A Simple Chaotic Flow with a Plane of Equilibria

Abstract

Using a systematic computer search, a simple four-dimensional chaotic flow was found that has the unusual feature of having a plane of equilibria. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Current-Mode Network Structures Dedicated for Simulation of Dynamical Systems with Plane Continuum of Equilibrium

“…Quite recently, chaotic mathematical models with multiple line equilibrium have been proposed. One such example can be found in report 47 and can be described in general form (2) together with the following functions:…”

“…Both the cases are provided in Ref. 47 and a comprehensive study of another di®erent \hyperbolic" case can be found in Ref. 48.…”

“…Another example of a dynamical system with circle equilibrium can be found in Ref. 47. This system belongs into class (2) with functions…”

“…Surprisingly, deformation of the singular xed points into higher-dimensional objects does not preclude evolution of a strange attractor. A signi¯cant number of research articles have been devoted to the mathematical model with chaotic behavior and equilibrium in the form of surface objects such as one or several lines (two parallel), [45][46][47] hyperbola, 48 circle 49 and ellipse, square, 50 other conic-sectioned equilibrium 51 or a general curve equilibrium. [52][53][54] However, it seems that only a portion of line, circle or square is responsible for chaos generation.…”

Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping

“…Recently there has been an increasing effort in constructing new chaotic attractors with pre-designed types of equilibria [1][2][3][4][5][6][7][8][9][10][11][12]. These systems include dynamical systems with no equilibrium points [13][14][15][16][17][18][19][20][21], with only stable equilibria [22][23][24][25][26][27], with curves of equilibria [28][29][30], with surfaces of equilibria [8,9], and with non-hyperbolic equilibria [31,32].…”