Homoclinic orbits: Since Poincaré till today

The history and the contemporary results in homoclinic orbits are reported

Blue sky catastrophe in singularly perturbed systems

We show that the blue sky catastrophe, which creates a stable periodic orbit whose length increases with no bound, is a typical phenomenon for singularly-perturbed (multi-scale) systems with at least two fast variables. Three distinct mechanisms of this bifurcation are described. We argue that it is behind a transition from periodic spiking to periodic bursting oscillations

On some mathematical topics in classic synchronization

A few mathematical problems arising in the classical synchronization theory are discussed, especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies done by Cartwright and Littlewood. Today we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive descriptiob of bifurcational phenomena in the two-parameter case. Of a particular value among ones is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori

A new simple bifurcation of a periodic orbit of "blue sky catastrophe'' type

In this paper, we study a global bifurcation of codimension one connected with the disappearance (for positive values of a parameter μ) of a saddle-node periodic orbit L0 under the condition that all orbits from the locally unstable manifold Wu of L0 tend to L0 as t → +∞. Conditions are presented which guarantee the blue sky catastrophe: the appearance of a stable periodic orbit Lμ which exists for any small positive values of μ but its length and period unboundedly increase as μ → + 0

Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial

A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique allows for uncovering the stunning complexity and universality of bi-parametric structures and detect their organizing centers - codimension-two T-points and separating saddles in the kneading-based scans of the iconic Lorenz equation from hydrodynamics, a normal model from mathematics, and a laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201

On dynamical properties of diffeomorphisms with homoclinic tangencies

We study bifurcations of a homoclinic tangency to a saddle fixed point without non-leading multipliers. We give criteria for the birth of an infinite set of stable periodic orbits, an infinite set of coexisting saddle periodic orbits with different instability indices, non-hyperbolic periodic orbits with more than one multiplier on the unit circle, and an infinite set of stable closed invariant curves (invariant tori). The results are based on the rescaling of the first-return map near the orbit of homoclinic tangency, which is shown to bring the map close to one of four standard quadratic maps, and on the analysis of the bifurcations in these maps

Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

We show that maps with infinitely many homoclinic tangencies of arbitrarily high orders are dense among real-analytic area-preserving diffeomorphisms in the Newhouse regions

On some mathematical topics in classical synchronisation

A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies done by Cartwright and Littlewood. Today we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive description of bifurcational phenomena in the two-parameter case. Of a particular value among ones is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we 1 prove a number of theorems on birth and breakdown of nonsmooth invariant tori