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

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions

Skew-product maps and heterodimensional cycles

Nesta tese estudam-se as dinâmicas geradas pela criação de ciclos heterodimensionais, seja do tipo parcialmente hiperbólicas com folheações invariantes e dinâmica central unidimensional, seja associada a produtos torcidos. Num primeiro cenário, considera-se uma família, a um parâmetro, de difeomorfismos exibindo um desdobramento de um ciclo heterodimensional associado a duas selas com diferentes índices e cuja dinâmica central e dada por um difeomorfismo côncavo. O estudo da dinâmica semi-local desta família, depois do desdobramento do ciclo, e então reduzido a análise de um sistema iterado de funções, obtido pela composição de potências da aplicação côncava com uma translação. Motivado pelo estudo deste tipo de sistemas iterados de funções, introduz-se um modelo mais geral de sistemas parcialmente hiperbólicos: os produtos torcidos associados a aplicação shift de Bernoulli de n-símbolos. Em ambos os casos, obtêm-se condições que garantem a prevalência de hiperbolicidade ou, em sentido contrário, a prevalência de não hiperbolicidade. No caso dos produtos torcidos e assumindo hipóteses de não hiperbolicidade, prova-se a existência de uma medida invariante, ergódica e não-hiperbólica com um suporte não trivial. Encontra-se ainda um limite superior para o crescimento do número de orbitas periódicas. Introduz-se ainda uma família modelo de difeomorfismos, a dois parâmetros, em que um dos parâmetros está relacionado com o desdobramento do ciclo heterodimensional do tipo descrito acima, e o outro associado a uma função côncava especial que fornece a dinâmica central. Neste caso e possível localizar, em função dos dois parâmetros, intervalos escalonados de hiperbolicidade e de não hiperbolicidade e em simultâneo descrever as bifurcações secundárias associadas a transição das regiões de hiperbolicidade para as de não hiperbolicidade. In this thesis we study the dynamics generated by the creation of heterodimensional cycles, either of the partially hyperbolic type, with invariant foliations and onedimensional central dynamics, or associated to skew-product maps. In the rst scenario, we consider a one-parameter family unfolding a heterodimensional cycle associated to two saddles of di erent indices and such that the central dynamics is given by a concave di eomorphism. The study of the semi-local dynamics of this family, after the unfolding of the cycle, is then reduced to the analysis of a system of iterated functions, obtained by compositions of powers of the concave map with a translation. Motivated by the study of the this kind of iterated systems of functions, we introduce a more general model for partially hyperbolic systems: the skew-product maps associated to the bernoulli shift of n-symbols. In both cases we obtain conditions which ensure prevalence of hyperbolicity or, in the opposite direction, prevalence of non-hyperbolicity. In the skew-products case and under some non-hyperbolicity hypothesis, we prove the existence of an invariant ergodic and non-hyperbolic measure with an uncountable support. We also obtain an upper bound for the growth of the number of periodic orbits. We also introduce a two-parameter family model of di eomorphisms, being one the parameters associated to the unfolding of a heterodimensional cycle of the type described above, and the other associated to a special concave function that gives the central dynamics. In this case, depending on the two parameters, we are able to identify scalled intervals of hyperbolicity and of non-hyperbolicity, and furthermore describe the secondary bifurcations associated to the transition from hyperbolicity to non-hyperbolicity

Hausdorff dimension of heteroclinic intersections for some partially hyperbolic sets

We introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ which have two transitive hyperbolic sets, one is of index 1 (the dimension of the unstable subbundle) and the other is of index 2. We prove that: the unstable set of the first hyperbolic set and the stable set of the second are of Hausdorff dimension nearly 2; the intersection of these unstable and stable sets contains a set of Hausdorff dimension nearly 1.Comment: 28 pages, 8 figure

Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered

Finding first foliation tangencies in the Lorenz system

This is the final version of the article. Available from SIAM via the DOI in this record.Classical studies of chaos in the well-known Lorenz system are based on reduction to the one-dimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations on a particular Poincar e section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the rst time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincar e return map. We consider how the three-dimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits | before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to nd the rst tangency between the stable foliation and the intersection curves in the Poincar e section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP set-up that we are able to detect and readily continue the locus of rst foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point or T-point bifurcations on the locus of rst foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to nd other rst foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor