Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry

doi:10.1016/j.physd.2013.01.010

Physica D: Nonlinear Phenomena

Volume 250, 1 May 2013, Pages 34-46

https://doi.org/10.1016/j.physd.2013.01.010 Get rights and content

Abstract

Three-dimensional symmetric piecewise linear differential systems near the conditions corresponding to the fold-Hopf bifurcation for smooth systems are considered. By introducing one small parameter, we study the bifurcation of limit cycles in passing through its critical value, when the three eigenvalues of the linear part at the origin are at the imaginary axis of the complex plane.

The simultaneous bifurcation of three limit cycles is proved. Conditions for stability of these limit cycles are provided, and analytical expressions for their period and amplitude are obtained.

Finally, we apply the achieved theoretical results to a generalized version of Chua’s circuit, showing that the fold-Hopf bifurcation takes place for a certain range of parameters.

Highlights

► Continuous 3D symmetric piecewise linear differential systems are considered. ► One-parameter fold-Hopf bifurcation unfoldings are studied. ► The critical system exhibits a bounded set foliated by periodic orbits. ► From this set, the simultaneous bifurcation of three limit cycles is proved. ► Such fold-Hopf bifurcation is detected in a generalized version of Chua’s circuit.

Section snippets

Introduction and preliminary results

The study of limit cycles of a differential system is, after the analysis of equilibrium solutions, one of the most important problems in the qualitative analysis of dynamical systems. Several tools from bifurcation theory are available in the case of smooth systems in order to guarantee the bifurcation and existence of limit cycles, see [1], [2], [3]. The situation is worse in the case of piecewise smooth systems, see [4], so that new results concerning this class of differential non-smooth

Statement of main results

Our first result concerns the possible bifurcation of symmetrical periodic orbits using the three zones. We introduce a new parameter $δ = d - t ω^{2}$ , which characterizes the criticality of the bifurcation, in a similar way to what happens in the cases considered in [11], [12].

Theorem 3

Let us consider system (1)–(2) under conditions (4)where it is assumed $ρ \neq 0$ and $δ = d - t ω^{2} \neq 0$ and fixed. For $ε = 0$ the system (1)–(2) undergoes a tri-zonal limit cycle bifurcation, that is, from the configuration of periodic orbits that

Realization in a generalized Chua’s circuit

In this section we consider the generalized version of Chua’s circuit that appears in Fig. 6(a), where a negative resistance device $R_{N}$ has been introduced with respect to the standard model (see for example [20]). To obtain more information about negative resistance devices, see [21].

The state equations of the circuit of Fig. 6 are (see [22]) $\frac{d v_{1}}{d τ} = \frac{1}{C_{1}} [G (v_{2} - v_{1}) - f (v_{1})],$ $\frac{d v_{2}}{d τ} = \frac{1}{C_{2}} [G (v_{1} - v_{2}) + i_{3}],$ $\frac{d i_{3}}{d τ} = - \frac{1}{L} (v_{2} + R_{N} i_{3}),$ where $v_{1}$ and $v_{2}$ are the voltages across the capacitors $C_{1}$ and $C_{2}$ , $i_{3}$ is the current

Proof of the main results

We start by giving the proof of Proposition 2.

Proof of Proposition 2

From the definition of system (1)–(2) under conditions (4) in the central zone, it is direct to assure that the unique equilibrium point for $ε \neq 0$ and $| x | < 1$ is the origin.

From the third component of equations of system (1)–(2), in the zone $x > 1$ , the $x$ -coordinate of the equilibrium point for $ε \neq 0$ is equal to $x_{ε}^{+} = 1 + \frac{ε}{d} (ρ^{2} ε^{2} + ω^{2})$ . If $d ε < 0$ then $x_{ε}^{+} < 1$ and the equilibrium point is not a real equilibrium but it is a virtual one. The same is true for the zone $x < - 1$

Acknowledgments

Authors are partially supported by the Ministerio de Ciencia y Tecnología, Plan Nacional I+D+I, in the frame of projects MTM2009-07849, MTM2010-20907, MTM2012-31821, and by the Consejería de Educación y Ciencia de la Junta de Andalucía under grants TIC-0130 and P08-FQM-03770.

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Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry

Abstract

Highlights

Section snippets

Introduction and preliminary results

Statement of main results

Realization in a generalized Chua’s circuit

Proof of the main results

Acknowledgments

Physica D

J. Math. Anal. Appl.

Limit cycle bifurcation from center in symmetric piecewise-linear systems

Internat. J. Bifur. Chaos

Invariant cones for non-smooth dynamical systems

Math. Comput.

Generalized Hopf bifurcation for non-smooth planar systems

Philos. Trans. R. Soc. Ser. A

Generalized Hopf bifurcations for planar filippov systems continuous at the origin

J. Nonlinear Sci.

Limit cycle bifurcation in 3D continuous piecewise linear systems with two zones. Application to Chua’s circuit

Internat. J. Bifur. Chaos

The focus-center-limit cycle bifurcation in symmetric 3D piecewise linear systems

SIAM J. Appl. Math.