Analytical representation of stroboscopic maps of ordinary nonlinear differential equations.
01 Jun 1987-European Physical Journal B (Springer Science and Business Media LLC)-Vol. 68, Iss: 2, pp 253-258
TL;DR: The stroboscopic map of nonlinear dynamical systems can be described by means of a series expansion with only few non-trivial coefficients, provided that the frequency of the stroboscope coincides with the basic frequency of an oscillator.
Abstract: The stroboscopic map of some nonlinear dynamical systems can be described by means of a series expansion with only few non-trivial coefficients, provided that the frequency of the stroboscope coincides with the basic frequency of the oscillator. An analytic representation of such a ‘simple’ map can be obtained by the following two methods: (i) analytical integration of the ordinary differential equation, or (ii) numerical integration on a discrete grid scheme and subsequent approximation by an appropriate series of functions.
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TL;DR: In this article, Shah and Ramachandran considered weak space charge effects in an rf trap and obtained an analytic expression for the time varying distribution function of the 1D plasma, which was shown to be a Maxwellian up to the lowest order in nonlinearity.
Abstract: Exact solutions for one-dimensional (1D) plasma dynamics in an rf trap are known when space charge effects are neglected [K. Shah and H. S. Ramachandran, Phys. Plasmas 15, 062303 (2008)]. In this work, weak space charge effects in an rf trap are considered. An analytic expression for the time varying distribution function of the 1D plasma is obtained. It is shown that the plasma is a Maxwellian up to the lowest order in nonlinearity and that the spatially constant temperature periodically oscillates in time at the same rate as the rf frequency. It was shown by Krapchev [Phys. Rev. Lett. 42, 497 (1979)] that the time averaged distribution function is double humped with respect to velocity beyond a certain threshold in space. The time average of the complete time varying distribution function is obtained and some of the predictions of Krapchev are recovered, while also finding discrepancies. The relationship between stroboscopic orbits and the time averaged ponderomotive orbit are obtained for such traps.
16 citations
TL;DR: The interpolated Poincare map is proposed in this article based on numerical integration in one period or less and spline interpolation, which is illustrated via a study of mixing of fluids in a co-rotating discontinuous cavity flow.
Abstract: The interpolated Poincare map is proposed based on numerical integration in one period or less and spline interpolation. Efficiency and applicability of this method is illustrated via a study of mixing of fluids in a co-rotating discontinuous cavity flow.
5 citations
TL;DR: In this article, an iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODEs with not necessarily small nonlinearities is presented.
1 citations
TL;DR: In this paper, a method for nonlinear system identification, based on the technique of interpolated mapping, is formulated, where the input to the procedure is a map, taking initial conditions on a regular grid to their images after a fixed time step.
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TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
16,554 citations
TL;DR: In this article, a large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function.
Abstract: A large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum 2. With f(2) - f(x) ~ Ix - 21" (for Ix - 21 sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio c~ (a = 2.5029078750957... for z = 2). This structure is determined by a universal function g*(x), where the 2"th iterate off, f("~, converges locally to ~-"g*(~nx) for large n. For ithe class of f's considered, there exists a A~ such that a 2"-point stable limit cycle including :7 exists; A~ - ~ ~ ~-" (~ = 4.669201609103... for z = 2). The numbers = and have been computationally determined for a range of z through their definitions, for a variety off's for each z. We present a recursive mechanism that explains these results by determining g* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.
2,984 citations
TL;DR: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed in this article, where the authors propose a mechanism for generating turbulence in a dissipative system with respect to dissipative energy.
Abstract: A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.
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TL;DR: On montre qu'il existe 2 regimes d'agregation cinetique, irreversible, de colloides aqueux, determinee par le potentiel interparticulaire a courte distance, avec controle de la probabilite de collage lors de l'approche de 2 particules.
Abstract: We show that there are two regimes of irreversible, kinetic aggregation of aqueous colloids, determined by the short-range interparticle potential, through its control of the sticking probability upon approach of two particles. Each regime has different rate-limiting physics, aggregation dynamics, and cluster-mass distributions, and results in clusters with different fractal dimensions. These results set limits on the fractal dimension, ${d}_{f}$, for gold aggregates of $1.75l~{d}_{f}l~2.05$ (\ifmmode\pm\else\textpm\fi{}0.05).
600 citations