31 research outputs found
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 -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. ...
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