Hyperbolic Points Research Articles

On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system

In the present paper, we give a new treatment of the mechanism of generation of chaotic dynamics in a perturbed conservative system in a neighborhood of the separatrix contour of a hyperbolic singular point of the unperturbed system. We theoretically prove and justify by three numerical examples of classical Hamiltonian systems with one and a half degrees of freedom and by an example of a simply conservative three-dimensional system that the complication of the dynamics in a conservative system as the perturbation increases is caused by a nonlocal effect of multiplication of hyperbolic and elliptic cycles (and the tori surrounding them), which has nothing in common with the mechanism of separatrix splitting in classical Hamiltonian mechanics.

Non-resonance [formula omitted] homoclinic bifurcation with an inclination flip

Local active coordinates approach is employed to study the bifurcation of a non-resonance three-dimensional smooth system which has a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues - α , - β , 1 satisfying α > β > 0 . A homoclinic orbit is called an inclination-flip homoclinic orbit if the strong inclination property of the stable manifold is violated. In this paper, we show the existence of 1-homoclinic orbit, 1-periodic orbit, 2 n -homoclinic orbit and 2 n -periodic orbit in the unfolding of an inclination-flip homoclinic orbit. And we figure out the bifurcation diagram based on the existence region of the corresponding bifurcation.

Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps

We consider a one-parameter family of area preserving maps in a neighbourhood of an elliptic fixed point. As the parameter evolves hyperbolic and elliptic periodic orbits of different periods are created. The exceptional resonances of order less than 5 have to be considered separately. The invariant manifolds of the hyperbolic periodic points bound islands containing the elliptic periodic points. Generically, these manifolds split. It turns out that the inner and outer splittings are different under suitable conditions. We provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as a function of geometric properties. The numerical agreement is illustrated using mainly the Hénon map as an example.

A general view of pseudoharmonics and pseudoantiharmonics to calculate external arguments of Douady and Hubbard

Harmonics give us a compact formula and a powerful tool in order to calculate the external arguments of the last appearance hyperbolic components and Misiurewicz points of the Mandelbrot set in some particular cases. Antiharmonics seem however to have no application. In this paper, we give a general view of pseudoharmonics and pseudoantiharmonics, as a generalization of harmonics and antiharmonics. Pseudoharmonics turn out to be a more powerful tool than harmonics since they allow the calculation of external arguments of the Mandelbrot set in many more cases. Likewise, unlike antiharmonics, pseudoantiharmonics turn out to be a powerful tool to calculate external arguments of the Mandelbrot set in some cases.

Semiclassical analysis of spin systems near critical energies

The spectral properties of $su(2)$ Hamiltonians are studied for energies near the critical classical energy $\epsilon_c$ for which the corresponding classical dynamics presents hyperbolic points (HP). A general method leading to an algebraic relation for eigenvalues in the vicinity of $\epsilon_c$ is obtained in the thermodynamic limit, when the semi-classical parameter $n^{-1}=(2s)^{-1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near $\epsilon_c$, are also computed and shown to be in agreement with the numerical results.

SLIP-SQUASHING FACTORS AS A MEASURE OF THREE-DIMENSIONAL MAGNETIC RECONNECTION

A general method for describing magnetic reconnection in arbitrary three-dimensional magnetic configurations is proposed. The method is based on the field-line mapping technique previously used only for the analysis of magnetic structure at a given time. This technique is extended here so as to analyze the evolution of magnetic structure. Such a generalization is made with the help of new dimensionless quantities called "slip-squashing factors". Their large values define the surfaces that border the reconnected or to-be-reconnected magnetic flux tubes for a given period of time during the magnetic evolution. The proposed method is universal, since it assumes only that the time sequence of evolving magnetic field and the tangential boundary flows are known. The application of the method is illustrated for simple examples, one of which was considered previously by Hesse and coworkers in the framework of the general magnetic reconnection theory. The examples help us to compare these two approaches; they reveal also that, just as for magnetic null points, hyperbolic and cusp minimum points of a magnetic field may serve as favorable sites for magnetic reconnection. The new method admits a straightforward numerical implementation and provides a powerful tool for the diagnostics of magnetic reconnection in numerical models of solar-flare-like phenomena in space and laboratory plasmas.

West Adriatic coastal water excursions into the East Adriatic

A pool of less saline surface waters was observed in late June 2006 at the northern edge of the South Adriatic Pit (SAP). Three possible sources were considered: (1) Albanian rivers, (2) local Croatian rivers, or (3) relatively fresh West Adriatic Current (WAC) waters. Available CTD and ADCP data, together with satellite images indicate that WAC waters are the most likely source. This requires an excursion of WAC water across the width of the Adriatic and is especially surprising as low winds and stable atmospheric conditions prevailed in mid/late June. However, quite strong NNW winds occurred during the first 12 days of June, with peak winds close to the western shore. These winds were the result of the translation of the Azorean high to the British Isles, producing strong pressure gradients over the Adriatic. The winds enhanced the WAC during early June 2006, preconditioning a cross-basin eddy circulation that appeared during the wind relaxation and calm conditions. As the unusually calm conditions persisted for more than two weeks, the WAC eddies and filaments grew freely and had enough time to reach middle east Adriatic waters. Navy Coastal Ocean Model (NCOM) simulations, using high-resolution Adriatic bathymetry and realistic atmospheric forcing show that such excursions are plausible and can occur when eddies and instabilities push WAC waters across the hyperbolic flow point separating the WAC and Eastern Adriatic Currents near the Palagruža Sill. During the latter half of June 2006, NCOM simulations show that the hyperbolic point was particularly well formed as an anticyclonic WAC, a cyclonic SAP rim flow, an anticyclonic cell southeast of Lastovo Island, and a cyclonic cell over the centre of the Palagruža Sill all bordered on each other. A simplified channel model suggests that the presence of the escarpment is a critical factor for producing cross-basin exchange of the coastal current following the relaxation of strong winds with a cyclonic wind-stress curl. However, the introduction of the Gargano Peninsula in the simulations was critical to the production of mesoscale eddies in the exchange flow, and such eddies qualitatively agree with the convoluted structures observed in satellite images.

Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by product we prove that the Julia set of a non-renormalizable polynomials with only hyperbolic periodic points is locally connected, and the Branner-Hubbard conjecture. The main tools are the enhanced nest construction (developed in a previous joint paper with Weixiao Shen) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case).

Computational unsolvability of domains of attraction of nonlinear systems

Let S be the domain of attraction of a computable and asymptotically stable hyperbolic equilibrium point of the non-linear system x = f(x). We show that the problem of determining S is computationally unsolvable. We also present an upper bound of the degree of unsolvability of this problem.

C1-STABLE INVERSE SHADOWING CHAIN COMPONENTS FOR GENERIC DIFFEOMORPHISMS

Let f be a diffeomorphism of a compact <TEX>$C^{\infty}$</TEX> manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of <TEX>$C^1$</TEX>-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set <TEX>$\cal{R}(f)$</TEX> of f has <TEX>$C^1$</TEX>-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) <TEX>$C^1$</TEX>-generically, the chain component <TEX>$C_f(p)$</TEX> of f associated to p is hyperbolic if and only if <TEX>$C_f(p)$</TEX> has the <TEX>$C^1$</TEX>-stable inverse shadowing property.

Preserving Synchronization under Characteristic Polynomial Modifications

In this article we present a methodology under which stability and synchronization of a dynamical master/slave system configuration are preserved under specific modifications made to its Jacobian matrix's characteristic polynomial. We propose to modify the coefficients of the associated characteristic polynomial by calculating their value to the m-th power, with m an odd, positive integer. The objective is to show that under these modifications, hyperbolic critical points are preserved along the stable and unstable manifolds. It is also shown that a consequence of this approach is the preservation of the signature of the Jacobian matrix associated with the dynamical system. To illustrate the results we present several examples of well known chaotic attractors.

Robustly expansive homoclinic classes are generically hyperbolic

Let $f: M \to M$ be a diffeomorphism defined in a $d$-dimensional compact boundary-less manifold $M$. We prove that generically$C^1$-robustly expansive homoclinic classes $H(p)$, $p$ an$f$-hyperbolic periodic point, are hyperbolic.

On dynamical systems close to a product of $m$ rotations

We consider one parameter families of analytic vector fields anddiffeomorphisms, including for a parameter value, say $\varepsilon =0$, the product of rotations in $\R^{2m}\times \R^n$ such that forpositive values of the parameter the origin is a hyperbolic point ofsaddle type. We address the question of determining the limit stableinvariant manifold when $\varepsilon$ goes to zero as a subcenterinvariant manifold when $\varepsilon = 0$.

Linearization of families of germs of hyperbolic vector fields

In this article, we develop some techniques to linearize families of smooth vector fields in a neighbourhood of a hyperbolic equilibrium point. In particular, we present the linearizing conjugacy in an explicit way and describe the smoothness of the conjugacy in terms of the eigenvalues of the vector fields.

Propagation of singularities for the wave equation on manifolds with corners

In this paper we describe the propagation of C°° and Sobolev singularities for the wave equation on C°° manifolds with corners M equipped with a Riemannian metric g. That is, for X = M x R t , P = D 2 t - Δ M , and u ∈ H 1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C°° counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web

A model map $Q$ for the Hopf-saddle-node (HSN) bifurcation of fixed points of diffeomorphisms is studied. The model is constructed to describe the dynamics inside an attracting invariant two-torus which occurs due to the presence of quasi-periodic Hopf bifurcations of an invariant circle, emanating from the central HSN bifurcation. Resonances of the dynamics inside the two-torus attractor yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps, where a web of hyperbolic periodic points is expected to occur inside the two-torus attractor. It is conjectured that heteroclinic intersections of the invariant manifolds of the saddle periodic points may give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. To understand this phenomenon, a simple model map of the standard two-torus is developed and studied and the relations with the starting model map $Q$ are discussed.

Magnetic Effects on the Coalescence of Kelvin-Helmholtz Vortices

We simulate the coalescence process of MHD-scale Kelvin-Helmholtz vortices with the electron inertial effects taken into account. Reconnection of highly stretched magnetic field lines within a rolled-up vortex destroys the vortex itself and the coalescence process, which is well known in ordinary fluid dynamics, is seen to be inhibited. When the magnetic field is initially antiparallel across the shear layer, on the other hand, multiple vortices are seen to coalesce continuously because another type of magnetic reconnection prevents the vortex decay. This type of reconnection at the hyperbolic point also changes the field line connectivity and thus leads to large-scale plasma mixing across the shear layer.

Effective diffusion of scalar fields in a chaotic flow

The advection of a tracer field in a fluid flow can create complex scalar structures and increase the effect of weak diffusion by orders of magnitude. One tool to quantify this is to measure the flux of scalar across contour lines of constant scalar. This gives a diffusion equation in area coordinates with an effective diffusion that depends on the structure of the scalar field and, in particular, takes large values when scalar contours become very extended. The present paper studies the properties of this effective diffusion using a mixture of analytical and numerical tools. First the presence of hyperbolic stationary points, that is, saddles, in the scalar concentration field is investigated analytically, and it is shown that these give rise to singular spikes in the effective diffusion. This is confirmed in numerical simulations in which complex scalar fields are generated using a time-periodic flow. Issues of numerical resolution are discussed and results are given on the dependence of the effective diffusion on grid resolution and discretization in area or scalar values. These simulations show complex dependence of the effective diffusion on time as saddle points appear and disappear in the scalar field. It is found that time averaging (in the presence of an additional scalar source term) removes this dependence to leave robust results for the effective diffusion.

Local product structure for expansive homeomorphisms

Let f : M → M be an expansive homeomorphism with dense topologically hyperbolic periodic points, M a closed manifold. We prove that there is a local product structure in an open and dense subset of M. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus.

Slow decay of concentration variance due to no-slip walls in chaotic mixing

Chaotic mixing in a closed vessel is studied experimentally and numerically in different two-dimensional (2D) flow configurations. For a purely hyperbolic phase space, it is well known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.