Melnikov distance

One of the main tools for determining the existence of (or non-existence of) chaos in a perturbed Hamiltonian system is Melnikov theory. In this theory, the distance between the stable and unstable manifolds of the perturbed system is calculated up to the first-order term. Consider a smooth dynamical system ${\ddot x}=f(x)+\epsilon g(t)$ , with $\epsilon \geq 0$ and $g(t)$ periodic with period $T$ . Suppose for $\epsilon = 0$ the system has a hyperbolic fixed point x₀ and a homoclinic orbit $\phi (t)$ corresponding to this fixed point. Then for sufficiently small $\epsilon \neq 0$ there exists a T-periodic hyperbolic solution. The stable and unstable manifolds of this periodic solution intersect transversally. The distance between these manifolds measured along a direction that is perpendicular to the unperturbed homoclinc orbit $\phi (t)$ is called the Melnikov distance. If $d(t)$ denotes this distance, then $d(t)=\epsilon (M(t)+O(\epsilon ))$ . The function $M(t)$ is called the Melnikov function.

Melnikov’s distance can be used to predict chaotic vibrations ^[1]. In this method, critical amplitude is found by setting the distance between homoclinic orbits and stable manifolds equal to zero. The Melnikov’s method provides necessary but not sufficient condition for chaos.

References

Guckenheimer, John; Holmes, Philip (1983), Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, New York: Springer-Verlag, doi:10.1007/978-1-4612-1140-2, ISBN 0-387-90819-6, MR 0709768 .

↑ Alemansour, Hamed; Miandoab, Ehsan Maani; Pishkenari, Hossein Nejat (March 2017). "Effect of size on the chaotic behavior of nano resonators". Communications in Nonlinear Science and Numerical Simulation. 44: 495–505.

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[1] Alemansour, Hamed; Miandoab, Ehsan Maani; Pishkenari, Hossein Nejat (March 2017). "Effect of size on the chaotic behavior of nano resonators". Communications in Nonlinear Science and Numerical Simulation. 44: 495–505.