Bifurcation for Non-smooth Dynamical Systems via Reduction Methods (original) (raw)
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Aspects of Bifurcation Theory for Nonsmooth Dynamical Systems
Invariant manifolds play an important role in the study of Dynamical Systems, since they help to reduce the essential dynamics to lower dimensional objects. In that way, a bifurcation analysis can easily be performed. In the classical approach, the reduction to invariant manifolds requires smoothness of the system which is typically not given for nonsmooth systems. For that reason, techniques have been developed to extend such a reduction procedure to nonsmooth systems. In the present paper, we present such an approach for systems involving sliding motion. In addition, an analysis of the reduced equation shows that the generation of periodic orbits through nonlinear perturbations which is usually related to Hopf bifurcation follows a different type of bifurcation if nonsmooth elements are present, since generically symmetry is broken by the nonsmooth terms. Keywords: Invariant manifold, Sliding motion, Nonlinear piecewise dynamical systems, Non-smooth systems, Invariant cones, Perio...
Analysis of discontinuous bifurcations in nonsmooth dynamical systems
Regular and Chaotic Dynamics, 2012
Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddlenode, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known dry friction oscillator, which serves as a popular model for the description of self-excited frictional oscillations of a braking system. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.
Invariant manifolds for nonsmooth systems with sliding mode
Mathematics and Computers in Simulation, 2014
Invariant manifolds play an important role in the study of Dynamical Systems, since they help to reduce the essential dynamics to lower dimensional objects. In that way, a bifurcation analysis can easily be performed. In the classical approach, the reduction to invariant manifolds requires smoothness of the system which is typically not given for nonsmooth systems. For that reason, techniques have been developed to extend such a reduction procedure to nonsmooth systems. In the present paper, we present such an approach for systems involving sliding motion. In addition, an analysis of the reduced equation shows that the generation of periodic orbits through nonlinear perturbations which is usually related to Hopf bifurcation follows a different type of bifurcation if nonsmooth elements are present, since generically symmetry is broken by the nonsmooth terms.
A general class of prototype dynamical systems is introduced, which allows to study the generation of complex bifurcation cascades of limit cycles, including bifurcations breaking spontaneously a symmetry of the system, period doubling and homoclinic bifurcations, and transitions to chaos induced by sequences of limit cycle bifurcations. The prototype systems are adaptive, with friction forces f(V(x)) being functionally dependent exclusively on the mechanical potential V(x), characterized in turn by a finite number of local minima. We discuss several low-dimensional systems, with friction forces f(V) which are linear, quadratic or cubic polynomials in the potential V. We point out that the zeros of f(V) regulate both the relative importance of energy uptake and dissipation respectively, serving at the same time as bifurcation parameters, hence allowing for an intuitive interpretation of the overall dynamical behavior. Starting from simple Hopf-and homoclinic bifurcations, complex sequences of limit cycle bifurcations are observed when the energy uptake gains progressively in importance. The term 'prototype dynamical system' is employed for generic, but otherwise reduced systems, allowing to study and to understand a certain relevant phenomenon (like dynamical behavior and/or bifurcation scenario). For this, the dynamical behavior of the system should be dominated by the prime phenomenon of interest, with the system being otherwise simple enough to allow for straightforward numerical and (at least partial) analytic investigations 1–4. Additionally, their dynamical behavior can often be understood in terms of general concepts, such as energy balance, symmetry breaking, etc. Examples of prototype systems are the normal forms of standard bifurcation analysis 5,6 and classical systems, like the Van der Pol oscillator 5 , or the Lorenz model 7 , which have been of central importance for the development of dynamical systems (systems) theory. As an example we consider the Liénard equation, + () + () = , () ̈ x f x x g x 0 1 a generic adaptive mechanical system, which includes the Van der Pol oscillator and the Takens-Bogdanov system 8,9. The periodically forced extended Liénard systems with a double-well potential have also been studied by many authors (see e.g. the double-well Duffing oscillator 10–12). In this paper we propose a new class of autonomous Liénard-type systems, which allow to study cascades of limit cycle bifurcations, using a bifurcation parameter controlling directly the balance between energy dissipation and uptake, and hence the underlying physical driving mechanism. Though there are a range of alternative construction methods for dynamical systems in the literature (see e.g. 13–15), they generally involve abstract concepts, such as implicitly defined manifolds, or mathematical tools accessible only to researchers with an in-depth math training. In contrast to these methods, we provide here a mechanistic design procedure, based on the construction of attractors through the interaction of generalized friction forces with potential forces, an intuitive concept especially suitable for interdiscipli-nary investigations (e.g. in modeling cardiovascular systems 16 or for solving optimization problems 17),
Fold bifurcation of T-singularities and invariant manifolds in 3D piecewise-smooth dynamical systems
Physica D: Nonlinear Phenomena, 2019
One of the most interesting typical singularity observed in 3D piecewise-smooth dynamical systems is the so called T-singularity (Teixeira Singularity). Throughout this paper, we study a Fold bifurcation involving two T-singularities that, as far as we know, has not yet been fully addressed in the literature. In this bifurcation both T-singularities collide and then disappear when a system parameter is varied. At the collision point appears a type of non-generic T-singularity. For this study we use a normal form and we apply the standard analysis for such class of systems, based on the analysis of the sliding dynamics (Filippov's sliding vector field) and the crossing dynamics (first return map) occurring on the switching boundary. We fully describe the unfolding dynamics of the Fold bifurcation under study and also analyze the existence, stability and bifurcations of invariant manifolds in this scenario.
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.
Nonstandard Bifurcations in Oscillators with Multiple Discontinuity Boundaries
Nonlinear Dynamics, 2004
The model of a double-belt friction oscillator is proposed, which exhibitsmultiple discontinuity boundaries in the phase space. The system consists of a simpleoscillator dragged by two different rough supports moving with constant driving velocitiesand subjected to an elastic restoring force and viscous damping. Self-sustained oscillationshave been observed to occur, with nonstandard attracting properties. By consideringthe problem from a nonsmooth dynamical systems perspective, the evolution ofsteady state attractors as the velocities of the belts are varied is described. The nonsmoothnesssets of the system at hand and, in particular, the presence of multiple discontinuityboundaries, lead to nonstandard bifurcations which are studied here by meansof analytical and numerical tools.