Hyperbolic Saddle Research Articles

Direct observation of homoclinic orbits in human heart rate variability.

Homoclinic trajectories of the interbeat intervals between contractions of ventricles of the human heart are identified. The interbeat intervals are extracted from 24-h Holter ECG recordings. Three such recordings are discussed in detail. Mappings of the measured consecutive interbeat intervals are constructed. In the second and in some cases in the fourth iterate of the map of interbeat intervals homoclinic trajectories associated with a hyperbolic saddle are found. The homoclinic trajectories are often persistent for many interbeat intervals, sometimes spanning many thousands of heartbeats. Several features typical for homoclinic trajectories found in other systems were identified, including a signature of the gluing bifurcation. The homoclinic trajectories are present both in recordings of heart rate variability obtained from patients with an increased number of arrhythmias and in cases in which the sinus rhythm is dominant. The results presented are a strong indication of the importance of deterministic nonlinear instabilities in human heart rate variability.

Design and ultimate behavior of RC plates and shells: two case studies

Two cases of design are performed for the hyperbolic paraboloid saddle shell (Lin-Scordelis saddle shell) and the hyperbolic cooling tower (Grand Gulf cooling tower) to check the design strength against a consistent design load, therefore to verify the adequacy of the design algorithm. An iterative numerical computational algorithm is developed for combined membrane and flexural forces, which is based on equilibrium consideration for the limit state of reinforcement and cracked concrete. The design algorithm is implemented in a finite element analysis computer program developed by Mahmoud and Gupta. The amount of reinforcement is then determined at the center of each element by an elastic finite element analysis with the design ultimate load. Based on ultimate nonlinear analyses performed with designed saddle shell, the analytically calculated ultimate load exceeded the design ultimate load from 7% to 34% for analyses with various magnitude of tension stiffening. For the cooling tower problem the calculated ultimate load exceeded the design ultimate load from 26% to 63% with similar types of analyses. Since the effective tension stiffening would vary over the life of the shells due to environmental factors, a degree of uncertainty seems inevitable in calculating the actual failure load by means of numerical analysis. Even though the ultimate loads are strongly dependent on the tensile properties of concrete, the calculated ultimate loads are higher than the design ultimate loads for both design cases. For the cases designed, the design algorithm gives a lower bound on the design ultimate load with respect to the lower bound theorem. This shows the adequacy of the design algorithm developed, at least for the shells studied. The presented design algorithm for the combined membrane and flexural forces can be evolved as a general design method for reinforced concrete plates and shells through further studies involving the performance of multiple designs and the analyses of differing shell configurations.

Un critère d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteurs

We consider dimension 3 vector fields (resp. diffeomorphisms), in a neighborhood of a hyperbolic saddle. We give a criterion to decide if the imaginary part (resp. the angular part) of the eigenvalues of the linear part of the dynamical system at the fixed point is a topological conjugacy invariant if we assume that the conjugacy maps a non-spiraling curve onto another one. We apply this result to the situation of diffeomorphisms and vector fields with a quasi-transversal homoclinic orbit. To cite this article: E. Dufraine, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 53–58

Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three

Dans cet article, nous etudions l'application de Dulac associee a un champ de vecteurs definie au voisinage d'une singularite de type col, autrement dit l'application de passage de coin d'une section transverse a la variete stable de dimension 2 a une section transverse a la variete instable de dimension 1. Nous montrons, qu' a une fonction (dite de Mourtada) multiplicative pres, cette application s'ecrit comme si le champ etait localement lineaire.

Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D

The ideal incompressible magnetohydrodynamic equations in two dimensions are studied analytically and numerically. We focus on a standard scenario of singularity formation where level sets of the magnetic stream function contain a hyperbolic saddle. Analytically we obtain an exponential estimate on the closing of the angle under the assumption that the velocity remains bounded. This result is supported by high-resolution numerical simulations using adaptive structured mesh refinement. © 2000 John Wiley & Sons, Inc.

Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D

The ideal incompressible magnetohydrodynamic equations in two dimensions are studied analytically and numerically. We focus on a standard scenario of singularity formation where level sets of the magnetic stream function contain a hyperbolic saddle. Analytically we obtain an exponential estimate on the closing of the angle under the assumption that the velocity remains bounded. This result is supported by high-resolution numerical simulations using adaptive structured mesh refinement. © 2000 John Wiley & Sons, Inc.

Abnormal escape rates from nonuniformly hyperbolic sets

Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log |J^u f|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug.

On the Integrability of Two-Dimensional Flows

This paper deals with the notion of integrability of flows or vector fields on two-dimensional manifolds. We consider the following two key points about first integrals: (1) They must be defined on the whole domain of definition of the flow or vector field, or defined on the complement of some special orbits of the system; (2) How are they computed? We prove that every local flow ϕ on a two-dimensional manifold M always has a continuous first integral on each component of M\\Σ where Σ is the set of all separatrices of ϕ. We consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two-dimensional flow or vector field. Finally, we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus, of a non-resonant hyperbolic node, and of a Siegel hyperbolic saddle.

Communicating with chaos using two-dimensional symbolic dynamics

Symbolic representations of controlled chaotic orbits produced by signal generators can be used for communicating. In this Letter, communicating with chaos is investigated by using more realistic dynamical systems described by two-dimensional invertible maps. The major difficulty is how to specify a generating partition so that a good symbolic dynamics can be defined. A solution is proposed whereby hyperbolic chaotic saddles embedded in the attractor are exploited for digital encoding. Issues addressed include the channel capacity and noise immunity when these saddles are utilized for communication.

On the behavior of hyperbolic neutral points in two-dimensional ideal magnetohydrodynamics.

We study ideal incompressible magnetohydrodynamics in two dimensions. We obtain an exponential estimate on the closing of the angle at hyperbolic saddle points of the magnetic stream function under the assumption that the velocity remains bounded. The analytic results are supported by numerical simulations. These results give evidence against a standard scenario for singularity formation for these equations.

Nonexistence of Simple Hyperbolic Blow-Up for the Quasi-Geostrophic Equation

The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak [7] suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible. Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler. These results were announced in [9], but with a slight difference in the definition of a simple hyperbolic saddle. The definition given in Section 4 generalizes the one given in the announcement. See also Constantin [4], discussed in Section 7, Remark 5 below. The whole work described in this paper is basically part of the author's thesis. I am particularly grateful to my thesis advisor Charles Fefferman for his attention, support, guidance and advice. I am indebted to D. Christodoulou and P. Constantin for helpful corrections and suggestions. I wish to thank A. Majda for suggesting the subject and E. Tabak for discussions and com-

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 ∈ ℂ. Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze 2πi )=H○S(z) (1). IfH λ is an unfolding of diffeomorphisms depending on λ ∈ (ℂ,0), withH 0=Id, one introduces its ideal $$\mathcal{I}_H$$ . It is the ideal generated by the germs of coefficients (a i (λ), 0) at 0 ∈ ℂ k , whereH λ(z)−z=Σa i (λ)z i . Then one can find a parameter solutionS λ (z) of (1) which has at each pointz 0 belonging to the domain of definition ofS 0, an expansion in seriesS λ(z)=z+Σb i (λ)(z−z 0) i with $$(b_i ,0) \in \mathcal{I}_H$$ , for alli. This result may be applied to the bifurcation theory of vector fields of the plane. LetX λ be an unfolding of analytic vector fields at 0 ∈ ℝ2 such that this point is a hyperbolic saddle point for each λ. LetH λ(z) be the holonomy map ofX λ at the saddle point and $$\mathcal{I}_H$$ its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals σ, τ, transversal to the separatrices of the saddle point, such that the difference between the transition mapD λ(z) and the identity is divisible in the ideal $$\mathcal{I}_H$$ . Finally, suppose thatX λ is an unfolding of a saddle connection for a vector fieldX 0, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz∈σ, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX λ, is finite and can be computed explicity in terms of the Bautin ideal.

Cyclicity of planar polycycles and ensembles with codimension 3 degeneration through a saddle-node and two hyperbolic saddles

The cyclicity of four classes of codimension 3 planar polycycles and ensembles containing a saddle node and two hyperbolic saddles is dealt with. The exact cyclicity or cyclicity bound of them is obtained by finitely smooth normal form theory.

On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Ultimate behavior of RC hyperbolic paraboloid saddle shell

The ultimate behavior of a reinforced concrete hyperbolic paraboloid saddle shell under uniformly distributed vertical load is investigated using an inelastic, large displacement finite-element program originally developed at North Carolina State University. Unlike with the author\'s previous study which shows that the saddle shell possesses a tremendous capacity to redistribute the stresses, introducing tension stiffening in the model the cracks developed are no longer through cracks and formed as primarily bending cracks. Even though with small tension stiffening effect, the behavior of the shell is changed markedly from the one without tension stiffening effect. The load-deflection curves are straight and the slope of the curves is quite steep and remains unchanged with varying the tension stiffening parameters. The failure of the shell took place quite suddenly in a cantilever mode initiated by a formation of yield lines in a direction parallel to the support-to-support diagonal. The higher the tension stiffening parameters the higher is the ultimate load. The present study shows that the ultimate behavior of the shell primarily depends on the concrete tensile characteristics, such as tensile strength (before cracking) and the effective tension stiffening (after cracking). As the concrete characteristics would vary over the life of the shell, a degree of uncertainty is involved in deciding a specified ultimate strength of the saddle shell studied. By the present study, however, the overload factors based on ACI 318-95 are larger than unity for all the cases studied except that the tension stiffening parameter is weak by 3 with and without the large displacement effect, which shows that the Lin-Scordelis saddle shell studied here is at least safe.

Smoothness property for bifurcation diagrams

Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to sub-analytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with the line of saddle connections. It is possible to prove that the flatness is smooth and to compute its asymptotic properties.

HOMOCLINIC ORBITS FOR 3−DIMENSIONAL SYSTEMS

Suppose a dynamical system dx/dt=F(x;μ), x∈Rs, μ∈Rm, has a hyperbolic saddle at x=0 with a homoclinic loop, for μ=μ0. When μ varies from μ0, the loop will be destroyed in general. For s=2, Perko proved that, if μ varies on an (m−1) dimensional hypersurface, then the system remains to admit homoclinic orbit. We consider here the same problem for s=3. The result is: if μ varies on an (m−2) hypersurface, then the system remains to admit homoclinic orbit.

Chaos in the Duffing System Subjected to Parametric and External Excitation. 1st Report. Homoclinic Behavior near the Unperturbed Separatrices.

Chaotic dynamics of the Duffing oscillator with parametric and external forcing are studied. The unperturbed system, in which the damping and external forcing terms are removed and the parametric forcing amplitude or frequency is zero, has a hyperbolic saddle with a pair of separatrices. The following two cases are analyzed : (i) the parametric forcing amplitude is small ; and (ii) the parametic forcing amplitude is large but the frequency is small. The external forcing amplitude and damping coefficient are assumed to be small for both cases. Using a version of Melnikov's method, we obtain conditions under which the stable and unstable manifolds of a normally hyperbolic invariant torus intersect transversely and chaotic dynamics may occur near the unperturbed separatrices. Such chaotic motions are appropriately characterized by a generalization of the Bernoulli shift. We also describe the influence of parametric forcing on conditions for the occurrence of chaos. Numerical simulation results are given to demonstrate the theoretical results. Sustained chaotic motions wandering near the unperturbed homoclinic orbits are found.

Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents the temperature evolving on the two dimensional boundary of a rapidly rotating half space with small Rossby and Ekman numbers and constant potential vorticity. The possibility of frontogenesis within this approximation is an important issue in the context of geophysical flows. A striking mathematical and physical analogy is developed between the structure and formation of singular solutions of this quasi-geostrophic active scalar in two dimensions and the potential formation of finite time singular solutions for the 3-D Euler equations. Detailed mathematical criteria are developed as diagnostics for self-consistent numerical calculations indicating strong front formation. These self-consistent numerical calculations demonstrate the necessity of nontrivial topology involving hyperbolic saddle points in the level sets of the active scalar in order to have singular behaviour; this numerical evidence is strongly supported by mathematical theorems which utilize the nonlinear structure of specific singular integrals in special geometric configurations to demonstrate the important role of nontrivial topology in the formation of singular solutions.

Degenerate and non-trivial hyperbolic 2-polycycles: appearance of two independent Ecalle-Roussarie compensators and Khovanskii's theory

In this paper, we study bifurcations of limit cycles in the neighborhood of hyperbolic polycycles with two vertices which are resonant hyperbolic saddle points and such that the product of their ratios of hyperbolicity is equal to one. We will be interested in hyperbolic polycycles with a first return map which is not equal to the identity map and we will show that their cyclicity in sufficiently differentiable unfoldings is finite and this cyclicity is expressed as an explicit function of the non-triviality order.