Chua's Singularities: Great Miracle in Circuit Theory

Abstract

In the work [Chua, 1992], a deep intuition of its author gave rise to the choice of singularities corresponding to Chua's circuit. Therefore, it is the only one probably exhibiting three saddle points named Chua's singularities in this paper. One of the singularities is a saddle in forward time (dt > 0) of integration, whereas the other two are saddles in backward time (dt < 0) of integration. In the following, the term Chua's Chaos denotes chaos related to Chua's singularities. These singularities are the source of all special surfaces that are the subject of this contribution. We named the surface to which all other surfaces are bound as the Double-Arm Stable Manifold (DASM). The beauty and multifunctionality of this surface represents the unfathomable Intelligence in the sense of [Tolle, 2003]. The presence of the DASM in the state space is a sufficient condition for the generation of Chua's chaos or corresponding periodic windows. Since Chua's singularities are not limited by circuit morphology or the order of state equations, the research on Chua's chaos seems to be still very promising.

Boundary surface and stable manifold in sequential circuits

“…The procedure for calculating the depiction surfaces shows that although there is no uniform method for calculation and depiction of surfaces separating attractors from each other in the state space, combination of grid method and backward integration step technique makes possible the depiction of such new phenomena as the double-arm stable manifold, published in work [21]. Mentioned analysis and circuit simulation can help to understand the behavior of sequential or chaos-generating circuits in case of their failure or other problems.…”

“…Based on [6], [21], the plane of the EBS is defined by linear transformation of system (1) as follows 1 1 1 1 2 1 1 3 2 0 y i u u (5) where 1k are the eigenvectors and the appropriate u 1 , u 2 , i corresponds to incremental quantities of state variables. Values of eigenvectors and coordinates of singularities for N1 and N2 are listed in Table 2 In Fig.…”

On the piecewise-linear approximation of the polynomial chaotic dynamics

“…Let recall that mentioned procedures for dynamical system classification is not a complete listing. The basin of attractions and stable manifolds should be established by some trivial method provided in [9].…”

Dynamical Tangles in Third-Order Oscillator with Single Jump Function

“…As will be clarified later, additional information must be obtained before the start of the searching procedure, such as location of the fixed points, eigenspaces, boundary planes, attraction sets, and corresponding basins. The description of procedure solving this problem for the famous Chuas equations [5] can be found in publication [6]. Many associated problems like vector field geometry of the so-called double-hook or dual double-scroll attractors [7] are solved in the interesting book [8].…”