Hyperbolic Stationary Point Research Articles

Bi-Orbital States in Hyperbolic Traps

We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is the sum of a hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under the resonance 1 : (−1). In this case, anharmonic terms define the dynamics and an effective Hamiltonian on the space of motion constants of the ideal harmonic operator. We show that if the anharmonic part is symmetric, then the effective Hamiltonian has unstable equilibriums and separatrix, which define distinct classically allowed regions in the space of motion constants of the ideal trap. The corresponding stationary states of the trapped charge can form bi-orbital states, i.e., a state localized on two distinct classical trajectories. We obtain semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space and express it in terms of complex instantons.

Asymptotic completeness of scattering in the nonlinear Lamb system for nonzero mass

We establish the asymptotic completeness in the nonlinear Lamb system with nonzero mass for hyperbolic stationary states. For the proof, we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point, using the Banach space Inverse Function Theorem and a priori estimates. We give a counterexample showing that the hyperbolicity condition is essential.

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.

On asymptotic completeness of scattering in the nonlinear Lamb system, II

We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ordinary differential equation) converging to a hyperbolic stationary point using the inverse function theorem in a Banach space. We give counterexamples which show nonexistence of such trajectories for nonhyperbolic stationary points.

Linearization in the large of nonlinear systems and Koopman operator spectrum

According to the Hartman–Grobman Theorem, a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point. Here, we extend this linearization around stable (unstable) equilibria or periodic orbits to the whole basin of attraction, for both discrete diffeomorphisms and flows. We discuss the connection of the linearizing transformation to the spectrum of Koopman operator.

The Burgers equation with affine linear noise: Dynamics and stability

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.

Multiscroll attractors by switching systems

In this paper, we present a class of three-dimensional dynamical systems having multiscrolls which we call unstable dissipative systems (UDSs). The UDSs are dissipative in one of its components but unstable in the other two. This class of systems is constructed with a switching law to display various multiscroll strange attractors. The multiscroll strange attractors result from the combination of several unstable "one-spiral" trajectories by means of switching. Each of these trajectories lies around a saddle hyperbolic stationary point. Thus, we describe how a piecewise-linear switching system yields multiscroll attractors, symmetric or asymmetric, with chaotic behavior.

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.

A higher order local linearization method for solving ordinary differential equations

The local linearization (LL) method for the integration of ordinary differential equations is an explicit one-step method that has a number of suitable dynamical properties. However, a major drawback of the LL integrator is that its order of convergence is only two. The present paper overcomes this limitation by introducing a new class of numerical integrators, called the LLT method, that is based on the addition of a correction term to the LL approximation. In this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the LL method. In particular, it is proved that the LLT method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence. The performance of the introduced method is further illustrated through computer simulations.

A mechanism of chaotic mixing in an elementary deterministic flow

A typical mechanism of the chaotic mixing of a passive impurity was theoretically and numerically elucidated using an elementary model of interaction between a point vortex and a periodic plane flow. The chaotic transfer and mixing, appearing as a result of the transversal intersection of stable and unstable manifolds of a hyperbolic stationary point, may lead to far-reaching consequences in geophysical flows.

Runge–Kutta Solutions of Stiff Differential Equations Near Stationary Points

Runge–Kutta methods applied to stiff systems in singular perturbation form are shown to give accurate approximations of phase portraits near hyperbolic stationary points. Over arbitrarily long time intervals, Runge–Kutta solutions shadow solutions of the differential equation and vice versa. Precise error bounds are derived. The proof uses attractive invariant manifolds to reduce the problem to the nonstiff case, which was previously studied by Beyn.

Fresnel zones of hyperbolic type from the physical point of view

The distinguishing features of Fresnel zones have been studied for hyperbolic stationary points. It has been found expedient to overdetermine Fresnel saddle zones as 1 2 π-neighbourhoods of stationary points in contrast to π-neighbourhoods for elliptic points. It has been shown that though formally a Fresnel zone has infinite “tails” stretched along the separatrix, the actual field formation region is always bounded. Moreover, the analysis of the ray localization process using simple (rectangular, rhombic, or elliptic) shaped holes or with a Gaussian window, has indicated that the field formation region in the neighbourhood of a stationary saddle point is almost similar to that for an elliptic stationary point. An analysis has been made on the strength of measures suggested in this paper of localizing rays in a given plane. Consideration of long “tails” of Fresnel zones is necessary only in exotic situations.

Topological versus differentiable hyperbolicity: Extending orbit conjugacies

A series of recent examples by Dean Neumann. Dennis Pixton, and the author 19. IO, 13. 151 of “topologically hyperbolic” stationary points in C”flows. not Locally orbit-conjugate to differentiably hyperbolic stationary points, motivate broad necessary conditions on the size of certain Morsetheoric canceling diffeomorphisms. These and a theorem of Wesley Wilson [ 121 are used to state necessary conditions for a diffeomorphism of the coreless solid torus to decompose into two homeomorphisms, each of which is extendible onto the core. Philip Hartman, Jaco Palis, and others have proven the classic result that all differentiably hyperbolic stationary points (eigenvalues of the derivative map off the unit circle), with the same stable-unstable splitting, are locally orbit-conjugate [3, 121; that is, a local homeomorphism carries orbits near one to orbits near the other. Topologically and differentiably hyperbo!ic stationary points cannot be distinguished by the natures of their nearby ingressing and egressing sets (for definitions see [4]). Courtney Coleman conjectured that the two were in fact locally orbit-conjugate. Wilson formalized the question as follows [ 171: Assume B,,., = Dm x D” is an isolating block [2] of the stationary point p = (01 x iO in the C”-flow, 4, on Rmf”. So the orbits of ail other points of B m,n meet %B,*,. Denote by b+ = t3Dm x D” the ingressing set for #\I?,,,.,: and by 6= D* x 3D” the egressing set [4], 5 = %D’” x r3D” is then the external tangency set. Further require that A + = D” X {O} be the set of points of B,~, which do not exit B, n those points which do not exit B,:,L in positive time; A = {O X D” be in negative time. Then Coleman’s conjecture is that #iB,,, is orbit-conjugate on B,., to the semi-flow #,,, on B m.n generated by

A uniform continuity condition which is equivalent to Coleman's conjecture

At the Fifth International Conference on Nonlinear Oscillations (Kiev, 1970), Coleman [l] raised the question of whether or not a flow with a topologically hyperbolic stationary point is necessarily locally conjugate with some flow near a differentiably hyperbolic (generic) stationary point. We shall prove an affirmative result in the case where either the stable manifold or the unstable manifold has dimension one and we shall prove the equivalence of a uniform continuity condition in the general case. These affirmative results are contrasted with the recent counterexample by Neumann [2], which has both the stable manifold and the unstable manifold with dimension 2. Using Hartman’s theorem [3], it is easy to show that every differentiably hyperbolic stationary point on [w N is locally conjugate to the flow F~;,,~-,~ of the Standard Example, where