Strange attractors in periodically-kicked degenerate Hopf bifurcations

We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodically-kicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a new multi-parameter version of the theory of rank one maps developed by Wang and Young.Comment: 16 page

Unfolding chaotic quadratic maps --- parameter dependence of natural measures

We consider perturbations of quadratic maps $f_a$ admitting an absolutely continuous invariant probability measure, where $a$ is in a certain positive measure set $\mathcal{A}$ of parameters, and show that in any neighborhood of any such an $f_a$, we find a rich fauna of dynamics. There are maps with periodic attractors as well as non-periodic maps whose critical orbit is absorbed by the continuation of any prescribed hyperbolic repeller of $f_a$. In particular, Misiurewicz maps are dense in $\mathcal{A}$. Almost all maps $f_a$ in the quadratic family is known to possess a unique natural measure, that is, an invariant probability measure $\mu_a$ describing the asymptotic distribution of almost all orbits. We discuss weak*-(dis)continuity properties of the map $a\mapsto \mu_a$ near the set $\mathcal{A}$, and prove that almost all maps in $\mathcal{A}$ have the property that $\mu_a$ can be approximated with measures supported on periodic attractors of certain nearby maps. On the other hand, for any $a \in \mathcal{A}$ and any periodic repeller $\Gamma_a$ of $f_a$, the singular measure supported on $\Gamma_a$ can also approximated with measures supported on nearby periodic attractors. It follows that $a\mapsto \mu_a$ is not weak*continuous on any full-measure subset of $(0,2]$. Some of these results extend to unimodal families with critical point of higher order, and even to not-too-flat flat topped families

Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms

We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction)

Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D, N=2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results.Comment: 55 pages; Latex fil

The boundary of hyperbolicity for Henon-like families

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 hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to this bifurcation parameter and even, to some extent, at the bifurcation parameter.Comment: 32 pages, 11 figures. Several minor revisions, additional figures, clarifications of some argument

From limit cycles to strange attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.Comment: 27 page