Abstract
This work is dedicated to the study of an approach that allows the generation of hidden attractors based on a class of piecewise-linear (PWL) systems. The systems produced with the approach present the coexistence of self-excited attractors and hidden attractors such that hidden attractors surround the self-excited attractors. The first part of the approach consists of the generation of self-excited attractors based on pairs of equilibria with heteroclinic orbits. Then, additional equilibria are added to the system to obtain a bistable system with a second self-excited attractor with the same characteristics. It is conjectured that a necessary condition for the existence of the hidden attractor in this class of systems is the rupture of the trajectories that resemble heteroclinic orbits that join the two regions of space that surround the pairs of equilibria; these regions resemble equilibria when seen on a larger scale. With the appearance of a hidden attractor, the system presents a multistable behavior with hidden and self-excited attractors.
Highlights
Academic Editor: Xianggui Guo is work is dedicated to the study of an approach that allows the generation of hidden attractors based on a class of piecewiselinear (PWL) systems. e systems produced with the approach present the coexistence of self-excited attractors and hidden attractors such that hidden attractors surround the self-excited attractors. e first part of the approach consists of the generation of self-excited attractors based on pairs of equilibria with heteroclinic orbits. en, additional equilibria are added to the system to obtain a bistable system with a second self-excited attractor with the same characteristics
It is conjectured that a necessary condition for the existence of the hidden attractor in this class of systems is the rupture of the trajectories that resemble heteroclinic orbits that join the two regions of space that surround the pairs of equilibria; these regions resemble equilibria when seen on a larger scale
Introduction ere are two classes of attractors according to [1], which are defined as follows: the first class is given by those classical attractors excited from unstable equilibria called self-excited attractors whose basin of attraction intersects at least a neighborhood of an equilibrium point [2], and they are not difficult to find via numerical methods, and the second class is called hidden attractors whose basin of attraction does not contain neighborhoods of equilibria. e localization of this last class represents a more difficult task which has led to interesting approaches as the analytical-numerical algorithm suggested in [1] for the localization of hidden attractors of Chua’s circuit
Summary
Where x (x1, x2, x3)T ∈ R3 is the αij ∈ R3×3 is a linear operator, state vector, and B (β1, β2, β3)T. E vector f(x)B is a constant vector in each atom Pi such that the equilibria are given by x∗eqi i 1, . With the atoms of a P partition containing a saddle equilibrium point in each of them as defined above, it is possible to generate heteroclinic orbits. To generate a heteroclinic orbit, at least two equilibria are required. Erefore, consider a partition with two atoms P P1, P2, the constant vector B ∈ R3 is defined as follows: B. E points where the stable and unstable manifolds intersect at SW are given by xin. Us, the heteroclinic orbits are defined as follows: HO1 x ∈ φxin, t: t ∈ (− ∞, ∞),. Us, one can find initial conditions for the simulation of the heteroclinic orbits as close to the equilibria as desired.
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