Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronization, a classical paradigm of order. We survey the main theory behind complete, generalized and phase synchronization phenomena in...

Topics: Chaotic Dynamics, Nonlinear Sciences

Source: http://arxiv.org/abs/1703.08296

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of...

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1512.02148

We prove Knudsen's law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles. Dynamics are randomly perturbed and the corresponding random map studied under a skew-type deterministic representation which is shown to be ergodic and exact.

Source: http://arxiv.org/abs/1006.4831v1

We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.

Source: http://arxiv.org/abs/1212.1931v1