Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

On the Electro-Mechanical Stability of Elastomeric Coaxial Fibers

The stability of cylindrical coaxial fibers made from soft elastomeric materials is studied for electro-static loadings. The general configuration considered is a three-component axisymmetric fiber having a conducting core bonded to a dielectric annulus in turn bonded to an outer conducting annular sheath. A voltage difference between the conducting components is imposed. The stresses and actuated elongation in the perfectly concentric fiber are analyzed, and the critical voltage at which stability of the concentric configuration is lost is determined via solution of the non-axisymmetric bifurcation problem. The role of the geometry and moduli contrasts among the components is revealed, and the sub-class of two-component fibers is also analyzed. The idealized problem of a planar layer with conducting surfaces that is bonded to a stiff substrate on one surface and free on the other exposes the importance of short wavelength surface instability modes.

Nonlinear Behavior of a Novel Switching Jerk System

This paper proposes a novel chaotic jerk system, which is defined on four domains, separated by codimension-2 discontinuity surfaces. The dynamics of the proposed system are conveniently described and analyzed through a generalization of the Poincaré map which is constructed via an explicit solution of each subsystem. This provides an approach to formulate a robust bifurcation problem as a nonlinear inhomogeneous eigenvalue problem. Also, we establish some criteria for the existence of a period-doubling bifurcation and discuss some of the interesting categories of complex behavior such as multiple period-doubling bifurcations and chaotic behavior when the trajectory undergoes a segment of sliding motion. Our results emphasize that the sharp switches in the behavior are mainly responsible for generating new and unique qualitative behavior of a simple linear system as compared to the nonlinear continuous system.

Bifurcation analysis for degenerate problems with mixed regime and absorption

We are concerned with the study of a bifurcation problem driven by a degenerate operator of Baouendi–Grushin type. Due to its degenerate structure, this differential operator has a mixed regime. Studying the combined effects generated by the absorption and the reaction terms, we establish the bifurcation behavior in two cases. First, if the absorption nonlinearity is dominating, then the problem admits solutions only for high perturbations of the reaction. In the case when the reaction dominates the absorption term, we prove that the problem admits nontrivial solutions for all the values of the parameter. The analysis developed in this paper is associated with patterns describing transonic flow restricted to subsonic regions.