On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations

At a border-collision bifurcation a fixed point of a piecewise-smooth map intersects a surface where the functional form of the map changes. Near a generic border-collision bifurcation there are two fixed points, each of which exists on one side of the bifurcation. A simple eigenvalue condition indicates whether the fixed points exist on different sides of the bifurcation (this case can be interpreted as the persistence of a single fixed point), or on the same side of the bifurcation (in which case the bifurcation is akin to a saddle-node bifurcation). A similar eigenvalue condition indicates whether or not there exists a period-two solution on one side of the bifurcation. Previously these conditions have been combined to obtain five distinct scenarios for the existence and relative coexistence of fixed points and period-two solutions near border-collision bifurcations. In this Letter, it is shown that one of these scenarios, namely that two fixed points exist on one side of the bifurcation and a period-two solution exists on the other side of the bifurcation, cannot occur. The remaining four scenarios are feasible. Therefore there are exactly four distinct scenarios for fixed points and period-two solutions near border-collision bifurcations.Comment: 8 pages, 1 figure, submitted to Appl. Math. Let

The structure of mode-locking regions of piecewise-linear continuous maps

The mode-locking regions of a dynamical system are the subsets of the parameter space of the system within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a curious chain structure with points of zero width called shrinking points. In this paper we perform a local analysis about an arbitrary shrinking point. This is achieved by studying the symbolic itineraries of periodic solutions in nearby mode-locking regions and performing an asymptotic analysis on one-dimensional slow manifolds in order to build a comprehensive theoretical framework for the local dynamics. We obtain leading-order quantitative descriptions for the shape of nearby mode-locking regions, the location of nearby shrinking points, and the key properties of these shrinking points. We apply the results to the three-dimensional border-collision normal form, nonsmooth Neimark-Sacker-like bifurcations, and grazing-sliding bifurcations in a model of a dry friction oscillator

Scaling laws for large numbers of coexisting attracting periodic solutions in the border-collision normal form

A wide variety of intricate dynamics may be created at border-collision bifurcations of piecewise-smooth maps, where a fixed point collides with a surface at which the map is nonsmooth. For the border-collision normal form in two dimensions, a codimension-three scenario was described in previous work at which the map has a saddle-type periodic solution and an infinite sequence of stable periodic solutions that limit to a homoclinic orbit of the saddle-type solution. This paper introduces an alternate scenario of the same map at which there is an infinite sequence of stable periodic solutions due to the presence of a repeated unit eigenvalue in the linearization of some iterate of the map. It is shown that this scenario is codimension-four and that the sequence of periodic solutions is unbounded, aligning with eigenvectors corresponding to the unit eigenvalue. Arbitrarily many attracting periodic solutions coexist near either scenario. It is shown that if $K$ denotes the number of attracting periodic solutions, and $\varepsilon$ denotes the distance in parameter space from one of the two scenarios, then in the codimension-three case $\varepsilon$ scales with $\lambda^{-K}$, where $\lambda > 1$ denotes the unstable stability multiplier associated with the saddle-type periodic solution, and in the codimension-four case $\varepsilon$ scales with $K^{-2}$. Since $K^{-2}$ decays significantly slower than $\lambda^{-K}$, large numbers of attracting periodic solutions coexist in open regions of parameter space extending substantially further from the codimension-four scenarios than the codimension-three scenarios.Comment: 37 pages, 5 figures, submitted to: Int. J. Bifurcation Chao

On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. The purpose of this paper is to explore the concept of perturbing the vector field so that forward evolution is well-defined, and characterising the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. Three forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. In each case, the limit leads to a novel probabilistic notion of forward evolution from the two-fold

A general framework for boundary equilibrium bifurcations of Filippov systems

As parameters are varied a boundary equilibrium bifurcation (BEB) occurs when an equilibrium collides with a discontinuity surface in a piecewise-smooth system of ODEs. Under certain genericity conditions, at a BEB the equilibrium either transitions to a pseudo-equilibrium (on the discontinuity surface) or collides and annihilates with a coexisting pseudo-equilibrium. These two scenarios are distinguished by the sign of a certain inner product. Here it is shown that this sign can be determined from the number of unstable directions associated with the two equilibria by using techniques developed by Feigin. A new normal form is proposed for BEBs in systems of any number of dimensions. The normal form involves a companion matrix, as does the leading order sliding dynamics, and so the connection to the stability of the equilibria is explicit. In two dimensions the parameters of the normal form distinguish, in a simple way, the eight topologically distinct cases for the generic local dynamics at a BEB. A numerical exploration in three dimensions reveals that BEBs can create multiple attractors and chaotic attractors, and that the equilibrium at the BEB can be unstable even if both equilibria are stable. The developments presented here stem from seemingly unutilised similarities between BEBs in discontinuous systems (specifically Filippov systems as studied here) and BEBs in continuous systems for which analogous results are, to date, more advanced

Twenty Hopf-like bifurcations in piecewise-smooth dynamical systems

For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such `Hopf-like' bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincare maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator-prey systems, ocean circulation, and the McKean and Wilson-Cowan neuron models

Grazing-sliding bifurcations creating infinitely many attractors

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed

Equilibrium-Independent Dissipativity with Quadratic Supply Rates

Equilibrium-independent dissipativity (EID) is a recently introduced system property which requires a system to be dissipative with respect to any forced equilibrium configuration. This paper is a detailed examination of EID with quadratic supply rates for a common class of nonlinear control-affine systems. We provide an algebraic characterization of EID for such systems in the spirit of the Hill-Moylan lemma, where the usual stability condition is replaced by an incremental stability condition. Based on this characterization, we state results concerning internal stability, feedback stability, and absolute stability of EID systems. Finally, we study EID for discrete-time systems, providing the relevant definitions and an analogous Hill-Moylan-type characterization. Results for both continuous-time and discrete-time systems are illustrated through examples on physical systems and convex optimization algorithms.Comment: Revised version, 15 page

The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations

An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear model of a chaotic electrical circuit. The variation of a secondary parameter reveals a period-doubling cascade to chaos with windows of periodicity. The dynamics is well approximated by a one-dimensional unimodal map which explains this bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms

Dimension reduction for slow-fast, piecewise-smooth, continuous systems of ODEs

The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems of ODEs occurs on critical manifolds that are piecewise-linear. At points of non-differentiability, such manifolds are not normally hyperbolic and so the fundamental results of geometric singular perturbation theory do not apply. In this paper it is shown that if the critical manifold is globally stable then the system is forward invariant in a neighbourhood of the critical manifold. It follows that in this neighbourhood the dynamics is given by a regular perturbation of the dynamics on the critical manifold and so dimension reduction can be achieved. If the attraction is instead non-global, additional dynamics involving canards may be generated. For boundary equilibrium bifurcations of piecewise-smooth, continuous systems, the results are used to establish a general methodology by which such bifurcations can be analysed. This approach is illustrated with a three-dimensional model of ocean circulation