Less Is More I: A Pessimistic View of Piecewise Smooth Bifurcation Theory

Glendinning, Paul

doi:10.1007/978-3-319-55642-0_13

Cited by 4 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They hold for a good range of models, they are informative, but there is much extra detail that they do not provide and they do not attempt a complete topological classifications. Given the hazards created by the proliferation of bifurcations in PWS systems outlined in [4], we consider the existence of these results a cause for optimism, and they provide a template for the expression of further descriptions of PWS dynamics.…”

Section: Resultsmentioning

confidence: 99%

“…The parameter µ is considered to be the bifurcation parameter and some results for these maps are described in [4]. Banerjee et al [1] show that the border collision normal form has parameters with (a) a trapping region; and (b) transverse intersections of stable and unstable manifolds and hence quasi-one-dimensional attractors: this has been called robust chaos.…”

Section: The Border Collision Normal Form: Young's Theoremmentioning

confidence: 99%

“…However, as argued in [4], the number of bifurcations in piecewise smooth (PWS) systems increases alarmingly with the dimension of the ambient phase space or the complexity of the system, and this may mean that complete descriptions, in the same spirit as would be given for smooth systems, become infeasible and certainly become unwieldy. This creates a problem for mathematicians with a background in smooth bifurcation theory: there are many potentially beautiful problems such as the existence of Shilnikov homoclinic bifurcations with sliding segments in local bifurcations of stationary points of PWS systems [4], but if the general result is that for the boundary equilibrium bifurcation in R n then the local dynamics can contain analogues of any bifurcation of smooth systems in R n , as may well be the case, then it is unclear how to proceed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Less Is More II: An Optimistic View of Piecewise Smooth Bifurcation Theory

Glendinning¹

2017

Trends in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Resultsmentioning

confidence: 99%

Section: The Border Collision Normal Form: Young's Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Less Is More II: An Optimistic View of Piecewise Smooth Bifurcation Theory

Glendinning¹

2017

Trends in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…This suggests that statements regarding limit cycles in higher dimensions will need to be weaker than those presented here. Attempts to make strong statements may lead to a proliferation of cases [76] (if 20 HLBs are not enough already!). But certainly the basic geometric mechanisms described here for the creation of a limit cycle (e.g.…”

Section: Discussionmentioning

confidence: 99%

A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

Simpson

2018

Physics Letters A

View full text Add to dashboard Cite

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 statedependent 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 Poincaré 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.

show abstract

“…In order to understand other invariant sets created in BEBs, it is in general not possible to employ dimension reduction techniques that are invaluable for high-dimensional smooth systems of ODEs. BEBs do not involve centre manifolds and so, as with maps [11], it appears that in n-dimensional systems BEBs can be inextricably n-dimensional [12].…”

Section: Introductionmentioning

confidence: 99%

A general framework for boundary equilibrium bifurcations of Filippov systems

Simpson

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

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.

show abstract

Less Is More I: A Pessimistic View of Piecewise Smooth Bifurcation Theory

Cited by 4 publications

References 14 publications

Less Is More II: An Optimistic View of Piecewise Smooth Bifurcation Theory

Less Is More II: An Optimistic View of Piecewise Smooth Bifurcation Theory

A compendium of Hopf-like bifurcations in piecewise-smooth dynamical systems

A general framework for boundary equilibrium bifurcations of Filippov systems

Contact Info

Product

Resources

About