Bifurcations in Nonsmooth Dynamical Systems

Bifurcations in Nonsmooth Dynamical Systems ∗ Mario

A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local...

Hidden Bifurcations and Attractors in Nonsmooth Dynamical Systems

Two-folds in nonsmooth dynamical systems

In a three dimensional dynamical system with a discontinuity along a codimension one switching manifold, orbits of the system may be tangent to both sides of the switching manifold generically at isolated points. It is perhaps surprising, then, that examples of such ‘two-fold’ singularities are difficult to find amongst physical models. They occur where the relative curvature between the flow f...

Gap bifurcations in nonlinear dynamical systems.

We investigate the dynamics generated by a type of equation which is common to a variety of physical systems where the undesirable effects of a number of self-consistent nonlinear forces are balanced by an externally imposed controlling harmonic force. We show that the equation presents a new sequence of bifurcations where periodic orbits are created and destroyed in such a nonsimultaneous way ...

Discontinuity Induced Bifurcations of Nonhyperbolic Cycles in Nonsmooth Systems

We analyze three codimension-two bifurcations occurring in nonsmooth systems, when a nonhyperbolic cycle (fold, flip, and Neimark–Sacker cases, in both continuous and discrete time) interacts with one of the discontinuity boundaries characterizing the system’s dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of n...