Formation and Localization of Four-wing Attractor in Phase space

A chaotic attractor is formed in a finite region in phase space by the long-term trajectory of a three or higher-dimensional dissipative system. The attractor is a fractional-dimensional geometry, whose dimension is a fraction but less than the dimension of the phase space. The geometry of an attractor can be as complex as a multi-wing geometry. The emergence and confinement of such a complex geometrical attractor can be understood by the Nambu mechanics without numerically solving the governing equations of the dynamics. In this article, we show that the four-wing geometry of an attractor appears in the phase space by the intersection of two energy-like Hamiltonian functions. We further show that the dynamical equations require the localization range of these surfaces so that their intersection is confined to a certain region of the phase space. We analytically find the required conditions based on the system parameters for the localization of the attractor.