This property of the system is called multistability.

The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. {\displaystyle dX/dt=AX} At some values of r, the attractor is a single point (a "fixed point"), at other values of r two values of x are visited in turn (a period-doubling bifurcation); at yet other values of r, any given number of values of x are visited in turn; finally, for some values of r, an infinitude of points are visited.

A Ann. Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems.

If Xis not compact, then it will be convenient to compactify by adding a point at infinit… And the matrix system {\displaystyle r=2.6} Attractors are portions or subsets of the phase space of a dynamical system. = < An attractor is a subset A of the phase space characterized by the following three conditions: Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. We think of T as the transition map of a discrete-time dynamical system with state space X. Namely, starting from a state x, the system evolves by first going to state T (x), then to state T 2 (x), and so on and so forth. The relationships and measure theoretic properties of boundaries in dynamical systems are analyzed. An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor (see strange attractor below). Leonov Department of Applied Cybernetics, Saint-Petersburg State University, Russia Department of Mathematical Information Technology, University of Jyv¨skyl¨, Finland (e-mail: [email protected]) a a Abstract: From a computational point of … x

(Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.)

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Copyright © 2020 Elsevier B.V. or its licensors or contributors. Secondary 11B05. The dynamical systems arise from dissipative systems, their motion cease if there is no driving forces. Some attractors are known to be chaotic (see #Strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system.

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