Related papers: Higher dimensional topology and generalized Hopf b…

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of…

In this paper we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to find the parameters of a parametric singular system that they play the role of universal unfolding…

We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity…

We consider coupled diffusions in $n$-dimensional space and on a compact manifold and the resulting effective advective-diffusive motion on large scales in space. The effective drift (advection) and effective diffusion are determined as a…

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part…

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and…

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor-repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast…

In this paper we study local and global symmetric Hopf bifurcation in abstract parabolic systems by means of the twisted equivariant degree.

A degenerate dynamical system is characterized by a state-dependent multiplier of the time derivative of the state in the time evolution equation. It can give rise to Hamiltonian systems whose symplectic structure possesses a non-constant…

The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of…

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions $D >2$ and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable…

The Poincar\'e-Hopf Theorem relates the Euler characteristic of a 2-dimensional compact manifold to the local behavior of smooth vector fields defined on it. However, despite the importance of Filippov vector fields, concerning both their…

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a…

A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich…

We construct integrable holomorphic G-structures and flat holomorphic Cartan geometries on every complex Hopf manifold, without using the normal forms given by the Poincar\'e-Dulac Theorem. We provide a new proof of the latter using charts…

We describe some families of differentiable vector fields with the Hopf bifurcation at infinity, without assuming the continuous differentiability. These vector fields have isolated singular points on the plane, and the initial families are…

The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of…

In aggregation-fragmentation processes, a steady state is usually reached in the long time limit. This indicates the existence of a fixed point in the underlying system of ordinary differential equations. The next simplest possibility is an…

Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…

We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish…