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Register a new userHopf-like boundary equilibrium bifurcations involving two foci in Filippov systems
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David J.W. Simpson
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This paper concerns two-dimensional Filippov systems --- ordinary differential equations that are discontinuous on one-dimensional switching manifolds. In the situation that a stable focus transitions to an unstable focus by colliding with a switching manifold as parameters are varied, a simple sufficient condition for a unique local limit cycle to be created is established. If this condition is violated, three nested limit cycles may be created simultaneously. The result is achieved by constructing a Poincare map and generalising analytical arguments that have been employed for continuous systems. Necessary and sufficient conditions for the existence of pseudo-equilibria (equilibria of sliding motion on the switching manifold) are also determined. For simplicity only piecewise-linear systems are considered.
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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.
Boundary equilibria bifurcation (BEB) arises in piecewise-smooth systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which limit as some $epsilon to 0$ to piecewise-smooth (PWS) systems which undergo a BEB. This work completes a classification for codimension-1 singularly perturbed BEB in the plane initiated by the present authors in [19], using a combination of tools from PWS theory, geometric singular perturbation theory (GSPT) and a method of geometric desingularization known as blow-up. After deriving a local normal form capable of generating all 12 singularly perturbed BEBs, we describe the unfolding in each case. Detailed quantitative results on saddle-node, Andronov-Hopf, homoclinic and codimension-2 Bogdanov-Takens bifurcations involved in the unfoldings and classification are presented. Each bifurcation is singular in the sense that it occurs within a domain which shrinks to zero as $epsilon to 0$ at a rate determined by the rate at which the system loses smoothness. Detailed asymptotics for a distinguished homoclinic connection which forms the boundary between two singularly perturbed BEBs in parameter space are also given. Finally, we describe the explosive onset of oscillations arising in the unfolding of a particular singularly perturbed boundary-node (BN) bifurcation. We prove the existence of the oscillations as perturbations of PWS cycles, and derive a growth rate which is polynomial in $epsilon$ and dependent on the rate at which the system loses smoothness. For all the results presented herein, corresponding results for regularized PWS systems are obtained via the limit $epsilon to 0$.
The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if and only if it is exponentially stable for $F$ when the constant component of $F$ is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for $F$. We also show exponential stability is preserved under small perturbations to the pieces of $F$. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of $F$ have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of $x^*$ from the general Filippov system $f$ to the simpler system $F$. Yet in general this problem remains difficult. We provide a four-dimensional example of $F$ for which orbits appear to converge to $x^*$ in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of $F$ can in this case be reduced to the combination of a one-dimensional return map and a scalar function.
We consider smooth systems limiting as $epsilon to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < epsilon ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $epsilon-$dependent domain which shrinks to zero as $epsilon to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $epsilon-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.
We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR^[2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed interaction networks, of which networks of chemical reactions are a special case.
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