Hyperbolic Points Research Articles

Successive approximation: A survey on stable manifold of fractional differential systems

Abstract This paper outlines a reliable strategy to approximate the local stable manifold near a hyperbolic equilibrium point for nonlinear systems of differential equations of fractional order. Furthermore, the local behavior of these systems near a hyperbolic equilibrium point is investigated based on the fractional Hartman-Grobman theorem. The fractional derivative is described in the Caputo sense. The solution existence, uniqueness, stability and convergence of the proposed scheme is discussed. Finally, the validity and applicability of our approach is examined with the use of a solvable model method.

Expansive flows of the three-sphere

In this article we show that the three-dimensional sphere admits transitive expansive flows in the sense of Komuro with hyperbolic equilibrium points. The result is based on a construction that allows us to see the geodesic flow of a hyperbolic three-punctured two-dimensional sphere as the flow of a smooth vector field on the three-dimensional sphere.

Codimension one structurally stable chain classes

The well-known stability conjecture of Palis and Smale states that if a diffeomorphism is structurally stable, then the chain recurrent set is hyperbolic. It is natural to ask if this type of result is true for an individual chain class, that is, whether or not every structurally stable chain class is hyperbolic. Regarding the notion of structural stability, there is a subtle difference between the case of a whole system and the case of an individual chain class. The latter is more delicate and contains additional difficulties. In this paper we prove a result of this type for the latter, with an additional assumption of codimension 1. Precisely, let f f be a diffeomorphism of a closed manifold M M and let p p be a hyperbolic periodic point of f f of index 1 or dim ⁡ M − 1 \dim M-1 . We prove if the chain class of p p is structurally stable, then it is hyperbolic. Since the chain class of p p is not assumed in advance to be locally maximal, and since the counterpart of it for the perturbation g g is defined not canonically but indirectly through the continuation p g p_g of p p , the proof is quite delicate.

Transport and buckling dynamics of an elastic fibre in a viscous cellular flow

We study, using both experiment and theory, the coupling of transport and shape dynamics for elastomeric fibres moving through an inhomogeneous flow. The cellular flow, created electromagnetically in our experiment, comprises many identical cells of counter-rotating vortices, with a global flow geometry characterized by a backbone of stable and unstable manifolds connecting hyperbolic stagnation points. Our mathematical model is based upon slender-body theory for the Stokes equations, with the fibres modelled as inextensible elastica. Above a certain threshold of the control parameter, the elasto-viscous number, transport of fibres is mediated by their episodic buckling by compressive stagnation point flows, lending an effectively chaotic component to their dynamics. We use simulations of the model to construct phase diagrams of the fibre state (buckled or not) near stagnation points in terms of two variables that arise in characterizing the transport dynamics. We show that this reduced statistical description quantitatively captures our experimental observations. By carefully reproducing the experimental protocols and time scales of observation within our numerical simulations, we also quantitatively explain features of the measured buckling probability curve as a function of the effective flow forcing. Finally, we show within both experiment and simulation the existence of short and long time scales in the evolution of fibre conformation.

SRB measures for almost Axiom A diffeomorphisms

We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems – a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, area-preserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the “singular sets” of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the “chaotic saddle” for autonomous and nonautonomous versions of the Hénon map.

Condition Monitoring of an Induction Motor Stator Windings Via Global Optimization Based on the Hyperbolic Cross Points

The objective of condition monitoring of induction machines is to detect the incipient stage of a fault before serious damage occurs with high associated cost. Although the condition monitoring techniques have been intensively investigated in the last decades, research is still carried out in reducing cost and improving accuracy. This paper proposes a novel method that enables efficient and accurate monitoring of the stator winding circuit fault. The proposed method is based on the sparse grid optimization method applied in the least squares estimation of the circuit parameters that characterize the condition of a fault incipient. The kernel of the method is the efficient search for the objective function minimum on the grid created by using the hyperbolic cross points (HCPs). The system cost and complexity are minimized since the proposed method only requires voltage and current signals recorded at a machine terminal without any invasive or additional hardware circuitry. The proposed HCP algorithm is robust to supply voltage unbalance and motor loading state. The validity and effectiveness of the proposed scheme is experimentally tested on a three-phase 800-W 380-V induction motor.

Existence of chaos in the plane [formula omitted] and its application in macroeconomics

The Devaney, Li–Yorke and distributional chaos in the plane R2 can occur in the continuous dynamical system generated by Euler equation branching. Euler equation branching is a type of differential inclusion ẋ∈{f(x),g(x)}, where f,g:X⊂Rn→Rn are continuous and f(x)≠g(x) in every point x∈X. Stockman and Raines (2010) defined the so-called chaotic set in the plane R2 whose existence leads to the existence of Devaney, Li–Yorke and distributional chaos. In this paper, we follow up on Stockman and Raines (2010) and we show that chaos in the plane R2 is always admitted for hyperbolic singular points in both branches not lying in the same point in R2. But the chaos existence is also caused by a set of solutions of Euler equation branching. We research this set of solutions. In the second part we create the new overall macroeconomic equilibrium model called IS-LM/QY-ML model. This model is based on the fundamental macroeconomic equilibrium model called IS-LM model. We model an inflation effect, an endogenous money supply, an economic cycle etc. in addition to the original IS-LM model. We research the dynamical behaviour of this new IS-LM/QY-ML model and show when a chaos exists with relevant economic interpretation.

On hyperbolic points and periodic orbits of symplectomorphisms

We prove the existence of infinitely many periodic orbits of symplectomorphisms isotopic to the identity if they admit at least one hyperbolic periodic orbit and satisfy some condition on the flux. Our result is proved for a certain class of closed symplectic manifolds and the main tool we use is a variation of Floer theory for symplectomorphisms, the Floer-Novikov theory. The proof relies on an important result on hyperbolic orbits, namely, that a \emph{Floer-Novikov} trajectory which converges to an iteration $k$ of the hyperbolic orbit and crosses its fixed neighborhood has energy bounded bellow by a strictly positive constant independent of $k$. The main theorem follows from this feature of hyperbolic orbits and certain properties of quantum homology on the class of symplectic manifolds we work with.

Finite size Lyapunov exponent at a saddle point

A simple stochastic system is considered modeling Lagrangian motion in a vicinity of a hyperbolic stationary point in two dimensions. We address the dependence of the Finite Size Lyapunov Exponent (FSLE) λ on the diffusivity D and the direction of the initial separation θ. It is shown that there is an insignificant difference between the curves λ=λ(θ) for pure dynamics (D=0) and for infinitely large noise (D=∞). For small D a well known boundary layer asymptotic is employed and compared with numerical results.

Topology and self-assembly of defect-colloidal superstructure in confined chiral nematic liquid crystals.

We describe formation of defect-colloidal superstructures induced by microspheres with normal surface anchoring dispersed in chiral nematic liquid crystals in confinement-unwound homeotropic cells. Using three-dimensional nonlinear optical imaging of the director field, we demonstrate that some of the induced defects have nonsingular solitonic nature while others are singular point and line topological defects. The common director structures induced by individual microspheres have dipolar symmetry. These topological dipoles are formed by the particle and a hyperbolic point defect (or small disclination loop) of elementary hedgehog charge opposite to that of a sphere with perpendicular boundary conditions, which in cells with thickness over equilibrium cholesteric pitch ratio approaching unity are additionally interspaced by a looped double-twist cylinder of continuous director deformations. The long-range elastic interactions are probed by holographic optical tweezers and videomicroscopy, providing insights to the physical underpinnings behind self-assembled colloidal structures entangled by twisted solitons. Computer-simulated field and defect configurations induced by the colloidal particles and their assemblies, which are obtained by numerically minimizing the Landau-de Gennes free energy, are in agreement with the experimental findings.

Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures

We present here a construction of horseshoes for any $\mathcal{C}^{1+\alpha}$ mapping $f$ preserving an ergodic hyperbolic measure $\mu$ with $h_{\mu}(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+\alpha}$ mapping $f$ is greater than or equal to $h_{\mu}(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.

Response to Comment on “Modeling and Validation of Nanoparticle Charging Efficiency of a Single-Wire Corona Unipolar Charger”

Fernandez-Diaz and Domat's preceeding Letter to the Editor argued that the geometry of the discharge electrode of the charger of our previous article entitled “Modeling and Validation of Nanoparticle Charging Efficiency of a Single-Wire Corona Unipolar Charger” was a needle but not a wire. Therefore, the boundary condition for the ionic charge density at the discharge wire surface used in the simulation was suspected to be not completely correct. In the present work, the exact shape of the tip of the discharge electrode was examined and found as a hyperbolic point. The experimental onset voltage was further compared with the theoretical onset values calculated by either wire-in-tube or point-to-plane assumption. Results show that the theory of needle-type discharge electrode is also not appropriate either for the boundary condition at the tip because of higher calculated onset voltages than the experimental data.Copyright 2014 American Association for Aerosol Research

Diverting homoclinic chaos in a class of piecewise smooth oscillators to stable periodic orbits using small parametrical perturbations

Abstract This paper investigates the mechanisms of small parametrical perturbations in controlling chaos in a class of non-autonomous piecewise smooth oscillators, which describe a large class of nonlinear dynamical systems in the real world. The analytical expressions of two homoclinic orbits of unperturbed piecewise smooth oscillators, which connect the same hyperbolic saddle point are solved analytically. Firstly, when there are no small parametrical perturbations, by using Melnikov׳s approach, it is rigorously proven that the homoclinic chaos in the Smale horseshoes sense exists when the system׳s parameters are selected above the threshold for chaos occurrence. Secondly, under the small parametrical perturbations, by using Melnikov׳s approach, a sufficient criterion is derived, serving as designing the parameters of the control signal, i.e., amplitude and phase position. In the process of computing Melnikov׳s functions, it is found that the expressions of Melnikov׳s functions could not be solved analytically because the homoclinic orbits are highly complicated. To this end, a numerical algorithm is proposed. Numerical simulations are presented to verify the theoretical results. The results of this paper can be used to explore the underlying chaotic behaviors of the inertial neural network model.

Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u 0, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10, 11]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.

Modal and non-modal stability of two-dimensional Taylor–Green vortices

The linear stability of the two-dimensional Taylor–Green vortices at moderate Reynolds number Re = 2500 is studied by modal and non-modal stability analysis. The properties of modal stability depend on the symmetry type of the disturbances. The stability of penetrating disturbances depends weakly on the axial wavenumber, while non-penetrating disturbances are subjected to the elliptic instability which depends crucially on the wavenumber. The features of transient growth are revealed by non-modal stability analysis. Its mechanism for small optimization time is the stretching near the hyperbolic stagnation points; it does not depend strongly on the symmetry type and the wavenumber. For large optimization time, however, the transient growth is controlled by the strain field around the elliptic stagnation points as the amplified state is similar to unstable modes of elliptic instability. The optimization time at which the properties of the transient growth change is smaller than that of a vortex pair studied by Donnadieu et al (2009 Phys. Fluids 21 094102).

Homoclinic points for convex billiards

In this paper we investigate some generic properties of a billiard system on a convex table. We show that C∞ generically, every hyperbolic periodic point admits some homoclinic orbit.

Merging of two helical vortices

Vortex merging is a basic fluid phenomenon which has been much studied for two-dimensional flows. Here we extend such a study to a specific class of three-dimensional flows, namely to vortices possessing a helical symmetry. In addition to the standard Reynolds number, this introduces another dimensionless control number, the pitch, which quantifies the periodicity length along the helix axis. Helical vortices with large pitches merge very much as in a two-dimensional setting. However, their rotation speed is reduced and the merging period is delayed. These effects, caused by the presence of a self-induced velocity in curved three-dimensional vortices, are understood by computing the streamfunction in the frame of reference rotating with the two vortices, and by inspecting the locations of its hyperbolic points. At intermediate pitch values, only viscous diffusion acts, resulting in a slow viscous type of merging. Finally for small pitches, the system is unstable resulting, at the nonlinear stage, in a different type of merging which breaks the initial central symmetry.

Chain components with stably limit shadowing property are hyperbolic

Let f be a diffeomorphism on a closed smooth manifold M. In this paper, we show that f has the -stably limit shadowing property on the chain component of f containing a hyperbolic periodic point p, if and only if is a hyperbolic basic set. MSC:37C50, 34D10.

Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number

AbstractA standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor$\mathit{Pe}^{1/2}$larger than molecular diffusivity when the Péclet number$\mathit{Pe}$is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time$t$by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances$|\boldsymbol {x}|$from the scalar release point that satisfy$|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the$O(\mathit{Pe}^{1/2})$effective diffusivity, are recovered from our analysis in the limit$|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when$|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances$|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.