Unstable dimension variability and heterodimensional cycles in the border-collision normal form.

Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to emphasize the existence of these phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise linear, they are relatively amenable to an exact analysis. We explicitly identify parameter values for heterodimensional cycles and argue that the existence of heterodimensional cycles between two given saddles can be dense in parameter space. We numerically identify key bifurcations associated with unstable dimension variability by studying a one-parameter subfamily that transitions continuously from where periodic solutions are all saddles to where they are all repellers. This is facilitated by fast and accurate computations of periodic solutions; indeed the piecewise-linear form should provide a useful testbed for further study.

Creation of discontinuities in circle maps.

Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and 'threshold' systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

Hierarchy and polysynchrony in an adaptive network.

We describe a simple adaptive network of coupled chaotic maps. The network reaches a stationary state (frozen topology) for all values of the coupling parameter, although the dynamics of the maps at the nodes of the network can be nontrivial. The structure of the network shows interesting hierarchical properties and in certain parameter regions the dynamics is polysynchronous: Nodes can be divided in differently synchronized classes but, contrary to cluster synchronization, nodes in the same class need not be connected to each other. These complicated synchrony patterns have been conjectured to play roles in systems biology and circuits. The adaptive system we study describes ways whereby this behavior can evolve from undifferentiated nodes.

Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems.

In this paper we study the qualitative dynamics of piecewise-smooth slow-fast systems (singularly perturbed systems) which are everywhere continuous. We consider phase space topology of systems with one-dimensional slow dynamics and one-dimensional fast dynamics. The slow manifold of the reduced system is formed by a piecewise-continuous curve, and the differentiability is lost across the switching surface. In the full system the slow manifold is no longer continuous, and there is an O(ɛ) discontinuity across the switching manifold, but the discontinuity cannot qualitatively alter system dynamics. Revealed phase space topology is used to unfold qualitative dynamics of planar slow-fast systems with an equilibrium point on the switching surface. In this case the local dynamics corresponds to so-called boundary-equilibrium bifurcations, and four qualitative phase portraits are uncovered. Our results are then used to investigate the dynamics of a box model of a thermohaline circulation, and the presence of a boundary-equilibrium bifurcation of a fold type is shown.

Imperfect homoclinic bifurcations.

Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.

Melnikov analysis of chaos in a simple epidemiological model.

Melnikov's method is applied to an SIR model of epidemic dynamics with a periodically modulated nonlinear incidence rate. This analysis establishes mathematically, for the first time, the existence of chaotic motion in these models. A related technique also makes it possible to prove that homoclinic bifurcations occurs in the model.