We search for rotational, four-dimensional maps of standard type (x(n+1)-2x(n)+x(n-1) = epsilonf(x, epsilon)) possessing one or two polynomial integrals. There are no nontrivial maps corresponding to cubic oscillators...
We search for rotational, four-dimensional maps of standard type (x(n+1)-2x(n)+x(n-1) = epsilonf(x, epsilon)) possessing one or two polynomial integrals. There are no nontrivial maps corresponding to cubic oscillators, but we find a four-parameter family of such maps corresponding to quartic oscillators. This seems to be the only such example.
For an area preserving map, each chaotic orbit appears numerically to densely cover a region (an irregular component) of nonzero area. Surprisingly, the measure approximated by a long segment of such an orbit deviates...
For an area preserving map, each chaotic orbit appears numerically to densely cover a region (an irregular component) of nonzero area. Surprisingly, the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component. Most prominently, there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles. We show that this phenomena is transient, and therefore numerical ergodicity on the irregular component eventually obtains, though the times involved are extremely long - 10(10) iterates. A Markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class - for example, a rotational circle. However, the density spikes do occur in a Markov tree model that includes the effects of islands-around-islands,
Suppose we have two chemical reactions occurring simultaneously. Then theamount y of a reactant changes due to both processes and behaves as a function of time t as y(t) =x_1e~(α_1t) + x_2e~(α_2t), where x_1, x_2, ...
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Suppose we have two chemical reactions occurring simultaneously. Then theamount y of a reactant changes due to both processes and behaves as a function of time t as y(t) =x_1e~(α_1t) + x_2e~(α_2t), where x_1, x_2, α_1, and α_2 are fixed parameters. Typically, weobserve the function y(t) for m fixed t values, perhaps t = 0, Δt, 2Δt, ..., t_(final).
The kinematical and dynamical issues involved in streching and alignment in chaotic and turbulent flows are examined. Dynamical systems tools, such as Lyapunov exponents, are dicussed in the context of fluid mechanica...
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The kinematical and dynamical issues involved in streching and alignment in chaotic and turbulent flows are examined. Dynamical systems tools, such as Lyapunov exponents, are dicussed in the context of fluid mechanical problems and their connection with stretching rates in turbulent flows. Formalisms for the stretching and alignment of passive scalars, passive vectors and nonpassive vectors are developed and the interplay between kinematics and dynamics emphasized. This enables us to compare and contrast the behavior of line elements, gradients of scalar fields, vorticity and magnetic field lines.
In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonhar...
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In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see (3]). In the case of an orthonormal basis, our estimate reduces to Kadec' optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.
A mathematical analysis of the time-dependent, baroclinic, Boussinesq, Eliassen balanced model of tropical cyclone dynamics is presented. An efficient numerical method is developed to solve the equations in momentum c...
A mathematical analysis of the time-dependent, baroclinic, Boussinesq, Eliassen balanced model of tropical cyclone dynamics is presented. An efficient numerical method is developed to solve the equations in momentum coordinates with cylindrical symmetry. Computer simulations illustrate the effects of an experimental parameterization for the heating due to condensation. The model reproduces realistic characteristics of a tropical cyclone and provides a useful tool for analyzing the dynamic relationship between the energy influx due to latent heating and the vortex structure.
Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only...
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Guided by the example of gauge transformations associated with classical Yang-Mills fields, a very general class of transformations is considered. The explicit representation of these transformations involves not only the independent and the dependent field variables, but also a set of position-dependent parameters together with their first derivatives. The stipulation that an action integral associated with the field variables be invariant under such transformations gives rise to a set of three conditions involving the Lagrangian and its derivatives, together with derivatives of the functions that define the transformations. These invariance identities constitute an extension of the classical theorem of Noether to general transformations of this kind. An application to the case of gauge fields demonstrates the existence of two distinct types of conservation laws for such fields.
Transport times for a chaotic system are highly sensitive to initial conditions and parameter values. We present a technique to find rough orbits (epsilon chains) that achieve a desired transport rapidly and which can...
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Transport times for a chaotic system are highly sensitive to initial conditions and parameter values. We present a technique to find rough orbits (epsilon chains) that achieve a desired transport rapidly and which can be stabilized with small parameter perturbations. The strategy is to build the epsilon chain from segments of a long orbit - the point is that long orbits have recurrences in neighborhoods where faster orbits must also pass. The recurrences are used as the switching points between segments. The resulting epsilon chain can be refined by gluing orbit segments over the switching points, providing that a local hyperbolicity condition is satisfied. As an example, we show that transport times for the standard map can be reduced by factors of 10(4). The techniques presented here can be easily generalized to higher dimensions and to systems where the dynamics is known only as a time-series.
A Hamiltonian difference scheme associated with the integrable nonlinear Schrodinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result...
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A Hamiltonian difference scheme associated with the integrable nonlinear Schrodinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result in a novel type of chaotic evolution. The chaotic solution is characterized by random bifurcations across standing wave states into left and right going traveling waves. In this class of problems where the solutions are not subject to even constraints, the traditional mechanism of crossings of the unperturbed homoclinic orbits/manifolds is not observed.
We consider a family of three-dimensional, volume preserving maps depending on a small parameter epsilon. As epsilon --> 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for sma...
We consider a family of three-dimensional, volume preserving maps depending on a small parameter epsilon. As epsilon --> 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for small epsilon the heteroclinic connection breaks up and that the splitting between its components scales with epsilon like epsilon(gamma) exp(-beta/epsilon). We estimate beta using the singularities of the epsilon --> 0+ heteroclinic orbit in the complex plane. We then estimate gamma using linearization about orbits in the complex plane. While these estimates are not proven, they are well supported by our numerical calculations. The work described here is a special case of the theory derived by Amick et al. which applies to q-dimensional volume preserving mappings.
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