Related papers: Visualizing Attractors of the Three-Dimensional Ge…
An account is given of the features, of the kind pertaining to q-statistics, of the dynamics at the one-dimensional critical attractors associated to the three familiar routes to chaos, intermittency, period doubling and quasiperiodicity.…
In this article, we study a two-parameter family of rotating rank-one maps defined on $\textbf{S}^1\times [1, 1+b]\times \textbf{S}^1$, with $b\gtrsim 0$, whose dynamics is characterised by a coupling of a family of planar maps exhibiting…
Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect…
A replicator equation with mutation processes is numerically studied. Without any mutations, two characteristics of the replicator dynamics are known: an exponential divergence of the dominance period, and hierarchical orderings of the…
We prove the saturation of a generalized partially hyperbolic attractor of a $C^2$ map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a $C^1$-generic diffeomorphism has zero volume. In…
We study the dynamics of the attractor of the doubling map with an asymmetrical hole by associating to each hole an element of the lexicographic world. A description of the topological entropy function is given. We show that the set of…
Many chaotic dynamical systems of physical interest present a strong form of nonhyperbolicity called unstable dimension variability (UDV), for which the chaotic invariant set contains periodic orbits possessing different numbers of unstable…
We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|\lambda\gamma| = 1$. For a proper three-parameter unfolding that splits the tangency, varies…
A one-parameter family of time-reversible systems on $\mathbb{T}^3$ is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the…
We present a multidimensional flow exhibiting a Rovella-like attractor: a transitive invariant set with a non-Lorenz-like singularity accumulated by regular orbits and a multidimensional non-uniformly expanding invariant direction.…
Motivated by bouncing motion of an inelastic particle on a vibrating board, a simple two-dimensional map is constructed and its behavior is studied numerically. In addition to the typical route to chaos through a periodic doubling…
We study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter (mean field and amplitude-frequency correlator) reduction. The main result of the paper is the exact solution of the…
In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for both discrete and continuous dynamical systems globally defined in $\mathbb{R}^3$. We also see that the techniques used…
We study in this paper global properties, mainly of topological nature, of attractors of discrete dynamical systems. We consider the Andronov-Hopf bifurcation for homeomorphisms of the plane and establish some robustness properties for…
We report on the experimental investigation of gluing bifurcations in the analog electronic circuit which models a dynamical system of the third order: Lorenz equations with an additional quadratic nonlinearity. Variation of one of the…
The generalized Floquet exponent and the attractiveness portrait (or A-portrait for short) of the attractor and of the smallest invariant closed set are suggested to be used for the study of dynamical systems. Based on the A-portrait, some…
The global bifurcation diagrams for two different one-parametric perturbations ($+\lambda x$ and $+\lambda x^2$) of a dissipative scalar nonautonomous ordinary differential equation $x'=f(t,x)$ are described assuming that 0 is a constant…
Time-periodic solitons of the parametrically driven damped nonlinear Schr\"odinger equation are obtained as solutions of the boundary-value problem on a two-dimensional spatiotemporal domain. We follow the transformation of the periodic…
We survey the global dynamics of semiflows generated by scalar semilinear parabolic equations which are $\mathbb{SO}(2)$ equivariant under spatial shifts of $x\in \mathbb{S}^1=\mathbb{R}/2\pi\mathbb{Z}$, i.e. $$ u_t = u_{xx} +…
We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter…