Bifurcation of switched nonlinear dynamical systems (original) (raw)
Related papers
Bifurcations in four-dimensional switched systems
Advances in Difference Equations
In this paper, the focus is on a bifurcation of period-K orbit that can occur in a class of Filippov-type four-dimensional homogenous linear switched systems. We introduce a theoretical framework for analyzing the generalized Poincaré map corresponding to switching manifold. This provides an approach to capturing the possible results concerning the existence of a period-K orbit, stability, a number of invariant cones, and related bifurcation phenomena. Moreover, the analysis identifies criteria for the existence of multi-sliding bifurcation depending on the sensitivity of the system behavior with respect to changes in parameters. Our results show that a period-two orbit involves multi-sliding bifurcation from a period-one orbit. Further, the existence of invariant torus, crossing-sliding, and grazing-sliding bifurcation is investigated. Numerical simulations are carried out to illustrate the results.
Discontinuity Induced Bifurcations in Nonlinear Systems
Procedia IUTAM, 2016
Nonlinear systems involving impact, friction, free-play, switching etc. are discontinuous and exhibit sliding and grazing bifurcations when periodic trajectories interact with the discontinuity surface which are classified into crossing sliding, grazing sliding, adding sliding and switching sliding bifurcations depending on the nature of the bifurcating solutions from the sliding surface. The sudden onset of chaos and the stick-slip motion can be explained in terms of these bifurcations. This paper presents numerical and numerical-analytical methods of studying the dynamics of harmonically excited systems with discontinuous nonlinearities representing them as Filippov systems. The switch model based numerical integration schemes in combination with the time domain shooting method are adopted to obtain the periodic solutions and the bifurcations.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2011
In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.
Multi-parametric bifurcations in a piecewise–linear discontinuous map
Nonlinearity, 2006
In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained.
Bifurcation Structures in a Bimodal Piecewise Linear Map
Frontiers in Applied Mathematics and Statistics
In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.
On special types of two- and three-parametric bifurcations in piecewise-smooth dynamical systems
2005
The aim of this paper is to present a brief overview about a special kind of two-parametric (or co-dimension two) bifurcations in piecewise-smooth dynamical systems. The characteristic property of these bifurcations is, that at the bifurcation point in a 2D parameter space an infinite number of bifurcation curves intersect. Several types of these bifurcations are discussed. Additionally, a new type of three parametric (or co-dimension three) bifurcations is reported.
Chaotic Behavior in a Switched Dynamical System
Modelling and Simulation in Engineering, 2008
We present a numerical study of an example of piecewise linear systems that constitute a class of hybrid systems. Precisely, we study the chaotic dynamics of the voltage-mode controlled buck converter circuit in an open loop. By considering the voltage input as a bifurcation parameter, we observe that the obtained simulations show that the buck converter is prone to have subharmonic behavior and chaos. We also present the corresponding bifurcation diagram. Our modeling techniques are based on the new French native modeler and simulator for hybrid systems called Scicos (Scilab connected object simulator) which is a Scilab (scientific laboratory) package. The followed approach takes into account the hybrid nature of the circuit.
On multi-parametric bifurcations in a scalar piecewise-linear map
Nonlinearity, 2006
In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that the structure of the 2D and 3D parameter spaces can be simply described using the concept of multiparametric bifurcations. It is demonstrated that an infinite number of twoparametric bifurcation lines starts at the origin of the 3D parameter space. Along each of these lines an infinite number of bifurcation planes starts, whereas the origin represents a three-parametric bifurcation.