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Various biological phenomena, like cell differentiation and pattern formation in multicellular organisms, are explained using the bifurcation theory. Molecular network motifs like positive feedback and mutual repressor exhibit bifurcation…
Generic high-dimensional bipartite pure states are overwhelmingly likely to be highly entangled. Remarkably, this ubiquitous phenomenon can already arise in finite-dimensional systems. However, unlike the bipartite setting, the entanglement…
This work extends the theory of reciprocal diagrams in graphic statics to frameworks that are invariant under finite group actions by utilizing the homology and representation theory of cellular cosheaves, recent tools from applied…
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…
We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses that are tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled…
Homotopy links have proven to be one of the most powerful tools of stratified homotopy theory. In previous work, we described combinatorial models for the generalized homotopy links of a stratified simplicial set. For many purposes, in…
We characterize synchronization phenomenon in discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. In terms of multiplicative ergodic properties of the induced linear…
We rigorously show that a class of systems of partial differential equations modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the center manifold(s). A direct consequence…
We show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially…
Linear complementarity problems provide a powerful framework to model nonsmooth phenomena in a variety of real-world applications. In dynamical control systems, they appear coupled to a linear input-output system in the form of linear…
We study the emergence of symmetric oscillatory behavior in multi-agent systems where each agent incorporates a continuous memory of its past states and past rates of change, modeled by distributed retarded and neutral delays. The…
We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can…
Two dynamical models that have been proposed to describe transitions between low and high confinement states (L-H transitions) in confined plasmas are analysed using singularity theory and stability theory. It is shown that the…
For a 2-D coupled PDE-ODE bulk-cell model, we investigate symmetry-breaking bifurcations that can emerge when two bulk diffusing species are coupled to two-component nonlinear intracellular reactions that are restricted to occur only within…
When the steady states at infinity become unstable through a pattern forming bifurcation, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B.Sandstede and…
Bifurcation with symmetry is considered in the case of an isotropy subgroup with a two-dimensional fixed point subspace and non-zero quadratic terms. In general, there are one or three branches of solutions, and five qualitatively different…
Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the…
For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global…
M. Field [5] refers to an unpublished work by J. Ize for a result that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. The main ingredient of the proof is the…
Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the…