Related papers: A classification of explosions in dimension one

We report the mechanism of the hysteresis in the transition between Regular and Mach reflections. A new discovery is that, the hysteresis loop is in fact the projection of a higher dimensional path, i.e. the valley lines in the surface of…

In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points…

We study main bifurcations of multidimensional diffeomorphisms having a non-transversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a…

The theoretical and numerical understanding of the key concept of topological entropy is an important problem in dynamical systems. Most studies have been carried out on maps (discrete-time systems). We analyse a scenario of global changes…

We present a mechanism for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the…

Many shear flows follow a route to turbulence that has striking similarities to bifurcation scenarios in low-dimensional dynamical systems. Among the bifurcations that appear, crisis bifurcations are important because they cause global…

We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix…

For dynamical systems that switch between different modes of operation, parameter variation can cause periodic solutions to lose or acquire new switching events. When this causes the eigenvalues (stability multipliers) associated with the…

We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest…

Without a model, it is impossible for a geophysicist to study the possibility of forecasting earth quakes. We will define a quantity, the event-degree, in the paper. This quantity plays an important role in the model of quakes forecasting.…

Supernovae explosions of massive stars are nowadays believed to result from a two-step process, with an initial gravitational core collapse followed by an expansion of matter after a bouncing on the core. This scenario meets several…

The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale…

We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor lambda, of the incident angle. These pinball billiards interpolate between a one-dimensional…

Tangent points, especial dynamics existing only in piecewise-smooth systems, usually have dynamical properties like equilibria of smooth systems. Loops connecting tangent points own partly properties of limit cycles and homoclinic loops of…

Triply degenerate fixed points appear in global bifurcations -- homoclinic and heteroclinic tangencies. In order to get Lorenz-like attractors, the dynamics of the first return map along the homoclinic or heteroclinic cycle should be…

Let a system of differential equations possess a saddle-node periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation…

The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in…

Patterns in reaction-diffusion systems near primary bifurcations can be studied locally and classified by means of amplitude equations. This is not possible for excitable reaction-diffusion systems. In this Letter we propose a global…

This article is motivated by the fact that there is a distinction between the descriptions of logical explosion from syntactic and semantic points of view. The discussion is illustrated using the concept of abstract model structures and the…

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider…