boundaries and are fractal. In other word, consider a 3D flow: it can show many different behaviors only based on initial conditions.
So I think at best we can find criteria that increase the possibility of finding chaotic systems.
References:
[1] Sprott JC. Some simple chaotic flows. Phys. Rev. E 1994;50:R647.
[2] Malihe Molaie, Sajad Jafari, J. C. Sprott, S. Mohammad Reza Hashemi Golpayegani, “Simple chaotic flows with one stable equilibrium,” International Journal of Bifurcation and Chaos, In press.
[3] Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H. [2013] “Elementary quadratic chaotic flows with no equilibria,” Phys. Lett. A, 377, 699-702.
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