Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Given a normally hyperbolic invariant manifold Λ for a map f, whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future.We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map is symplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions.We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type.We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (1) (2006) 64-188] using methods of fast-slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.     

Quasi-separatrix layers and three-dimensional reconnection diagnostics for line-tied tearing modes

In three-dimensional magnetic configurations for a plasma in which no closed field line or magnetic null exists, no magnetic reconnection can occur, by the strictest definition of reconnection. A finitely long pinch with line-tied boundary conditions, in which all the magnetic field lines start at one end of the system and proceed to the opposite end, is an example of such a system. Nevertheless, for a long system of this type, the physical behavior in resistive magnetohydrodynamics (MHD) essentially involves reconnection. This has been explained in terms comparing the geometric and tearing widths [1] and [2]. The concept of a quasi-separatrix layer [3] and [4] was developed for such systems. In this paper we study a model for a line-tied system in which the corresponding periodic system has an unstable tearing mode. We analyze this system in terms of two magnetic field line diagnostics, the squashing factor [5], [6] and [7] and the electrostatic potential difference [8] and [9] which has been used in kinematic reconnection studies. We discuss the physical and geometric significance of these two diagnostics and compare them in the context of discerning tearing-like (reconnection-like) behavior in line-tied modes.     

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four (that is, with its homogeneous part of degree four not identically zero) is nonisochronous. The proof uses the geometric properties of the period annulus and it requires the study of the Hamiltonian systems associated to a Hamiltonian function of the form H(x, y)=A(x)+B(x) y+C(x) y²+D(x) y³.     

Multidimensional Symplectic Separatrix Maps

Summary. We consider an a priori unstable (initially hyperbolic) near-integrable Hamiltonian system in a neighborhood of stable and unstable asymptotic manifolds of a family of hyperbolic tori. Such a neighborhood contains the most chaotic part of the dynamics. The main result of the paper is the construction of the separatrix map as a convenient tool for the studying of such dynamics. We present evidence that the separatrix map combined with the method of anti-integrable limit can give a large class of chaotic trajectories as well as diffusion trajectories. Received March 26, 2001; accepted November 5, 2001     

Drift in phase space: a new variational mechanism with optimal diffusion time

We consider nonisochronous, nearly integrable, a priori unstable Hamiltonian systems with a (trigonometric polynomial) O(μ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time T_d=O((1/μ)ln(1/μ)) by a variational method which does not require the existence of “transition chains of tori” provided by KAM theory. We also prove that our estimate of the diffusion time T_d is optimal as a consequence of a general stability result derived from classical perturbation theory.     

Large Normally Hyperbolic Cylinders in a priori Stable Hamiltonian Systems

We prove the existence of normally hyperbolic cylinders in a priori stable Hamiltonian systems the size of which is bounded from below independently of the size of the perturbation. This result should have applications to the study of Arnold’s diffusion.     

Dissipative and Hamiltonian Systems with Chaotic Behavior: An Analytic Approach

Some classes of dissipative and Hamiltonian distributed systems are described. The dynamics of these systems is effectively reduced to finite-dimensional dynamics which can be unboundedly complex in a sense. Yarying the parameters of these systems, we can obtain an arbitrary (to within the orbital topological equivalence) structurally stable attractor in the dissipative case and an arbitrary polynomial weakly integrable Hamiltonian in the conservative case. As examples, we consider Hopfield neural networks and some reaction–diffusion systems in the dissipative case and a nonlinear string in the Hamiltonian case.     

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three-body problem and the spatial circular restricted three-body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well-known “two-dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower-dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry-Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.     

Geometry of Discrete-Time Spin Systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space \((S^2)^n\). In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on \(T^*\mathbf {R}^{2n}\) for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.     

The gap lemma and geometric criteria for instability of viscous shock profiles

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for n × n conservation laws, n ≥ 2. © 1998 John Wiley & Sons, Inc.     

Notes on a new approximate solution of 2-D heat equation backward in time

In this paper, we consider a backward heat problem that appears in many applications. This problem is ill-posed. The solution of the problem as the solution exhibits unstable dependence on the given data functions. Using a new regularization method, we regularize the problem and get some new error estimates. Some numerical tests illustrate that the proposed method is feasible and effective. This work is a generalization of many recent papers, including the earlier paper [A new regularized method for two dimensional nonhomogeneous backward heat problem, Appl. Math. Comput. 215(3) (2009) 873–880] and some other authors such as Chu-Li Fu et al. ,  and , Campbell et al. [4].     

Uniform exponential attractors for first order non-autonomous lattice dynamical systems

In Abdallah (2008, 2009) [2] and [3], we have investigated the existence of exponential attractors for first and second order autonomous lattice dynamical systems. Within this work, in l², we carefully study the existence of a uniform exponential attractor for the family of processes associated with an abstract family of first order non-autonomous lattice dynamical systems with quasiperiodic symbols acting on a closed bounded set.     

Uniform persistence for a two-species ratio-dependent predator-prey system with diffusion and variable time delays

A two-species ratio-dependent predator-prey diffusion model with variable time delays is investigated. By constructing some suitable Lyapunov functionals and utilizing some analysis techniques, we obtain the permanence of this system. Our results generalize and improve the results of [10] and [11].     

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N?2. We address the question of well-posedness for large data having critical Besov regularity. Our result improves the analysis of R. Danchin (2007) in [13], of Q. Chen et al. (2010) in [8] and of B. Haspot (2009, 2010) in [15] and [16] inasmuch as we may take initial density in with 1?p<+∞. Our result relies on a new a priori estimate for the velocity, where we introduce a new unknown called effective velocity to weaken one the coupling between the density and the velocity. In particular for the first time we obtain uniqueness without imposing hypothesis on the gradient of the density.     

A note on Tang and He’s paper

In this paper, we establish new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which improve the recent results of Tang and He [X.H. Tang, X. He, Lower bounds for generalized eigenvalues of the quasilinear systems, J. Math. Anal. Appl. 385 (2012) 72-85] when 1 < p_i < 2 for i = 1, 2, … , n.     

A simple proof of the geometric fractional monodromy theorem

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ? ?² is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (?2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.     

Analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods

In this note we show the analytic solution of a class of fractional differential equations with variable coefficients by using operatorial methods. Taking inspiration from previous papers by Dattoli et al. [4], [5] and [6] about spectral properties of Laguerre derivative, we here generalize some of their results to fractional evolution equations. Besides that, we have two interesting generalized examples. One is about telegraph equation with time dependent coefficient. The other, that could be of some interest for realistic applications, is the fractional diffusion with a space-dependent diffusion coefficient.     

Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary conditions

This paper consider the multiple solutions for even Hamiltonian systems satisfying Sturm-Liouville boundary conditions. The gradient of Hamiltonian function is generalized asymptotically linear. The solutions obtained are shown to coincide with the critical points of a dual functional. Thanks to the index theory for linear Hamiltonian systems by Dong (2010) [1], we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by Chang (1993) [2].