Bifurcation diagrams which allow one to visualise changes in the behaviour of low dimensional nonlinear maps as a parameter is altered are common. Visualisation in higher dimensional systems is more difficult. A straightforward method to visualise bifurcations in flows of high dimensional nonlinear dynamical systems is presented, using the Lorenz ’96 systems with dimension 8 and dimension 40 as...

This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that +i should be simple eigenvalues of the linearized right-hand side at criticality. Then the Lyapunov-Schmidt method allows a reduction to a scalar equation G(x...

The spontaneous symmetry breaking in a vibro-fluidized low-density granular gas in three connected compartments is investigated. When the total number of particles in the system becomes large enough, particles distribute themselves unequally among the three compartments. Particles tend to concentrate in one of the compartments, the other two having the (relatively small) same average number of ...

Abstract. We study localised attractors of the parametrically driven, damped nonlinear Schrödinger equation. Time-periodic solitons of this equation are obtained as solutions of the boundary-value problem on a two-dimensional domain. Stability and bifurcations of periodic solitons and their complexes is classified. We show that the bifurcation diagram can be reproduced using a few-mode approxim...

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle-node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obt...

In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [3]. This relation is deeper than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci of diffe...

We study the Kaldor-Kalecki model of business cycles with delay in both the gross product and the capital stock. Simple-zero and double-zero singularities are investigated when bifurcation parameters change near certain critical values. By performing center manifold reduction, the normal forms on the center manifold are derived to obtain the bifurcation diagrams of the model such as Hopf, homoc...

Based on the work of domestic and foreign scholars and the application of chaotic systems theory, this paper presents an investigation simulation of retailer’s demand and stock. In simulation of the interaction, the behavior of the system exhibits deterministic chaos with consideration of system constraints. By the method of space’s reconstruction, the maximal Lyapunov exponent of retailer’s de...

A framework for understanding the global structure of near-integrable n DOF Hamiltonian systems is proposed. To this aim two tools are developed—the energy-momentum bifurcation diagrams and the branched surfaces. Their use is demonstrated on a few near-integrable 3 DOF systems. For these systems possible sources of instabilities are identified in the diagrams, and the corresponding energy surfa...

We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria ar...