相关论文: Width of homoclinic zone for quadratic maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the H\'enon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family…

We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincar\'e) maps and show that the…

For the family of Lozi maps, we study homoclinic points for the saddle fixed point $X$ in the first quadrant. Specifically, in the parameter space, we examine the boundary of the region in which homoclinic points for $X$ exist. For all…

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than $1$ but…

We study the fate of two-dimensional quadratic band crossing topological phases under a one-dimensional quasiperiodic modulation. By employing numerically exact methods, we fully characterize the phase diagram of the model in terms of…

We consider C^{2} Henon-like families of diffeomorphisms of R^{2} and study the boundary of the region of parameter values for which the nonwandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of…

The asymptotic distance between trajectories $d_{\infty}$, is studied in detail to characterize the occurrence of chaos. We show that this quantity is quite distinct and complementary to the Lyapunov exponents, and it allows for a…

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially…

In this paper, we deal with limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order 2 by studying expansions of the first order Melnikov functions near the loop and coefficients in these expansions. More…

I review recent work on the ``phase diagram'' of the one-dimensional complex Ginzburg-Landau equation for system sizes at which chaos is extensive. Particular attention is paid to a detailed description of the spatiotemporally disordered…

The purpose of this paper is to advance the knowledge of the dynamics arising from the creation and subsequent bifurcation of Poincar\'e heteroclinic cycles. The problem is central to dynamics: it has to be addressed if, for instance, one…

We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of the Duffing system with nonlinear fractional damping and external excitation. Using perturbation methods we have found a…

One- and two-parameter families of flows in $R^3$ near an Andronov-Hopf bifurcation (AHB) are investigated in this work. We identify conditions on the global vector field, which yield a rich family of multimodal orbits passing close to a…

We establish closed-form expansions for the number of colorings of a path or cycle on n vertices with colors from 1,...,x such that adjacent vertices are colored differently or with colors from y+1,...x.

We consider the family of quadratic H\'enon diffeomorphisms of the plane ${\bf R}^2$. A map will be said to be a "horseshoe" if its restriction to the nonwandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for…

The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their…

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincar\'e map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits…

The scope of this teaching package is to make a brief introduction to some notions and properties of chaotic systems. We first make a brief introduction to chaos in general and then we show some important properties of chaotic systems using…

We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this `cycling chaos' manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets…

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…