We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. In contrast to the usual method of computing the maximal Lyapunov exponent, our method is applied directly to the time series data and does not require phase space reconstruction. Moreover, the dimension of the dynamical system and the form of the underlying equations is irrelevant. The input is the time series data and the output is 0 or 1 depending on whether the dynamics is non-chaotic or chaotic. The test is universally applicable to any deterministic dynamical system, in particular to ordinary and partial differential equations, and to maps.Our diagnostic is the real valued function p(t) = t 0 φ(x(s)) cos(θ(s))ds where φ is an observable on the underlying dynamics x(t) and θ(t) = ct + t 0 φ(x(s))ds. The constant c > 0 is fixed arbitrarily. We define the mean-square-displacement M(t) for p(t) and set K = lim t→∞ log M(t)/ log t. Using recent developments in ergodic theory, we argue that typically K = 0 signifying nonchaotic dynamics or K = 1 signifying chaotic dynamics.

In this paper we address practical aspects of the implementation of the 0-1 test for chaos in deterministic systems. In addition, we present a new formulation of the test which significantly increases its sensitivity. The test can be viewed as a method to distill a binary quantity from the power spectrum. The implementation is guided by recent results from the theoretical justification of the test as well as by exploring better statistical methods to determine the binary quantities. We give several examples to illustrate the improvement.

We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons. PACS numbers: 5.45. Yv, 11.10.Ef, 11.10.Lm, Solitons Water wave theory first introduced solitons as solutions of unidirectional nonlinear wave equations, obtained via asymptotic expansions around simple wave motion of the Euler equations for shallow water in a particular Galilean frame [1]. Later developments identified some of these water wave equations as completely integrable Hamiltonian systems solvable by the inverse scattering transform (IST) method, see, e.g., [2]. We shall discuss the following 1+1 quadratically nonlinear equation in this class for unidirectional water waves with fluid velocity u(x, t),Here m = u − α 2 u xx is a momentum variable, partial derivatives are denoted by subscripts, the constants α 2 and γ/c 0 are squares of length scales, and c 0 = √ gh is the linear wave speed for undisturbed water at rest at spatial infinity, where u and m are taken to vanish. (Any constant value u = u 0 is also a solution.) Equation (1) was first derived by using asymptotic expansions directly in the Hamiltonian for Euler's equations in the shallow water regime and was thereby shown to be biHamiltonian and, thus, IST-integrable in [3]. Before [3], classes of integrable equations similar to (1) were known to be derivable from the theory of hereditary symmetries, [4]. However, these were not derived physically as water wave equations and their solution properties were not studied before [3]. See [5] for an insightful discussion of how the integrable equation (1) relates to the theory of hereditary symmetries.The interplay between the local and nonlocal linear dispersion in this equation is evident in its phase velocity relation, ω/k = (c 0 −γ k 2 ) /(1+α 2 k 2 ), for waves with frequency ω and wave number k linearized around u = 0. At low wave numbers, the constant dispersion parameters α 2 and γ perform rather similar functions. At high wave numbers, however, the parameter α 2 properly keeps the phase speed of the wave from becoming unbounded. The phase speed lies in the band ω/k ∈ (− γ/α 2 , c 0 ). Longer linear waves are the faster provided γ + c 0 α 2 ≥ 0. Equation (1) is not Galilean invariant. Upon shifting the velocity variable by u 0 and moving into a Galilean frame ξ = x − ct with velocity c, so that u(x, t) = u(ξ, t) + c + u 0 , this eq...

We derive the Camassa-Holm equation (CH) as a shallow water wave equation with surface tension in an asymptotic expansion that extends one order beyond the Korteweg-de Vries equation (KdV). We show that CH is asymptotically equivalent to KdV5 (the ÿfth-order integrable equation in the KdV hierarchy) by using the nonlinear/non-local transformations introduced in Kodama (Phys. Lett. A 107 (1985a) 245; Phys. Lett. A 112 (1985b) 193; Phys. Lett. A 123 (1987) 276). We also classify its travelling wave solutions as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.

Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent.In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the "tangent space" and "direct method" for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods.

The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G 2 (C 5) in choosing the numerical integrator, and the role of the Hodge star operator for relating 2 (C 5) and 3 (C 5) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.

The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution.

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24, 1361-1367 (doi:10.1088/0951-7715/ 24/4/018)) gives a rigorous proof of convergence of a fast-slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of superdiffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.