Ratchet universality in the bidirectional escape from a symmetric potential well

The present work discusses symmetry-breaking-induced bidirectional escape from a symmetric metastable potential well by the application of zero-average periodic forces in the presence of dissipation. We characterized the interplay between heteroclinic instabilities leading to chaotic escape and breaking of a generalized parity symmetry leading to directed ratchet escape to an attractor either at ∞ or at −∞ . Optimal enhancement of directed ratchet escape is found to occur when the wave form of the zero-average periodic force acting on the damped driven oscillator matches as closely as possible to a universal wave form, as predicted by the theory of ratchet universality. Speciﬁcally, the optimal approximation to the universal force triggers the almost complete destruction of the nonescaping basin for driving amplitudes which are systematically lower than those corresponding to a symmetric periodic force having the same period. We expect that this work could be potentially useful in the control of elementary dynamic processes characterized by multidirectional escape from a potential well, such as forced chaotic scattering and laser-induced dissociation of molecular systems, among others.

The present work discusses symmetry-breaking-induced bidirectional escape from a symmetric metastable potential well by the application of zero-average periodic forces in the presence of dissipation. We characterized the interplay between heteroclinic instabilities leading to chaotic escape and breaking of a generalized parity symmetry leading to directed ratchet escape to an attractor either at ∞ or at −∞. Optimal enhancement of directed ratchet escape is found to occur when the wave form of the zero-average periodic force acting on the damped driven oscillator matches as closely as possible to a universal wave form, as predicted by the theory of ratchet universality. Specifically, the optimal approximation to the universal force triggers the almost complete destruction of the nonescaping basin for driving amplitudes which are systematically lower than those corresponding to a symmetric periodic force having the same period. We expect that this work could be potentially useful in the control of elementary dynamic processes characterized by multidirectional escape from a potential well, such as forced chaotic scattering and laser-induced dissociation of molecular systems, among others. DOI: 10.1103/PhysRevE.103.022203

I. INTRODUCTION
Obtaining full control of the escape from a potential well is a problem of general interest in science, with broad technological implications in which the required energy to overcome the potential barrier can be supplied by both periodic and nonperiodic forces. Depending upon the force's features, escape can thus occur via the passage of the system over the potential barrier which separates the local potential minimum from one or several neighboring attracting domains. The energy required to surmount the potential barrier can be provided by different mechanisms, including the cases of noise-assisted and chaotic escapes. Diverse examples are known in distinct fields of chemical physics [1,2], electrical transport [3], astronomy and astrophysics [4][5][6], hydrodynamics [7][8][9], and quantum physics [10,11], among many others, in which escape phenomena can often be well described by a low-dimensional system of differential equations. Thus, a deterministic case that has been extensively studied in both dissipative and Hamiltonian systems is that where noise-free one-way escape is induced by an escape-inducing periodic force added to the low-dimensional model system, so that, before escape, chaotic transients of unpredictable duration (owing to the fractal character of the basin boundary) are usually observed for orbits starting from chaotic generic phase space regions (such as those surrounding separatrices). In this scenario, the effectiveness of secondary escape-taming periodic forces in suppressing one-way chaotic escape has been theoretically demonstrated for the case of the main resonance (between the two forces involved) in the context of dissipative systems capable of being studied by Melnikov analysis (MA) techniques [12,13], while its experimental effectiveness has also been demonstrated [14]. Moreover, the suppression of bidirectional chaotic escape from a symmetric quartic potential by secondary escape-taming forces has been demonstrated in the context of a damped-driven one-well Duffing oscillator [15]. In the last decades, the interest in the escape of chains of interacting oscillators out of metastable states [16][17][18][19] has grown in diverse scientific areas, although most of these studies have focused on the Hamiltonian limiting case.
The case of bidirectional escape from a symmetric potential well appears in diverse physical contexts, including solid-state turbulence in anisotropic solids [20], oscillating straight dislocation segments [21], and the boat capsize problem [22,23], and provides a natural scenario to explore the control of ratchet escape (i.e., directed escape from a symmetric potential well by symmetry breaking of zero-mean forces). In this regard, the theory of ratchet universality [24][25][26] predicts that there exists a universal force wave form which optimally enhances directed transport by symmetry breaking. For deterministic ratchets, the effectiveness of the theory of ratchet universality has been demonstrated in diverse physical contexts in which the driving forces are chosen to be biharmonic, such as in the cases of cold atoms in optical lattices [27,28], topological solitons [29], Bose-Einstein condensates exposed to a sawtoothlike optical lattice potential [30], matter-wave solitons [31], and one-dimensional granular chains [32]. Also, the interplay between thermal noise and symmetry breaking in the directed ratchet transport (DRT) of a Brownian particle moving on a periodic substrate subjected to a homogeneous temporal biharmonic force [33][34][35] as well as the cases of a driven Brownian particle subjected to a vibrating periodic potential [26], a driven Brownian particle in the presence of non-Gaussian noise [36], and coupled Brownian motors with stochastic interactions in the crowded environment [37] have been explained quantitatively in coherence with the degree-of-symmetry-breaking (DSB) mechanism, as predicted by the theory of ratchet universality [24,25].
In this present paper, we show that optimum enhancement of ratchet escape is achieved when maximal effective (i.e., critical) symmetry breaking occurs, i.e., when the wave form of the zero-average periodic force acting on the system matches as closely as possible to the exact universal wave form [24,25]. For the sake of clarity, we consider a simple paradigmatic model to discuss the bidirectional ratchet escape scenario: a damped-driven one-well Duffing oscillator described by the equation where all the variables and parameters are dimensionless [20] (δ, γ > 0), while F (t ) is a zero-average T -periodic external force. When the external force presents the shift symmetry, i.e., F (t + T /2) = −F (t ), as in the case of a harmonic force for example, the damped-driven oscillator presents the generalized parity symmetry i.e., if [x(t ), .
x(t )] is a solution of Eq. (1), then so is x(t + T /2)]. This means that nonsymmetric stationary solutions always occur in pairs, including those escaping to the attractors at ±∞. Here, we deliberately choose an external force breaking such a generalized parity symmetry to investigate the directed ratchet escape (DRE) scenario: where cn (·; m) and sn (·; m) are Jacobian elliptic functions [37] of parameter m, and ≡ 2K (m)/T , with K (m) being the complete elliptic integral of the first kind [38]. Fixing T , the wave form of F ellip (t ) ≡ F ellip (t; T, m) changes as the shape parameter m varies from 0 to 1 (see Fig. 1). Physically, the motivation for this choice is that F ellip (t; T, m = 0) = sin (2πt/T )/2 and that F ellip (t; T, m = 1) vanishes; i.e., in these two limits DRE is not possible, while it is expected for 0 < m < 1. Thus, one may expect that the strength of DRE to exhibit a maximum at a certain critical value m = m c as the shape parameter m is varied, the remaining parameters being held constant. The DSB mechanism implies that such a value m c corresponds to a particular force wave form which optimally enhances the ratchet effect. Furthermore, ratchet universality requires that such an optimal wave form should be closely related to that deduced for the case of a biharmonic force, in the sense of its Fourier series. Indeed, by using the Fourier series one could expect the critical value m c to be near m = 0.984 since the optimal values for the biharmonic approximation of the elliptic function are recovered at m = 0.984 (see Ref. [24] for additional details). The rest of the paper is organized as follows. In the next section we obtain analytical estimates of the regions of the parameter space where chaotic escape events prompted by heteroclinic bifurcations can occur by using MA. The analysis of the interplay between such heteroclinic instabilities leading to chaotic escape and the breaking of the generalized parity symmetry leading to DRE to an attractor either at ∞ or at −∞ is provided in Sec. III. Finally, Sec. IV is devoted to a discussion of the major findings and of some open problems.

II. CHAOTIC ESCAPE THRESHOLD
We assume that the complete system (1) satisfies the MA requirements, i.e., the dissipation and excitation terms are small-amplitude perturbations (0 < δ, γ 1) of the underlying conservative Duffing oscillator ..
x + x − 4x 3 = 0 (see Refs. [39,40] for general background). It should be emphasized that the criterion for a homoclinic (or heteroclinic) tangency-accurately predicted by MA in diverse systems [7,15,41]-is coincident with the change from a smooth to an irregular, fractal-looking, basin boundary [42]. It is worth noting that these results connect MA predictions with those concerning the erosion of the basin boundary in phase space.
Straightforward application of MA to Eqs. (1) and (4) yields the Melnikov function corresponding to the elliptic force: where the positive (negative) sign refers to the top (bottom) heteroclinic orbit of the underlying conservative Duffing oscillator: has a simple zero, i.e., there exists a value t 0 such that M ± ellip (t 0 ) = 0 and ∂M ± ellip (t 0 )/∂t 0 = 0, then a heteroclinic bifurcation occurs, signifying the possibility of bidirectional chaotic escape. From Eq. (5) one sees that a heteroclinic bifurcation is guaranteed if where the chaotic threshold function is It is worth noting that condition (10) is the same for DRE to an attractor either at +∞ or at −∞. In other words, the chaotic threshold condition [Eq. (10)] does not provide information relating to the effective scape direction for a given set of parameters and initial conditions. Clearly, this is because condition (10) is the same for the two heteroclinic orbits of the underlying conservative Duffing oscillator. From Eq. (11) one readily obtains U ellip (m, T → 0) = U ellip (m → 1, T ) = 0; i.e., in such limits chaotic escape is not expected (see Fig. 2). Let us consider the chaotic threshold as a function of m, holding T constant. Plots of U ellip (m, T = const) show that each curve presents a single maximum m max = m max (T ) such that m max (T ) increases as T is increased whenever T is larger than a certain value T * , while U ellip (m, T = const) is a monotonically decreasing function of m whenever T < T * [see Figs. 2(a) and 2(c)]. Therefore, if one considers fixing the parameters (δ, γ , T > T * ) for the system to lie at a periodic state (i.e., inside the well), then as m is increased a window of chaotic escape will appear provided the initial periodic state is sufficiently near the chaotic regime. We now study the chaotic threshold as a function of T , holding m constant. Plots of U ellip (m = const, T ) show that each curve asymptotically tends to a constant value which depends on m:

III. SYMMETRY-BREAKING-INDUCED ESCAPE
We explore in this section the effectiveness of the force F ellip (t ) [Eq. (3)] at controlling the strength of DRE in Eq. (1). It is worth recalling that the existence of a universal wave form for optimal enhancement of DRT is a direct consequence of the DSB mechanism: It is possible to consider a quantitative measure of the DSB on which the strength of directed transport by symmetry breaking must depend. This mechanism has led to the unveiling of a criticality scenario for DRT. Indeed, it has been shown that optimal enhancement of DRT is achieved when maximal effective (i.e., critical) symmetry breaking occurs, which is in turn a consequence of two reshaping-induced competing effects: the increase of the DSB and the decrease of the (normalized) maximal transmitted impulse over a halfperiod (I[ f ] ≡ | T /2 f (t )dt|; see Refs. [24,25] for additional details), thus implying the existence of a particular force wave form which optimally enhances DRT. The definitions of the DSB of the symmetries of a T -periodic zero-mean ac force f (t ) are included here for the sake of completeness: where increasing deviation of D s,+,− [ f ] from 1 (unbroken shift and reversal symmetries, respectively) indicates an increase in the DSB (see Refs. [24,25] for additional details). In the case of the elliptic force F ellip (t ), such reshaping-induced competing effects are clearly operating when m varies between 0 and 1, while the optimal enhancement of the ratchet effect is expected to occur at the critical value m c 0.984, as already explained in Sec. I. Indeed, we found that the impulse transmitted by the elliptic force per unit of period over a half-period (14) is a monotonously decreasing function of the shape parameter, while the quantifier of the DSB associated with its shift symmetry is a monotonously increasing function of the shape parameter [cf. Eq. (13)]: where E (m) is the complete elliptic integral of the second kind [38] (see Fig. 3).
For the bidirectional escape model (1), the initial conditions will determine, for a fixed set of its parameters, whether the system escapes to an attractor at ±∞ or settles into a bounded oscillation. Similarly to the case of noise-free one-way escape [7], there can exist a rapid and dramatic erosion of the safe basin (union of the basins of the bounded attractors) due to encroachment by the basins of the attractors at ±∞ (escaping basins). The basins of attraction were computed using a fourth-order Runge-Kutta algorithm with time steps in the range t = 0.005-0.01. To numerically x(t = 0) ∈ [−0.3535, 0.3535], and we selected those initial conditions inside the region bounded by the separatrix formed by the two heteroclinic orbits [Eq. (9)]. From this selected set of initial conditions, each integration is continued until either x (−x) exceeds 5, at which point the system is deemed to have escaped to the attractor at ∞ (−∞), or the maximum allowable number of cycles, here 20, is reached. To provide a quantitative measure of the DRE strength, we calculated the escape probabilities P ± associated with escape to the attractors at ±∞, respectively, and the total escape probability P T = P + + P − versus the shape parameter m. The escape probability is P ± ≡ N ± /N sep , where N ± is the (corresponding) number of starting points from which the system is deemed to have escaped and N sep = 106 673 is the number of starting points inside the separatrix according to the aforementioned criterion.
In the case of a shift-symmetric (harmonic) force (m = 0), we assume that the system presents a slight erosion of the nonescaping basin for a fixed set of parameters (δ, γ , T ) satisfying the chaotic threshold condition [cf. Eq. . Also, all the escape probabilities P + , P − , and P T present a decreasing behavior as m → 1 (see Fig. 5) because the impulse transmitted by the elliptic force per unit of period over a half-period [Eq. (14)] is a monotonously decreasing function of the shape parameter and F ellip (t; T, m = 1) vanishes. It is worth mentioning that we found similar results for other sets of parameters, i.e., the enhancement of the dramatic erosion and stratification of the nonescaping basin is a genuine feature of the DRE scenario associated with the universal force wave form [compare Figs. 4(a) and 4(d)]. Furthermore, we found that the optimal ratcheting force F ellip (t; T, m = m ratchet max ) triggers the almost complete destruction of the nonescaping basin for driving amplitudes which are systematically lower than those corresponding to the shift-symmetric (harmonic) force

IV. CONCLUSIONS
In summary, we have investigated the effectiveness of zeroaverage periodic forces at yielding directed ratchet escape from a symmetric potential well by considering an asymmetric external periodic force. Optimal enhancement of directed ratchet escape is predicted to occur when the wave form of the zero-average periodic force acting on a damped driven oscillator matches as closely as possible to a biharmonic universal wave form, as predicted by the theory of ratchet universality. Our numerical experiments confirmed those findings, as well as revealed the interplay between heteroclinic instabilities leading to chaotic escape and breaking of a generalized parity symmetry leading to directed ratchet escape to an attractor either at ∞ or at −∞. Specifically, the optimal approximation to the biharmonic universal force triggers the almost complete destruction of the nonescaping basin for driving amplitudes which are systematically lower than those corresponding to a symmetric periodic force having the same period. We should emphasize that the directed-ratchet-escape scenario we have discussed is general enough to be applied to many other dissipative nonautonomous systems. Specifically, such a scenario can be readily tested experimentally (for instance, in driven quantum Josephson circuits [11]) and can find application for improving the control of elementary dynamic processes characterized by multidirectional escape from a potential well, such as forced chaotic scattering [43] and transport phenomena in dissipative lattices as well as diverse atomic and molecular processes [44]. Additionally, a natural extension of this work would be to investigate the directed ratchet escape of a chain of coupled driven oscillators over the barriers of a metastable symmetric potential both in the presence and in the absence of dissipation (Hamiltonian limiting case). We would like to investigate this issue in the near future.