Degenerate Hopf Bifurcations in Chua's System

Abstract

In this paper we study the local codimension one, two and three Hopf bifurcations which occur in the classical Chua's differential equations with cubic nonlinearity. A detailed analytical description of the regions in the parameter space for which multiple small periodic solutions bifurcate from the equilibria of the system is obtained. As a consequence, a complete answer for the challenge proposed in [Moiola & Chua, 1999] is provided.

Hopf Bifurcations and Their Degeneracies in Chua's Equation

“…In particular, our work extends the study carried out for the Hopf bifurcations in [Messias et al, 2009] when γ = 0 and a = 1.…”

“…The presence of a cusp of periodic orbits can give rise to hysteretic phenomena. In particular, the recent paper [Messias et al, 2009] appears to be a particular case of our study when γ = 0 and a = 1.…”

“…Different cases appear in terms of the values of γ. In fact, it is convenient to distinguish cases γ = −1 and γ = 0 (partially studied by [Messias et al, 2009] for a = +1, α > 0 and β > 0) from the generic case γ = −1, 0.…”

Hopf bifurcation and the centers on center manifold for a class of three‐dimensional Circuit system

“…where 0 > 0 given in (17). According to the Routh-Hurwitz stability criterion, it has no purely imaginary eigenvalues, namely, at this time the equilibrium E is not a center or focus.…”

“…Because the Jacobian matrix at the origin O is the same as the one in the case of f(x) = c 2 x 2 for system (3), the critical conditions of Hopf bifurcation at the origin O still are (16) or (17). Very similar to the process in the previous section, we can also use the nondegenerate transformation: (x, ,ũ) ′ = P o (z, w, u) ′ given in (39) and let T = it to make system (3) become the following same form as the complex system (8):…”