# Showing papers in "Physica D: Nonlinear Phenomena in 1983"

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TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.

Abstract: We study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise. The exponent v is closely related to the fractal dimension and the information dimension, but its computation is considerably easier. Its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed. Algorithms for extracting v from the time series of a single variable are proposed. The relations between the various measures of strange attractors and between them and the Lyapunov exponents are discussed. It is shown that the conjecture of Kaplan and Yorke for the dimension gives an upper bound for v. Various examples of finite and infinite dimensional systems are treated, both numerically and analytically.

5,239 citations

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TL;DR: In this article, it was shown that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0, which correspond to exponents associated with ternary, quaternary and higher correlation functions.

Abstract: We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions Dq, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities Dq. We prove that lim q→0 Dq = fractal dimension (D), limq→1Dq = information dimension (σ) and Dq=2 = correlation exponent (v). Dq with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > Dq for any q′ > q. For homogeneous fractals Dq = Dq. A particularly interesting dimension is Dq=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D∞.

1,577 citations

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TL;DR: In this article, the authors show that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs.

Abstract: The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and their basins. This paper presents examples illustrating that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively higher dimension and include the one-dimensional quadratic map, the (two-dimensional) Henon map, systems of ordinary differential equations in three dimensions and a three-dimensional map. In the case of our study of the three-dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four, respectively. Based on the examples presented the following conjecture is proposed: almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises.

1,099 citations

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TL;DR: In this paper, the authors discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors, and conclude that dimension of the natural measure is more important than the fractal dimension.

Abstract: Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the “dimension of the natural measure”, and all of the metric dimensions take on a common value, which we call the “fractal dimension”. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.

1,000 citations

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TL;DR: In this paper, a semipopular account of the universal scaling theory for the period doubling route to chaos is presented, where the authors show that it can be seen as a form of a deterministic linear model.

Abstract: A semipopular account of the universal scaling theory for the period doubling route to chaos is presented.

668 citations

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TL;DR: A rigorous study of the ground states of one-dimensional models generalizing the discrete Frenkel-Kontorova model has been presented in this article, where the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits.

Abstract: We present a rigorous study of the classical ground-states under boundary conditions of a class of one-dimensional models generalizing the discrete Frenkel-Kontorova model. The extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits. For all boundary conditions, we select among all the extremum solutions of the energy of the model, those which correspond to the ground-states of the infinite system. We prove that these ground-states are either periodic (commensurate) or quasi-periodic (incommensurate) but are never chaotic. We also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions. The topological structure of the whole set of ground-states is described in details. In addition to physical applications, consequences for twist map homeomorphisms are mentioned. Part II (S. Aubry, P.Y. LeDaeron and G. Andre) will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map.

607 citations

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TL;DR: In this article, it was shown that Euler's equations are Lie-Poisson equations associated to the group of volume-preserving diffeomorphisms, and the dual of the Lie algebra is the space of vortices, and Kelvin's circulation theorem is interpreted as preservation of coadjoint orbits.

Abstract: This paper is a study of incompressible fluids, especially their Clebsch variables and vortices, using symplectic geometry and the Lie-Poisson structure on the dual of a Lie algebra. Following ideas of Arnold and others it is shown that Euler's equations are Lie-Poisson equations associated to the group of volume-preserving diffeomorphisms. The dual of the Lie algebra is seen to be the space of vortices, and Kelvin's circulation theorem is interpreted as preservation of coadjoint orbits. In this context, Clebsch variables can be understood as momentum maps. The motion of N point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit, and the standard canonical variables for them are a special kind of Clebsch variables. Point vortices with cores, vortex patches, and vortex filaments can be understood in a similar way. This leads to an explanation of the geometry behind the Hald-Beale-Majda convergence theorems for vorticity algorithms. Symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the KdV and the Schrodinger equations respectively.

517 citations

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TL;DR: In this paper, the authors present new numerical and theoretical results concerning kink-antikink collisions in the classical φ 4 field model in one-dimensional space, and describe the results of a new high-precision computer simulation that significantly extends and refines these observations of escape windows.

Abstract: We present new numerical and theoretical results concerning kink-antikink collisions in the classical (nonintegrable) φ 4 field model in one-dimensional space. Earlier numerical studies of such collisions revealed that, over a small range of initial velocities, intervals of initial relative velocity for which the kink and antikink capture one another alternate with regions for which the interaction concludes with escape to infinite separation. We describe the results of a new high-precision computer simulation that significantly extends and refines these observations of escape “windows”. We also discuss a simple theoretical mechanism that appears to account for this structure in a natural way. Our picture attributes the alternation phenomenon to a nonlinear resonance between the orbital frequency of the bound kink-antikink pair and the frequency of characteristic small oscillations of the field localized at the moving kink and antikink centers. Our numerical simulation also reveals long-lived small-scale oscillatory behavior in the time dependence of kink and antikink velocity following those collisions that do not lead to capture. We account for this fine structure in terms of the interaction between kink (and antikink) motion and small amplitude “radiation” generated during and after the collision. We discuss possible implications of our results for physical systems.

388 citations

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TL;DR: In this article, the exact results on the discrete Frenkel-Kontorova (FK) model and its extensions have been reviewed and a series of rigorous upper bounds for the stochasticity threshold of the standard map were obtained.

Abstract: This paper reviews exact results which we obtained on the discrete Frenkel-Kontorova (FK) model and its extensions, during the past few years. These models are associated with area preserving twist maps of the cylinder (or a part of it) onto itself. The theorems obtained for the FK model thus yields new theorems for the twist maps. We describe the exact structure of the ground-states which are either commensurate or incommensurate and assert the existence of elementary discommensurations under certain necessary and sufficient conditions. Necessary conditions for the trajectories to represent metastable configurations, which can be chaotic, are given. The existence of a finite Peierls-Nabarro barrier for elementary discommensurations is connected with a property of non-integrability of the twist map. We next prove that the existence of KAM tori corresponds to “undefectible” incommensurate ground-states and give a theorem which asserts that when the phonon spectrum of an incommensurate ground-state exhibits a finite gap, then the corresponding trajectory is dense on a Cantor set with zero measure length. These theorems, when applied to the initial FK model, allow one to prove the existence of the transition by “breaking of analyticity” for the incommensurate structures when the parameter which describes the discrepancy of the model from the integrable limit varies. These theorems also allow one to obtain a series of rigorous upper bounds for the stochasticity threshold of the standard map which for the fifth order approximation already approaches within 25% the value which is numerically known. Finally, we describe a theorem proving the existence of a devil's staircase for the variation curve of the atomic mean distance versus a chemical potential, for certain properties of the twist map which are generally satisfied.

355 citations

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TL;DR: In this article, the authors examined the effect of small stable regions or islands on the correlation function for the stochastic trajectories of the Hamiltonians of two degrees of freedom.

Abstract: The phase space for Hamiltonians of two degrees of freedom is usually divided into stochastic and integrable components. Even when well into the stochastic regime, integrable orbits may surround small stable regions or islands. The effect of these islands on the correlation function for the stochastic trajectories is examined. Depending on the value of the parameter describing the rotation number for the elliptic fixed point at the center of the island, the long-time correlation function may decay as t/sup -5/ or exponentially, but more commonly it decays much more slowly (roughly as t/sup -1/). As a consequence these small islands may have a profound effect on the properties such as the diffusion coefficient, of the stochastic orbits.

288 citations

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TL;DR: In this paper, two fixed points are found representing different types of universal behavior: a trivial fixed point for smooth motion and a nontrivial fixed point to represent the incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow.

Abstract: Dynamical systems with quasiperiodic behavior, i.e., two incommensurate frequencies, may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number. In this paper these curves are followed using renormalization group techniques which are applied to a one-dimensional system (circle) and also to an area-contracting map of an annulus. Two fixed points are found representing different types of universal behavior: a trivial fixed point for smooth motion and a nontrivial fixed point. The latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow. Fixed point functions are determined numerically and via an e-expansion and eigenvalues are calculated.

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TL;DR: In this article, an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value.

Abstract: An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C 1 conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency. When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.

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TL;DR: In this paper, phase space portraits have been constructed and analyzed for noisy (nonperiodic) data obtained in an experiment on a nonequilibrium homogeneous chemical reaction, where phase space trajectories define a limit set that is an "attractor" - following a perturbation, the trajectory quickly returns to the attracting set.

Abstract: Phase space portraits have been constructed and analyzed for noisy (nonperiodic) data obtained in an experiment on a nonequilibrium homogeneous chemical reaction. The phase space trajectories define a limit set that is an “attractor” - following a perturbation, the trajectory quickly returns to the attracting set. This attracting set is shown to be “strange” - nearby trajectories separate exponentially on the average. Moreover, the Poincare sections exhibit the stretching and folding that is characteristic of strange attractors.

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TL;DR: In this paper, the authors investigate the effects of such fluctuations coupled to deterministic chaotic systems, in particular, the metric entropy's response to the fluctuations, and find that the entropy increases with a power law in the noise level, and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory.

Abstract: One model of randomness observed in physical systems is that low-dimensional deterministic chaotic attractors underly the observations. A phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer, the amount of information available in observations, and just how this information affects predictions of the system's future behavior. In an effort to develop such a description, we discuss the information theory of highly discretized observations of random behavior. Metric entropy and topological entropy are well-defined invariant measures of such an attractor's “level of chaos”, and are computable using symbolic dynamics. Real physical systems that display low dimensional dynamics are, however, inevitably coupled to high-dimensional randomless, e.g. thermal noise. We investigate the effects of such fluctuations coupled to deterministic chaotic systems, in particular, the metric entropy's response to the fluctuations. We find that the entropy increases with a power law in the noise level, and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory. We also argue that in addition to the metric entropy, there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations: I0, the maximum average information obtainable about the initial condition that produces a particular sequence of measurements (or symbols).

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TL;DR: In this paper, a canonical Poisson bracket in the space of Clebsch potentials is constructed for ideal magnetohydrodynamics, multifluid plasmas, and elasticity.

Abstract: Poisson brackets are constructed by the same mathematical procedure for three physical theories: ideal magnetohydrodynamics, multifluid plasmas, and elasticity. Each of these brackets is given a simple Lie-algebraic interpretation. Moreover, each bracket is induced to physical space by use of a canonical Poisson bracket in the space of Clebsch potentials, which are constructed for each physical theory by the standard procedure of constrained Lagrangians.

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TL;DR: In this article, the authors make a connection between invariant circles and the renormalisation operator, and show that the stability of a simple fixed point of renormalization corresponds to a linear twist map.

Abstract: Kadanoff and Shenker introduced a renormalisation approach to invariant circles in area-preserving maps. This paper makes more precise the connection between invariant circles and the renormalisation operator. Restricting attention to noble rotation numbers, the stability of a simple fixed point of the renormalisation is analysed, corresponding to a linear twist map. It is found to be essentially attracting, so that noble circles persist under perturbation, giving a new view on KAM theory. Shenker and Kadanoff found evidence for another fixed point, corresponding to a map with a non-smooth noble circle. Further evidence is given in this paper. It has essentially only one unstable direction, and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle. Finally, noble circles are shown to be locally most robust, in an important sense.

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TL;DR: In this paper, the authors illustrate the following transition sequences; period doubling and the U-sequence, intermittency, the periodic-quasiperiodic-chaotic sequence, frequency locking, and an alternating periodic-chotic sequence.

Abstract: Experiments on nonlinear electrical oscillators, the Belousov-Zhabotinskii reaction, Rayleigh-Benard convection, and Couette-Taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom. Experimental results are presented illustrating the following transition sequences; period doubling and the U-sequence, intermittency, the periodic-quasiperiodic-chaotic sequence, frequency locking, and an alternating periodic-chaotic sequence.

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TL;DR: In this article, the authors established bounds on the number of modes which determine the solutions of the Navier-Stokes equations in 2-dimensional Rayleigh-Benard convection.

Abstract: New bounds are established on the number of modes which determine the solutions of the Navier-Stokes equations in two dimensions. The best bound available at present is nearly proportional to the generalized Grashof number (defined in the paper), and less than logarithmically dependent on the spatial structure, or the shape of the force driving the flow. To the extent than for the case of 2-dimensional Rayleigh-Benard convection, the generalized Grashof number may be identified with the usual Grashof number, the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds.

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TL;DR: In this paper, a simplified version of the experimentally determined Poincare map is proposed, and several features of the bifurcations of this map are described, including phase locking, period doubling and period chaotic dynamics at different values of the stimulation parameters.

Abstract: Periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking, period-doubling bifurcations and aperiodic “chaotic” dynamics at different values of the stimulation parameters. This behavior is analyzed by considering an experimentally determined one-dimensional Poincare or first return map. A simplified version of the experimentally determined Poincare map is proposed, and several features of the bifurcations of this map are described.

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TL;DR: In this paper, numerically the interactions of a kink and an antikink in a parametrically modified sine-Gordon model with potential V(φ) = (1 − r)2(1 − cosφ)/(1 + r2 + 2rcosφ).

Abstract: We study numerically the interactions of a kink (K) and an antikink (K) in a parametrically modified sine-Gordon model with potential V(φ) = (1 − r)2(1 − cosφ)/(1 + r2 + 2rcosφ). As the parameter r is varied from the pure sine-Gordon case (r = 0) to values for which the model is not completely integrable (r ≠ 0), we find that a rich structure arises in the KK collisions. For some regions of r(−0.20⪅r<0) this structure is very similar to that observed in KK interactions in the φ4 model, and we show that the theory recently suggested for these collisions also applies quantitatively to the modified sine-Gordon model. In other regions of r we observe new scattering phenomena, which we present in detail numerically and discuss in a qualitative manner analytically.

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TL;DR: In this paper, it was shown that the correlation properties of the quantum and corresponding classical motions are only similar for very short time intervals ts, and that the evolution of quantum system unlike the classical one is stable.

Abstract: Numerical studies are made of simple one- and two-dimensional quantum models which are stochastic in the classical limit. It is shown that the correlation properties of the quantum and corresponding classical motions are only similar for very short time intervals ts, and that the evolution of the quantum system, unlike the classical one, is stable. The diffusive excitation of the quantum system under a periodic perturbation is limited to a specific time interval t ∗ ≫ t s , during which the diffusion rate is similar to the corresponding classical diffusion rate. For the two-dimensional model, a continuous component in the correlation spectrum survives for an indefinite period t w ≫ t ∗ . It is shown that when the perturbation is quasiperiodic the interval t∗ increases sharply.

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TL;DR: In this paper, a unified treatment of polynomial eigenvalue soliton equations associated with sl(2) eigen value problems is given, where a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl( 2) is given.

Abstract: The soliton equations associated with sl(2) eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment; they are shown to be generated by a single family of commuting Hamiltonians on a subalgebra of the loop algebra of sl(2). The conserved densities and fluxes of the usual ANKS hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems. The Hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment.

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TL;DR: In this article, the interaction of autowave sources is experimentally studied in an active chemical medium with excitable kinetics, and three types of vortices are considered: 1) a spiral wave rotating in a simply-connected medium.

Abstract: The interaction of autowave sources is experimentally studied in an active chemical medium with excitable kinetics. Three types of vortices are considered: 1) A spiral wave (S) rotating in a simply-connected medium, 2) A spiral wave rotating around a hole (SH); and 3) A spiral wave (S N ) with topological charge N . It is found that S synchronizes SH (except very small holes), and spiral waves with lower topological charge synchronize those with higher topological charge. It is also found that the interaction of autowave sources displays some unique properties because of their ability to appear on inhomogeities, to vanish and to move in the medium. A new phenomen of induced drift of spiral waves is demonstrated. The drift was induced by high-frequency concentrational waves propagating in the medium. A similar drift is observed upon interaction of vortices. The mechanism of the induced drift is explained in terms of wave-break translocation from one wave to another. Using this effect, one can control the location of wave sources in an active medium.

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IBM

^{1}TL;DR: In this article, a 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated, which contains an open end, i.e., an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loudspeakers.

Abstract: A 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated. It contains an open end, i.e. an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loud-speakers. In the Poincare model of hyperbolic geometry, the solutions of Schrodinger's equation for the reflection of a particle which enters through the horn are easily constructed. The scattering phase shift as a function of the momentum is essentially given by the phase angle of Riemann's zeta function on the imaginary axis, at a distance of 1 2 from the famous critical line. This phase shift shows all the features of chaos, namely the ability to mimick any given smooth function, and great difficulty in its effective numerical computation. A plot shows the close connection with the zeros of Riemann's zeta function for low values of the momentum (quantum regime) which gets lost only at exceedingly large momenta (classical regime?) Some generalizations of this approach to chaos are mentioned.

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TL;DR: In this article, the authors studied the routes to chaos for a Rayleigh-Benard experiment in mercury, as a function of two parameters, the Rayleigh number (R ) and the Chandrasekhar number (Q ).

Abstract: We study the routes to chaos for a Rayleigh-Benard experiment in mercury, as a function of two parameters, the Rayleigh number ( R ) and the Chandrasekhar number ( Q ). For low Q the main route is a period-doubling cascade of bifurcations occurring at low R . For higher values of Q , two routes are observed, one related to a soft mode instability for moderate R , and a second one related to a three oscillators state, occurring at higher Rayleigh number values.

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TL;DR: In this article, a hierarchy of symmetries for the Kadomtsev-Petviashvili equation is presented, which depend on the space and time variables explicitly.

Abstract: We present a new hierarchy of symmetries for the Kadomtsev-Petviashvili equation. These new symmetries depend on the space and time variables explicitly. Together with the previously known classical symmetries, they constitute an infinite-dimensional Lie algebra.

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TL;DR: In this paper, the motion of the wave front is characterized in terms of α, β and γ, and the effect of fast absorption (b 0, 0 < γ < 1) that causes extinction within a finite time, may break the evolving pulse into several sub-pulses and causes the expanding front to reverse its direction.

Abstract: The quasi-linear parabolic equation ∂tu = a∂xxuα + b∂xuβ − cuγ exhibits a wide variety of wave phenomena, some of which are studied in this work; and some solvable cases are presented. The motion of the wave front is characterized in terms of α, β and γ. Among the interesting phenomena we note the effect of fast absorption (b 0, 0 < γ < 1) that causes extinction within a finite time, may break the evolving pulse into several sub-pulses and causes the expanding front to reverse its direction. In the convecting case (c 0, b ≠ 0) propagation has many features in common with Burgers equation, α = 1; particularly, if 0 < a ≪ 1, a shock-like transit layer is formed.

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TL;DR: In this paper, a theory of integrable Hamiltonian systems in two dimensions is presented, which admits to integrability of the orbits for magnetic or Coriolis forces as well as for forces derivable from a potential.

Abstract: A theory of integrable Hamiltonian systems in two dimensions is formulated and applied. The four-dimensional phase-space problem that is the vanishing of the Poisson bracket between another invariant and the Hamiltonian is here reduced to the solution of a series of two-dimensional configuration-space equations. (The theory is also applicable to more than two dimensions.) The constraints are found which admit to integrability of the orbits for magnetic or Coriolis forces as well as for forces derivable from a potential. When a system admits a given invariant, the invariant is found - for example, by quadrature. A number of examples including known and apparently previously unknown invariants are given. The theory of exact integrals of the motion also can be extended to the derivation of approximate invariants. The orbital structure of integrable or approximately integrable systems correlates with the degree (maximum power of the velocity) of a standardized invariant. The theory admits a variational principle, among other approximation techniques, for the computation of a “best” approximate invariant. The problem of the general cubic potential with one symmetric coordinate, V = 1 2 Ax 2 + 1 2 By 2 + Cx 2 y + 1 3 Dy 3 , (of which the well-studied Henon-Heiles potential is the special case for A = B and C = −D) is examined in detail.

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TL;DR: In this article, the authors derived an asymptotic nonlinear equation which directly describes the dynamics of the onset and stabilization of cellular structure: ƒ τ + τ +▿ 4 + ǫ 4 +ǫ α = 0.

Abstract: In the solidification of a dilute binary alloy, a planar solid-liquid interface is often found to be unstable, spontaneously assuming a cellular structure. If the solute rejection coefficient is close to unity, then, near the stability threshold, the characteristic cell size may significantly exceed the diffusional width of the solidification zone. This situation enables one to derive an asymptotic nonlinear equation which directly describes the dynamics of the onset and stabilization of cellular structure: ƒ τ +▿ 4 +▿[(2−ƒ)▿+αƒ=0 .