It is demonstrated that to optimally enhance directed transport by symmetry breaking of temporal forces there exists a universal force waveform which allows to deduce universal scaling laws that explain previous results for a great diversity of systems subjected to a standard biharmonic force and provide a universal quantitative criterion to optimize any application of the ratchet effect induced by symmetry breaking of temporal forces. PACS numbers: 05.60.-kUnderstanding the ratchet effect [1][2][3][4] induced by symmetry breaking of temporal forces is a fundamental issue that has remained unresolved for decades. While the dependence of the directed transport on each of the ratchet-controlling parameters has been individually investigated, there is still no general criterion to apply to the whole set of these parameters to optimally control directed transport in general systems without a ratchet potential .Consider a general deterministic system (classical or quantum, dissipative or non-dissipative, one-or multi-dimensional) subjected to a T -periodic zero-mean ac force f (t) where a ratchet effect is induced by solely violating temporal symmetries. A popular choice would be the simple case of a biharmonic force, f h1,h2 (t) = ǫ 1 har 1 (ωt + ϕ 1 ) + ǫ 2 har 2 (2ωt + ϕ 2 ), where har 1,2 represents indistinctly sin or cos. Clearly, the aforementioned symmetries are solely the shift symmetry of the force (f (t) = −f (t + T /2) , T ≡ 2π/ω) and the time-reversal symmetry of the system's dynamic equations. Of course, the breaking of the latter symmetry implies the breaking of some time-reversal symmetry of the force (f (−t) = ±f (t)) in some general case, but not in all cases [19]. The analysis of the breaking of these two fundamental symmetries allows to find the regions of the parameter space (ǫ 1 , ǫ 2 , ϕ 1 , ϕ 2 ) , ǫ 1 + ǫ 2 = const.,where the ratchet effect is optimal in the sense that the average of relevant observables (such as velocity and current, hereafter referred to as V ) is maximal, the remaining parameters

Conclusive mathematical arguments are presented supporting the ratchet conjecture [R. Chacón, J. Phys. A 40, F413 (2007)], i.e., the existence of a universal force waveform which optimally enhances directed transport by symmetry breaking. Specifically, such a particular waveform is shown to be unique for both temporal and spatial biharmonic forces, and general (non-perturbative) laws providing the dependence of the strength of directed transport on the force parameters are deduced for these forces. The theory explains previous results for a great diversity of systems subjected to such biharmonic forces and provides a universal quantitative criterion to optimize any application of the ratchet effect induced by symmetry breaking of temporal and spatial biharmonic forces.

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General results concerning maintenance or enhancement of chaos are presented for dissipative systems subjected to two harmonic perturbations (one chaos inducing and the other chaos enhancing). The connection with previous results on chaos suppression is also discussed in a general setting. It is demonstrated that, in general, a second harmonic perturbation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. Numerical results indicate that general theoretical findings concerning periodic chaos-inducing perturbations also work for aperiodic chaos-inducing perturbations, and in arrays of identical chaotic coupled oscillators.

A new route is described for eliminating chaos in nonlinear oscillators by changing only the shape of a weak nonlinear periodic perturbation and illustrated with the example of the Duffing-Holmes oscillator forced with the Jacobian elliptic function sn. Two techniques are used in the illustration: applying the Melnikov-Holmes analysis, and studying the behavior of the Lyapunov exponent from a simple recursion relation which models an unstable limit cycle. The connection with related previously described routes is also discussed in a general setting.PACS numbers: 05.45.-fb, 05.40.+J During the last decades, new phenomena have appeared from the inclusion of terms modeling periodic perturbations in the equations of nonlinear dissipative systems. The most ubiquitous is chaos [1]. It is also clear at first sight from the literature that it is the harmonic functions which have been overwhelmingly used to model the periodic perturbations. However, these functions are solutions of linear oscillators, and rarely of nonlinear equations. Nature is nonlinear, so we should also take the forcing mechanism to be a nonlinear system since this is the generic situation. In other words, it seems more appropriate to employ periodic functions that are solutions of nonlinear oscillators to construct more realistic perturbations. The simplest functions having this requirement are the Jacobian elliptic functions (JEF) [2]. If one considers polynomials to be the simplest nonlinear extension of linear oscillators, their solutions are known to be given in terms of JEF's. This is the case for the most studied nonlinear integrable oscillators, such as the Duffing or the Helmholtz. Also for nonlinearities in the form of harmonic functions, the pendulum, for instance [3], the solutions are in terms of JEF's. In comparison with the harmonic solutions, the JEF's add a new variable to the parameter space of the system: the elliptic parameter m that is responsible for the shape of the perturbation, i.e., for the temporal rate at which energy is transferred from the excitation mechanism to the system, having fixed the period. This fact leads us to expect new aspects of behavior of the system-unexplored in the harmonic case -when m is varied, the remaining parameters being left constant.In this Letter, we show how by altering solely the shape of a weak external nonlinear modulation, one may pass a dynamical system from a regular to a chaotic state, and vice versa. That suggests a possible explanation of some proposed mechanisms for controlling chaos. From previous work on the possibility of eliminating or reducing chaos in a dynamical system [4,5], it seems that the resonant property of the harmonic perturbation causing the regularization of the system is a necessary condition for a complete regularization. In particular, this is the case for the modelwhere asin(pcot) is responsible for the disappearance of chaos when a and p are suitably chosen-starting from chaos at a=0. [See Ref. [4] where /(x,x)=sinx + Gx-/; G,I constants.] The resonance condit...

We demonstrate that directed transport of bright solitons formed in a quasi-one-dimensional Bose-Einstein condensate can be reliably controlled by tailoring a weak optical lattice potential, biharmonic in both space and time, in accordance with the degree of symmetry breaking mechanism. By considering the regime where matter-wave solitons are narrow compared to the lattice period, (i) we propose an analytical estimate for the dependence of the directed soliton current on the biharmonic potential parameters that is in good agreement with numerical experiments, and (ii) we show that the dependence of the directed soliton current on the number of atoms is a consequence of the ratchet universality.

We show that directed ratchet transport of a driven overdamped Brownian particle subjected to a spatially periodic and symmetric potential can be reliably controlled by tailoring a biharmonic temporal force, in coherence with the degree-of-symmetry-breaking mechanism. We demonstrate that the effect of finite temperature on the purely deterministic ratchet scenario can be understood as an effective noise-induced change of the potential barrier which is in turn controlled by the degreeof-symmetry-breaking mechanism. Remarkably, we find that the same universal scenario holds for any symmetric periodic potential, while optimal directed ratchet transport occurs when the impulse transmitted (spatial integral over a half-period) by the symmetric spatial force is maximum.PACS numbers: 05.40.-a Directed transport without any net external force, the ratchet effect [1][2][3], has been an intensely studied interdisciplinary subject over the last few decades owing to its relevance in biology where ratchet mechanisms are found to underlie the working principles of molecular motors [4,5], and to its wide range of potential technological applications including micro-and nano-technologies. Directed ratchet transport (DRT) is today understood to be a result of the interplay of symmetry breaking [6], nonlinearity, and non-equilibrium fluctuations, in which these fluctuations include temporal noise [2], spatial disorder [7], and quenched temporal disorder [8]. In extremely small systems, including many of those occurring in biological and liquid environments as well as many nanoscale devices, DRT is often suitably described by overdamped ratchets, in which inertial effects are negligible in comparison with friction effects [2,[9][10][11]. Here, we show how ratchet universality [12] works subtly in the context of noisy overdamped ratchets by studying the dynamics of a universal model -a Brownian particle moving on a periodic substrate subjected to a biharmonic excitation [2,3],where γ is an amplitude factor, and the parameters η ∈ [0, 1] and ϕ account for the relative amplitude and initial phase difference of the two harmonics, respectively, while ξ (t) is a Gaussian white noise with zero mean and ξ (t) ξ (t + s) = δ (s), and σ = 2k b T with k b and T being the Boltzmann constant and temperature, respectively. For deterministic ratchets, this has been shown to also be the case for topological solitons [8] and matterwave solitons [13]. It is worth noting that, in spite of the abundance of numerical findings, the theoretical understanding of the directed transport phenomena represented by Eq. (1) remains far from being satisfactory [14] even about half a century after the earliest studies [15][16][17]. Indeed, all the earlier theoretical predictions (cf. Refs. [3,6,16,17,18]) indicate that the dependence of DRT velocity on the amplitudes of the two harmonics should scale asNote that this expression presents, as a function of η, a single maximum at η = 2/3, irrespective of the particular value of temperature, including the limiting val...

PREFACEAn exciting and extremely active area of multidisciplinary investigation during the past decade has been the problem of controlling chaotic systems. Indeed there have been a number of books written which have served to review a wide variety of chaos control theories, methods, and perspectives. The main reasons for such interest are the interdisciplinary character of the problem, the implicit promise of a better understanding of chaotic behavior, and the possibility of successful applications in such diverse areas of research as aerodynamics, biology, chemical engineering, epidemiology, electric power systems, electronics, fluid mechanics, laser physics, physiology, secure information processing, and so on. The subject of chaos control exhibits at present a huge spectrum of methods and techniques based on different perspectives. While useful, some of the aforementioned books have suffered from the too ambitious goal of attempting to discuss very concisely every method. On the other hand, many of the papers published on control (suppression/enhancement) of chaos by additional time-dependent excitations (forcing or parametric excitation) have been based on the results of computer simulations. That is why the author's goal was to write a monograph which would give a reasonably rigorous theory of a particular but highly relevant control technique: the suppression/enhancement of chaos by weak periodic excitations in low-dimensional, dissipative, and non-autonomous systems. Controlling chaos is therefore understood as a procedure which suppresses chaos when it is unwanted, and enhances existing chaos or gives rise to chaos in a dynamical system when it is useful. This book is not meant either to compete with the findings of other authors or to repeat known mathematical tools. Except in a very few places, results published by other authors are not reviewed.This monograph begins with an introduction where the method of controlling chaos by weak periodic excitations is approached from the general idea of the control of nonlinear dynamical systems. Some relevant aspects of the technique, such as its flexibility, robustness, scope, and experimental applicability are also discussed. Emphasis is put on the comparison between harmonic and non-harmonic excitations.Chapter 2 presents an intuitive argument to illustrate how added periodic excitations modify the stability of perturbed generic limit cycles. The class of chaotic, dissipative, and non-autonomous dynamical systems to be controlled is described as well as Melnikov's method, which is the analytical technique used to obtain the vii PREFACE viii theoretical results. For the sake of clarity, the cases with and without noise are studied separately. Also, it is shown that the maximum survival of the symmetries of solutions from a wide class of dynamical systems, subjected to both a primary chaos-inducing and a chaos-controlling excitation, corresponds to the optimal choice of the control parameters. For the purely deterministic case, the theory considers separately th...