David J. W. Simpson

Bifurcations in piecewise-smooth, continuous systems

Fundamentals of Piecewise-Smooth, Continuous Systems Discontinuous Bifurcations in Planar Systems Codimension-Two, Discontinuous Bifurcations The Growth of Saccharomyces cerevisiae Codimension-Two, Border-Collision Bifurcations Periodic Solutions and Resonance Tongues Neimark-Sacker-Like Bifurcations

Neimark–Sacker Bifurcations in Planar, Piecewise-Smooth, Continuous Maps

The multipliers of a fixed point of a piecewise-smooth, continuous map may change discontinuously as the fixed point crosses a discontinuity under smooth variation of parameters. We study the case when the multipliers “jump” from inside to outside the unit circle, and we assume the map is two-dimensional and piecewise-affine. The resulting dynamics is sometimes similar to the Neimark–Sacker bifurcation of a smooth map in which an attracting periodic or quasiperiodic orbit is created as the fixed point loses stability. However, the bifurcation is often much more complex, with multiple (chaotic) attractors, saddles, and repellors created or destroyed.

Mixed-mode oscillations in a stochastic, piecewise-linear system

Abstract We analyse a piecewise-linear FitzHugh–Nagumo model. The system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist. The addition of small noise induces mixed-mode oscillations (MMOs) in the vicinity of the canard point. We determine the effect of each model parameter on the stochastically driven MMOs. In particular we show that any parameter variation (such as a modification of the piecewise-linear function in the model) that leaves the ratio of noise amplitude to time-scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which MMOs occur. In that sense, the MMOs are robust. Furthermore, we show that the piecewise-linear model exhibits MMOs more readily than the classical FitzHugh–Nagumo model for which a cubic polynomial is the only nonlinearity. By studying a piecewise-linear model, we are able to explain results using analytical expressions and compare these with numerical investigations.

Discontinuity induced bifurcations in a model of Saccharomyces cerevisiae.

We perform a bifurcation analysis of the mathematical model of Jones and Kompala [K.D. Jones, D.S. Kompala, Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures, J. Biotechnol. 71 (1999) 105-131]. Stable oscillations arise via Andronov-Hopf bifurcations and exist for intermediate values of the dilution rate as has been noted from experiments previously. A variety of discontinuity induced bifurcations arise from a lack of global differentiability. We identify and classify discontinuous bifurcations including several codimension-two scenarios. Bifurcation diagrams are explained by a general unfolding of these singularities.

Simultaneous border-collision and period-doubling bifurcations

We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that with sufficient nondegeneracy conditions, a locus of period-doubling bifurcations emanates nontangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics is completely classified; in particular, we give conditions that ensure chaos.

Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form

The border-collision normal form is a piecewise-linear continuous map on ℝN that describes the dynamics near border-collision bifurcations of nonsmooth maps. This paper studies a codimension-three scenario at which the border-collision normal form with N = 2 exhibits infinitely many attracting periodic solutions. In this scenario there is a saddle-type periodic solution with branches of stable and unstable manifolds that are coincident, and an infinite sequence of attracting periodic solutions that converges to an orbit homoclinic to the saddle-type solution. Several important features of the scenario are shown to be universal, and three examples are given. For one of these examples, infinite coexistence is proved directly by explicitly computing periodic solutions in the infinite sequence.

Stochastic Regular Grazing Bifurcations

A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude

Unfolding a codimension-two, discontinuous, Andronov-Hopf bifurcation.

\varepsilon

The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise

. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms or, if there exists an attracting period-

Stochastic Perturbations of Periodic Orbits with Sliding

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