Nature of chaos in conservative and dissipative systems of the Duffing-Holmes oscillator

Bifurcation Theory of Dynamical Chaos

“…In Refs. [12,13] the given approach has been applied and strictly proved by continuation along parameter of solutions from dissipative into conservative areas by means of the Magnitskii method of stabilization of unstable periodic orbits [1] at research bifurcations and chaos in the Duffing-Holmes equation…”

“…Corresponding bifurcation diagrams in a plane (ε, μ) of existence of cycles of various periods down to a conservative case at μ = 0 are shown in [12][13][14]. Application of Magnitskii approach has revealed the essence of dynamical chaos in Hamiltonian and simply conservative systems.…”

“…where the derivative is taken over the variable ξ. Therefore, the separatrix of the heteroclinic contour of system (13) describes the switching wave of the system (12), the limit cycle and the separatrix loop of the singular point of system (13) describe the traveling wave and the running impulse of system (12). And diffusion chaos in system (12) is described by singular attractors of the system of ordinary differential Eq.…”

“…(13) in full accordance with the universal bifurcation Feigenbaum-Sharkovsky-Magnitskii theory. The greatest interest represents a case when c is a bifurcation parameter, describing a speed of wave distribution along an axis x, which is not included obviously into initial system (12). This case means, that system of a kind (12) with the fixed parameters can have infinitely number of various autowave solutions of any period running along a spatial axis with various speeds, and infinite number of modes of diffusion chaos.…”

“…The greatest interest represents a case when c is a bifurcation parameter, describing a speed of wave distribution along an axis x, which is not included obviously into initial system (12). This case means, that system of a kind (12) with the fixed parameters can have infinitely number of various autowave solutions of any period running along a spatial axis with various speeds, and infinite number of modes of diffusion chaos. One of such system, describing chemical turbulence in autocatalytic chemical reactions, is studied in [6,7,15].…”