Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Codimension 2 bifurcation of twisted double homoclinic loops

AbstractA local active coordinates approach is employed to obtain bifurcation equations of twisted double homoclinic loops. Under the condition of one twisted orbit, we obtain the existence and uniqueness and of the 1–1 double homoclinic loop, 2–1 double homoclinic loop, 2–1 right homoclinic loop, 1–1 large homoclinic loop, 2–1 large homoclinic loop and 2–1 large period orbit. For the case of double twisted orbits, we obtain the existence or non-existence of 1–1 double homoclinic loop, 1–2 double homoclinic loop, 2–1 double homoclinic loop, 2–2 double homoclinic loop, 2–1 large homoclinic loop, 1–2 large homoclinic loop, 2–2 large homoclinic loop, 2–2 right homoclinic loop, 2–2 large homoclinic loop, 2–2 left homoclinic loop and 2–2 large period orbit. Moreover, the bifurcation surfaces and their existence regions are given. Besides, bifurcation sets are presented on the 2 dimensional subspace spanned by the first two Melnikov vectors

Multidimensional Rovella-like attractors

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. Moreover, this attractor has a physical measure with full support but persists along certain0909.1033 submanifolds of the space of vector fields. As in the 3-dimensional Rovella-like attractor, this example is not robust. The construction introduces a class of multidimensional dynamics, whose suspension provides the Rovella-like attractor, which are partially hyperbolic, and whose quotient over stable leaves is a multidimensional endomorphism to which Benedicks-Carleson type arguments are applied to prove non-uniform expansion.Comment: 45 pages, 14 figures; improved introduction with more citations to other relevant related works. To appear in Journal of Differential Equation

Hyperchaotic attractors of three-dimensional maps and scenarios of their appearance

We study bifurcation mechanisms of the appearance of hyperchaotic attractors in three-dimensional maps. We consider, in some sense, the simplest cases when such attractors are homoclinic, i.e. they contain only one saddle fixed point and entirely its unstable manifold. We assume that this manifold is two-dimensional, which gives, formally, a possibility to obtain two positive Lyapunov exponents for typical orbits on the attractor (hyperchaos). For realization of this possibility, we propose several bifurcation scenarios of the onset of homoclinic hyperchaos that include cascades of both supercritical period-doubling bifurcations with saddle periodic orbits and supercritical Neimark-Sacker bifurcations with stable periodic orbits, as well as various combinations of these cascades. In the paper, these scenarios are illustrated by an example of three-dimensional Mir\'a map.Comment: 40 pages, 24 figure

Singular-hyperbolic attractors are chaotic

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $u$-Gibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.Comment: 55 pages, extra figures (now a total of 16), major rearrangement of sections and corrected proofs, improved introductio

Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics

The research presented in this PhD thesis within the framework of nonlinear deterministic dynamical systems depending on parameters. The work is divided into four Chapters, where the first is a general introduction to the other three. Chapter two deals with the investigation of a time-periodic three-dimensional system of ordinary differential equations depending on three parameters, the Lorenz-84 model with seasonal forcing. The model is a variation on an autonomous system proposed in 1984 by the meteorologist E. Lorenz to describe general atmospheric circulation at mid latitude of the northern hemisphere. ... Zie: Summary