Boundary Surfaces and Basin Bifurcations in Chua's Circuit

Abstract

One of the most important tasks in the analysis of a nonlinear system is to determine its global behavior and, in particular, to delineate the domains of attraction for asymptotically stable solutions. Stable manifolds often act as boundary surfaces between such domains in the state space. In this paper the morphology of boundary surfaces is studied in a single member of Chua's circuit family, although the techniques used apply equally well to many other nonlinear circuits. Of all the PWL circuits known so far which exhibit two stable states it is typical that their resistor characteristics each have at least three segments. Although bistability cannot be achieved via a 2-segment characteristic in the plane, complicated bistable behavior, including chaotic attractors, can occur locally at the boundary of two linear regions in the 3-D Chua's circuit, i.e. bistability is achieved with a minimal number of segments. By using a 3-segment characteristic, at least five attractors can be generated. Basin structure of the corresponding attractors is examined using numerical simulations. Period-adding, symmetry-breaking and remerging bifurcation phenomena are observed experimentally and numerically from an extended Chua's circuit. Dynamical properties of sequential circuits can be investigated by means of switching between the system's attractors, and boundary surfaces play a crucial role in the process of switching. The use of basin delineation in the triggering of multistable circuits is shown.

Boundary Surface of 5-Valued Memory

“…Time marks in Figure 3 (projection into 2 , 1 plane) illustrate the time interval of representative point movement on SLCs. L1, L2, and L3 are special because they are not tied to an unstable limit cycle as in the cases described in [14,22,28,29]. The symbol L denotes absolutely unstable limit cycle.…”

“…It is not possible without the knowledge of the morphological characteristics of the boundary surface (BS) that separates the regions of attraction of particular attractors. BS is used not only in memories [16][17][18][19][20], but also in chaos generating circuits [21][22][23][24]. Therefore, this paper presents morphology of BS in the form of 2D crosssections for two cases of five-valued elementary memory.…”

Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes

“…Pioneering work showing the presence of robust chaotic oscillation within dynamics of simple electronic circuit is [8]. So far, the so-called Chua´s oscillator was subject of laboratory demonstrations, deep numerical investigations and many research studies [9][10][11]. Several interesting strange attractors associated with different vector field local geometries have been localized within the dynamics of three-segment piecewise-linear Chua systems [12][13][14].…”

Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

“…In the past three decades, numerous works have been reported on this circuit, including realization schemes, experimental measurements, numerical observations, and theoretical proofs [2][3][4][5][6][7]. The classic Chua's system with a simple algebraic structure is a dimensionless form of Chua's equations, and its nonlinearity formed by Chua's diode, called Chua's nonlinearity in this paper, is three-segment piecewise-linear.…”