Bifurcation phenomena in control flows (original) (raw)

On Dynamic Bifurcations in Control Systems

IFAC Proceedings Volumes, 2001

We discuss controllability properties of control affine systems depending on a slowly varying parameter. Near a singular point, the control sets for fixed parameter values turn into subsets where controllability holds provided that the parameter variations are slow enough. Copyright (1) 2001lFAC

Bifurcations of 1-parameter Families of Control-affine Systems in the Plane

SIAM Journal on Control and Optimization, 2006

We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations at control-regular points. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. Such bifurcations form six different classes: two bifurcations of equilibrium sets, two bifurcations of critical sets, and two bifurcations of pairs of invariants. We also classify all generic 1-parameter families of control-affine systems with respect to orbital feedback equivalence.

Generic bifurcations of control-affine systems in the plane and their properties

Proceedings of the IEEE Conference on Decision and Control

We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations with nonvanishing control field. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. There are six such bifurcations: two bifurcations of equilibrium sets, two bifurcations of critical sets and two bifurcations of pairs of invariants. We also analyze how time optimality properties and stabilizability of generic planar families change when the parameter varies.

Local bifurcations of continuous dynamical systems under higher order conditions

Applied Mathematics Letters, 2010

In this paper we provide higher order conditions which imply the appearance of nonstandard local bifurcations in uniparametric families of one-dimensional continuous-time dynamical systems. By the Center Manifold Theory, they also describe generalizations of local bifurcations of uniparametric families of systems on R n .

Liapunov direct method in approaching bifurcation problems

Annali di Matematica Pura ed Applicata, Series 4, 1976

A D~IO GRAFFI nel suo 70 ° compleanno Summary. -A concept o] total stability/or continuous or discrete dynamical systems and a generalized de]inition o] bi/ureation are given: 4t is possible to show the link between an abrupt change o/the asymptotic behaviour o] a ]amily o] ]lows and the arising o] new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour o] ]lows near an invariant compact set. They are obtained by means o/an extension o/ Liapunov's direct method.

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...

Topological Invariants of Bifurcation

Trends in Mathematics

I will shortly discuss an approach to bifurcation theory based on elliptic topology. The main goal is a construction of an index of bifurcation points for C 1-families of Fredholm maps derived from the index bundle of the family of linearizations along the trivial branch. As illustration, I will present an application to bifurcation of homoclinic solutions of non-autonomous differential equations from a branch of stationary solutions.

Bifurcation Theory of Functional Differential Equations

Applied Mathematical Sciences, 2013

In this paper we survey the topic of bifurcation theory of functional differential equations. We begin with a brief discussion of the position of bifurcation and functional differential equations in dynamical systems. We follow with a survey of the state of the art on the bifurcation theory of functional differential equations, including results on Hopf bifurcation, center manifold theory, normal form theory, Lyapunov-Schmidt reduction, and degree theory.