Narrow Results

We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

In this paper, we present a novel method to analyze the behavior of discontinuous piecewise-smooth autonomous systems (denominated Filippov systems) in the planar neighborhood of the discontinuity boundary (DB). The method uses the evaluation of the vector fields on DB to analyze the nonsmooth local dynamics of the Filippov system without the integration of the ODE sets. The method is useful in the detection of nonsmooth bifurcations in Filippov systems. We propose a classification of the points, events and events combinations on DB. This classification is more complete in comparison with the others previously reported. Additional characteristics as flow direction and sliding stability are included explicitly. The lines and the points are characterized with didactic symbols and the exclusive conditions for their existence are based on geometric criterions. Boolean-valued functions are used to formulate the conditions of existence. Different problems are analyzed with the proposed methodology.

This paper discusses the time-dependent dynamics of electrostatic vibration energy harvesters (eVEHs) with linear and nonlinear mechanical resonators. These eVEHs are fundamentally nonlinear regardless of whether a linear or nonlinear resonator is being used. The model of the system under investigation has the form of a piecewise-smooth dynamical system of a Filippov type that has a specific discontinuity in the form of a hold-on term. We use a perturbation technique called the multiple scales method to develop a theory to analyze the steady-state dynamics of the system, be it with a linear or a nonlinear resonator. We then analyze the stability of the steady-state orbit to determine when the first doubling bifurcation occurs in the system. This gives an upper bound on the region of steady-state oscillations which allows us to determine a theoretical limit on the power convertible by the eVEH. We then turn our discussion to the nonlinear behavior we see in the system's transition to chaos. Since the eVEH studied here is a Filippov type system, sliding modes and sliding bifurcations are possible in the system. We discuss the evolution of the sliding region and give particular examples of sliding phenomena and sliding bifurcations. An understanding of sliding phenomena is required for analyzing the transition to chaos since segments of sliding motion appear on trajectories that undergo period-doubling bifurcations. The transition to chaos is explained in detail by the example of the system with a linear resonator, however we discuss examples of the system with mechanical nonlinearities and discuss the difference between the linear and nonlinear cases.

Discontinuities are a common feature of physical models in engineering and biological systems, e.g. stick-slip due to friction, electrical relays or gene regulatory networks. The computation of basins of attraction of such nonsmooth systems is challenging and requires special treatments, especially regarding numerical integration. In this paper, we present a numerical routine for computing basins of attraction (BA) in nonsmooth systems with sliding, (so-called Filippov systems). In particular, we extend the Simple Cell Mapping (SCM) method to cope with the presence of sliding solutions by exploiting an event-driven numerical integration routine specifically written for Filippov systems. Our algorithm encompasses a method for dynamic construction of the cell state space so that a lower number of integration steps are required. Moreover, we incorporate an adaptive strategy of the simulation time to render more efficiently the computation of basins of attraction. We illustrate the effectiveness of our algorithm by computing basins of attraction of a sliding control problem and a dry-friction oscillator.

We consider a nongeneric family of 3D Filippov linear systems with a discontinuity plane that have two parallel tangency lines, such that the region between them is the sliding region. We are interested in finding under what conditions the family has a crossing limit cycle, when the sliding region changes its stability. We call this phenomenon the pseudo-Hopf bifurcation. This class of systems is motivated by piecewise-linear control systems which have not yet been treated in the context of crossing limit cycles.

In this paper, we present a new generic local bifurcation in planar Filippov systems. The new bifurcation presented here is a visible–invisible two-fold bifurcation which does not appear in the list published by Kuznetsov et al. in 2003.