## Abstract

A review on the application of Melnikov's method to control homoclinic and heteroclinic chaos in low-dimensional, non-autonomous and dissipative oscillator systems by weak harmonic excitations is presented, including diverse applications, such as chaotic escape from a potential well, chaotic solitons in Frenkel–Kontorova chains and chaotic-charged particles in the field of an electrostatic wave packet.

## 1. Introduction

During the past 15 years or so, diverse techniques of non-feedback chaos control have been proposed (Chen & Dong 1998) that may be roughly classified into three types: (i) the parametric excitation of an experimentally adjustable parameter; (ii) entrainment to the target dynamics and (iii) the application of a coordinate-independent (or dipole) external periodic excitation. It is shown below that techniques (i) and (iii) may be unified in a general setting for the class of dissipative systems considered in this review. There exists numerical, experimental and theoretical evidence that the period of the most effective chaos-controlling excitations usually is a rational fraction of a certain period associated with the uncontrolled system, although the effectiveness of incommensurate excitations has also been demonstrated in some particular cases (Chacón & Martínez 2002). Indeed, resonances between the chaos-controlling excitation and (i) a periodic chaos-inducing excitation; (ii) an unstable periodic orbit embedded in the chaotic attractor; (iii) a natural period in a period-doubling route to chaos or (iv) a period associated with some peak in the power spectrum, have been considered in diverse successful chaos-controlling excitations. This is not really surprising since these types of resonances are closely related to each other. For instance, when a damped, harmonically forced oscillator exhibits a steady chaotic state, the power spectrum corresponding to a given state variable typically presents its main peaks at frequencies which are rational fractions of the chaos-inducing frequency for certain ranges of the chaos-inducing amplitude. The extensive literature concerning experimental, theoretical and numerical studies of non-feedback methods is frankly unapproachable owing to its volume in a review of the present type. Therefore, only the pioneering key work (from the author's viewpoint) is mentioned in the following. The effectiveness of periodic parametric excitations in suppressing chaos was shown by Alekseev & Loskutov (1987). Hübler & Lüscher (1989) discussed how a nonlinear oscillator can be driven toward a given target dynamics by means of resonant excitations. Braiman & Goldhirsch (1991) provided numerical evidence to show the possibility of inhibiting chaos by an additional periodic coordinate-independent excitation. Salerno (1991) showed, by the analysis of a phase-locked map, the possibility of suppressing chaos in long biharmonically driven Josephson junctions. Chacón & Díaz Bejarano (1993) discussed a new way to reduce or suppress steady chaotic states, by altering only the geometrical shape of weak periodic perturbations. Kivshar *et al*. (1994) showed analytically and numerically that the suppression of chaos may be achieved effectively by applying a high-frequency parametric force to a chaotic dynamical system. Experimental control of chaos by weak periodic excitations has been demonstrated in many diverse systems, including magnetoelastic systems (Ditto *et al*. 1990), ferromagnetic systems (Azevedo & Rezende 1991), electronic systems (Hunt 1991), laser systems (Roy *et al*. 1992; Meucci *et al*. 1994; Chizhevsky & Corbalán 1996; Uchida *et al*. 1998), chemical reactions (Petrov *et al*. 1993; Alonso *et al*. 2003), neurological systems (Schiff *et al*. 1994) and plasma systems (Ding *et al*. 1994).

This paper summarizes some main results concerning the application of Melnikov's method (Melnikov 1963; Arnold 1964; Guckenheimer & Holmes 1983; Wiggings 1990) to the problem of control of chaos in low-dimensional, non-autonomous and dissipative oscillator systems by small-amplitude harmonic perturbations. Specifically, the class of systems considered is described by the differential equation(1.1)where *U*(*x*) is a nonlinear potential; , a generic dissipative force which may include constant forces and time-delay terms; , a chaos-inducing excitation; and , an as yet undetermined suitable chaos-controlling excitation, with *F*_{c}(*t*), *F*_{s}(*t*) being harmonic functions of initial phases 0, *Θ* and frequencies *ω*, *Ω*, respectively. It is worth mentioning that Melnikov's method imposes some additional limitations on (1.1): the excitation, the time-delay and the dissipation terms are weak perturbations of the underlying conservative system , which has a separatrix. The original work of Melnikov (1963) was generalized by Arnold (1964) to a particular instance of a time-periodic Hamiltonian perturbation of a ‘two degrees of freedom’ integrable Hamiltonian system. Fifteen years later, Holmes (1979) was the first to apply Melnikov's method (to a damped, forced two-well Duffing oscillator) in the west. Thereafter, the method began to be popular. Chow *et al*. (1980) rediscovered Melnikov's results using alternative methods and emphasized that homoclinic and subharmonic bifurcations are closely related. Through the 1980s, a great variety of extensions and generalizations of Melnikov's approach was developed (Greenspan 1981; Homes & Marsden 1982; Lerman & Umanski 1984; Greundler 1985; Salam 1987; Schecter 1987; Wiggings 1987). The interested reader is referred to the books by Guckenheimer & Holmes (1983), Lichtenberg & Lieberman (1983), Wiggings (1988) and Arrowsmith & Place (1990) for more details and references. The work of Cai *et al*. (2002) provides the simplest extension of Melnikov's method to include perturbational time-delay terms.

The application of Melnikov's method to controlling chaos in low-dimensional systems by weak periodic perturbations began in about 1990. Indeed, Lima & Pettini (1990) provided a heuristic argument to extend the idea that parametric perturbations can modify the stability of hyperbolic or elliptic fixed points, in the phase space of linear systems, to the case of nonlinear systems, and hence that parametric perturbations could also provide a means to reduce or suppress chaos in nonlinear systems. They used, for the first time, the Melnikov method to analytically demonstrate this conjecture in the case of a damped, driven two-well Duffing oscillator subjected to a chaos-suppressing parametric excitation. However, their insufficient analysis of the corresponding Melnikov function led them into gross errors in their final results and conclusions. Specifically, they failed both theoretically to demonstrate the sensitivity of the suppression scenario to the initial phase of the chaos-suppressing excitation and to find it numerically. They also failed theoretically to predict the suppression of chaos in the case of subharmonic resonances (between the chaos-inducing and chaos-suppressing excitations) higher than the main one. Although a part of their erroneous analysis of the Melnikov function originated from a mistake in its calculation (Cuadros & Chacón 1993; Lima & Pettini 1993), its main weakness was in not providing a correct necessary and *sufficient* condition for the Melnikov function to always have the same sign (i.e. for the frustration of homoclinic bifurcations). For the two-well Duffing oscillator that they considered, such a correct necessary and sufficient condition was first deduced for the general case of subharmonic resonances by Chacón (1995*a*), where the extremely important role of the initial phase (of the chaos-suppressing excitation) on the suppression scenario was demonstrated theoretically. Cicogna & Fronzoni (1990) studied the suppression of chaos in the Josephson-junction model(1.2)where the parametric excitation *ξ* cos (*Ωt*+*θ*) sin *ϕ* is the chaos-suppressing excitation, for the single case of the main resonance *Ω*=*ω*, using Melnikov's method. Their insufficient analysis of the Melnikov function (in particular, that of the role played by the initial phase *θ*) also led them into gross mistakes in their final conclusions. On the contrary, it was demonstrated by Chacón (1995*b*) that the effect of the above parametric excitation in (1.2), for the general case of subharmonic resonances (*Ω*=*pω*, *p* an integer), is to suppress the chaotic behaviour when a *suitable* initial phase is used and only for *certain* ranges of its amplitude. It was also shown (Chacón 1995*b*) for the first time that such suitable initial phases are compatible with the surviving natural symmetry under the parametric excitation. It was also conjectured (Chacón 1998) that such maximum survival of the symmetries of solutions from a broad and relevant class of systems, subjected both to primary chaos-inducing and chaos-suppressing excitations, corresponds to the optimal choice of the suppressory parameters; specifically, to particular values of the initial phase differences between the two types of excitations for which the amplitude range of the suppressory excitation is maximum. Rajasekar (1993) applied Melnikov's method to study the suppression of chaos in the Duffing–van der Pol oscillator(1.3)where the additional forcing *η* cos (*Ωt*+*Ωϕ*) is the chaos-suppressing excitation, for the single case of the main resonance *Ω*=*ω*. He pointed out the relevant role of the initial phase (of the chaos-suppressing excitation) in the suppression scenario for the first time and also deduced the analytical expression of the boundaries of the regions in the *η*–*φ* phase plane where homoclinic chaos is inhibited. A generalization of Rajasekar's approach concerning the relative effectiveness of any two weak excitations in suppressing homoclinic/heteroclinic chaos is discussed in the work of Chacón (2002), where general analytical expressions are derived from the analysis of generic Melnikov functions, providing the boundaries of the regions as well as the enclosed area in the amplitude–initial phase plane of the chaos-suppressing excitation where homoclinic/heteroclinic chaos is inhibited. In addition, a criterion based on the aforementioned area was deduced and shown to be useful in choosing the most suitable of the possible chaos-suppressing excitations. Cicogna & Fronzoni (1993) analysed the Melnikov function associated with the family of systems(1.4)where *ϵg*(*x*) cos (*Ωt*+*θ*) is the chaos-suppressing excitation, for the single case of the main resonance *Ω*=*ω*. They deduced both the suitable suppressory values of the initial phase *θ* and the associated chaotic threshold function (*γ*/δ)_{th} when the chaos-suppressing excitation acts on the system. General results (Chacón 1999) concerning suppression of homoclinic/heteroclinic chaos were derived on the basis of Melnikov's for the family (1.1), for the general case of subharmonic resonance (*Ω*=*pω*, *p* an integer). There, a generic analytical expression was deduced for the maximum width of the intervals of the initial phase *Θ* for which homoclinic/heteroclinic bifurcations can be frustrated. It was also demonstrated that {0,*π*/2,*π*,3*π*/2} are, in general, the only optimal values of such initial phase, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation. The work of Chacón (2001*a*) presents general results concerning *enhancement* or maintenance of chaos for the family (1.1), where the connection with the results on chaos suppression was discussed in a general setting. It was also demonstrated that, in general, a second harmonic excitation can reliably play the role of an enhancer or an inhibitor by solely adjusting its initial phase. The work of Chacón (2001*b*) provides a preliminary Melnikov method-based approach concerning suppression of chaos by a chaos-suppressing excitation, which satisfies an *ultrasubharmonic* resonance condition with the chaos-inducing excitation. This approach was further applied to the problem of the inhibition of chaotic escape from a potential well by *incommensurate* escape-suppressing excitations (Chacón & Martínez 2002).

## 2. Basic theoretical approach

To illustrate the theoretical approach with a paradigmatic example, consider a single Josephson junction subjected to a nonlinear dissipative term and driven by two harmonic excitations (Chacón *et al*. 2001*b*)(2.1)where *x* and *t* are dimensionless variables and is proportional to the difference in potential between the two superconductors. It is also assumed that the dissipation and excitation terms are regarded as weak perturbations and *βF* sin (*Ωt*+*Θ*) is the chaos-suppressing excitation. The nonlinear dissipative term appears in the study of a single Josephson junction when the conditions are such that the interference effects between the pair and quasi-particle currents should be taken into account (Barone & Paterno 1982). The application of the Melnikov method to (2.1) yields the Melnikov function(2.2)with(2.3)Turning to the general case (1.1), let us assume that such a family of systems satisfies the requirements of the Melnikov method. Then, the application of the method to (1.1) provides the generic Melnikov function(2.4)where har (*τ*) means indistinctly sin (*τ*) or cos (*τ*) and *A* is a non-negative function, while *D* and *B* can be non-negative or negative functions, depending upon the respective parameters for each specific system. In particular, *D* contains the effect of the damping, time-delay terms and constant forces. In the absence of time-delay terms and constant drivings, *D*<0, while one has the three cases *D*≷0 when a constant driving and a time-delay term act on the system. In addition, *A* and *B* contain the effect of the chaos-inducing and chaos-controlling excitations, respectively. It can be noted that changing the sign of *B* is equivalent to having a fixed shift of the initial phase: , where the two signs before *π* apply to each of the sign superscripts of *Ψ*. Therefore, *B* will be considered (for example) as a positive function in the following. As phase and initial time *τ*_{0} are not fixed, one may study the simple zeros of by choosing quite freely the trigonometric functions in (2.4). Therefore, consider, for instance, the Josephson junction given by (2.1). It is worth noting that the Melnikov functions and *M*^{±}(*t*_{0}) (cf. (2.4) and (2.2), respectively) are connected by *linear* relationships, known for each specific system (1.1):(2.5)Therefore, the control theorems associated with any Melnikov function can be straightforwardly obtained from those associated with *M*^{±}(*t*_{0}; Chacón 1999).

### (a) Suppression of chaos

As is well known, the Melnikov method provides estimates in parameter space for the appearance of homoclinic (and heteroclinic) bifurcations and hence for transient chaos. This means that, in most of the cases, only *necessary* conditions for steady chaos (strange chaotic attractor) are obtained from the method. Therefore, one may always get sufficient conditions for the inhibition of even transient chaos (frustration of homoclinic/heteroclinic bifurcation) and, *a fortiori*, for the inhibition of the steady chaos that ultimately arises from such a homoclinic/heteroclinic bifurcation. This is the principal foundation of the utility of Melnikov method in predicting the suppression of (steady) chaos when a homoclinic/heteroclinic bifurcation occurs prior to its emergence. For the Josephson junction (2.1), one has the following theorem (Chacón *et al*. 2001*a*,*b*):

*Let Ω*=*pω*, *p* *an integer, such that, for* *M*^{+}(*t*_{0}) (*M*^{−}(*t*_{0})), *is satisfied for some integers* *m* *and* *n*. *Then,* *M*^{±}(*t*_{0}) *always has the same sign, specifically* *M*^{±}(*t*_{0})<0, *if and only if the following conditions are satisfied*:

(2.6)

Now, the following remarks are in order.

First, one can test the suppression theorem theoretically by considering the limiting Hamiltonian case (*α*=0). It can be noted that, in the absence of dissipation, (2.6) must be rewritten as *β*_{min}≤*β*≤*β*_{max}, *β*_{min}≡*R*, *β*_{max}≡*R*/*p*^{2}, since *β*_{min} cannot be zero now. Thus, one obtains (Chacón *et al*. 2001*a*,*b*) *Ω*=*ω*, *Θ*=*π* and *β*=1 as necessary and sufficient conditions for suppressing stochasticity. This result can be trivially obtained, to *first perturbative order*, from (2.2) and (2.3) with *α*=0, i.e. having *M*^{±}(*t*_{0})=0 for all *t*_{0}.

Second, the lower threshold for the chaos-suppressing amplitude, *β*_{min}, takes into account the *strength* of the initial chaotic state through the factor (1−*C*/*A*), since one usually finds that the corresponding maximal Lyapunov exponent *λ*^{+} increases as the ratio *C*/*A* decreases over a certain range of parameters. Therefore, for fixed chaos-inducing and chaos-suppressing frequencies (and hence *R* fixed), one would expect that *β*_{min} will increase as *λ*^{+} is increased. It can be noted that the corresponding upper threshold, *β*_{max}, does not verify this important property, because *β*_{max} arises from a *necessary* condition for the necessary condition yielding *β*_{min} also to be a sufficient condition. This means that *β*_{max} slightly underestimates the upper threshold for the chaos-suppressing amplitude, as is observed numerically and experimentally in different instances. It is worth noting that this remark holds for any Melnikov function (2.4).

Third, the asymmetry between the upper and lower homoclinic orbits (cf. (2.2) and (2.3), respectively) gives rise to two distinct sets of *optimal* initial phases that are suitable for suppressing chaos. The optimal suppressory values of *Θ* (hereafter denoted as *Θ*_{opt}) are those allowing the widest amplitude ranges for the chaos-suppressing excitation (the use of the adjective is justified below in the discussion of the suitable intervals of initial phase difference for taming chaos). Indeed, theorem 2.1 requires having *Θ*=*Θ*_{opt}≡*π*,*π*/2,0,3*π*/2 (*π*,3*π*/2,0,*π*/2) for *p*=4*n*−3,4*n*−2,4*n*−1,4*n* (*n*=1,2,…), respectively, in order to inhibit chaos when one considers orbits initiated near the upper (lower) homoclinic orbit. These distinct values are those compatible with the *surviving natural symmetry* under the additional forcing. Indeed, the dissipative, harmonically driven Josephson junction (*β*=0) is invariant under the transformation(2.7)where *n* is an integer, i.e. if is a solution of (2.1) with *β*=0, then so is . This pair of solutions may be essentially the same in the sense that they may differ by an integer number of complete cycles, i.e.(2.8)with *l* an integer, and termed symmetric. Otherwise, the time-shifting and sign reversal procedure yields a different solution, and the two solutions are termed broken-symmetric. When *β*>0 and *Θ* is arbitrary, the aforementioned natural symmetry is generally broken. The reason for this breaking is(2.9)for arbitrary *ω*, *Ω* and *Θ*. Assuming a resonance condition *Ω*=*pω*, the survival of the above symmetry implies(2.10)Obviously, this is only the case for an *odd* integer *p*. For an even integer *p*, one has the new transformation [*x*→−*x*, *t*→*t*+(2*n*+1)*π*/*ω*, *Θ*→*Θ*±*π*]. In other words, if is a solution for a value *Θ*, then so is for *Θ*±*π*. Thus, this explains the origin of the differences between the corresponding (at the same resonance order) allowed *Θ*_{opt} values for the two homoclinic orbits. Similar results have been found for the damped, driven pendulum mounted on a vertically oscillating point of suspension (Chacón 1995*b*). Therefore, this maximum symmetry principle appears to be the common background in the mechanism of regularization, by the application of resonant excitations.

Fourth, the width of the allowed interval [*β*_{min},*β*_{max}] for regularization is(2.11)with *R* given by (2.6). Since *R* is a positive function, there always exists a *maximum resonance order* *p*_{max} for suppression of homoclinic (and heteroclinic) chaos, for each fixed initial chaotic state (i.e. *C*/*A* fixed), which is(2.12)where the brackets indicate integer part. From (2.12), one sees that *p*_{max} increases as the *C*/*A* ratio is increased, which would be associated with the decrease in the corresponding maximal Lyapunov exponent over a certain range of parameters. For a given set of parameters satisfying theorem 2.1 hypothesis, as the resonance order *p* is increased, the allowed interval [*β*_{min},*β*_{max}] shrinks quickly for *low* frequencies. This means that initial chaotic states cannot necessarily be regularized to periodic attractors of arbitrary *long* period, since numerical experiments indicate that the regularized responses are typically a period-1 attractor for *p*=1 and a period-2 attractor for *p*=2. On the other hand, the asymptotic behaviour Δ*β*(*ω*→∞)=∞ (the remaining parameters being held constant) means that chaotic motion is not possible in this limit, as expected.

Fifth, to establish the suppression theorem corresponding to any Melnikov function (2.4), it is enough to transform into the form given by (2.2). Therefore, taking into account (2.5) and the aforementioned *Θ*_{opt} values, one finds that, in general, there exist at most four suitable optimal values for the suppressory initial phase difference between the two (commensurate: *Ω*=*pω*) excitations: 0,*π*/2,*π*,3*π*/2.

### (b) Enhancement of chaos

It has been mentioned earlier that the mechanism for suppressing homoclinic (and heteroclinic) chaos is the frustration of a homoclinic/heteroclinic bifurcation, which prevents the appearance of horseshoes in the dynamics. Chacón (2001*a*) showed that the enhancement of the initial chaos is achieved by moving the system from the homoclinic tangency condition *even more* than in the initial situation with no second periodic excitation. Thus, enhancement of chaos can mean increasing the duration of a chaotic transient, passing from transient to steady chaos or increasing the maximal Lyapunov exponent. Consider again that the family of systems modelled by (1.1) satisfies the requirements of the Melnikov method. Similar to the preceding discussion of the suppression of chaos, one can assume any particular form of the Melnikov function (2.4) to discuss the enhancement of chaos. Therefore, consider, for instance, the following nonlinearly damped, biharmonically driven, two-well Duffing oscillator,(2.13)where *η*, *Ω* and *Θ* are the normalized amplitude factor, frequency and initial phase, respectively, of the chaos-controlling parametric excitation (0<*η*≪1), and *β*,*δ*,*n*,*F* and *ω* are the normalized parameters of the potential coefficient, damping coefficient, damping exponent, chaos-inducing amplitude and chaos-inducing frequency, respectively (0<*δ*, *F*≪1, *β*>0, *n*=1,2, …). The application of the Melnikov method to (2.13) yields the Melnikov function(2.14)with(2.15)where the positive (negative) sign refers to the right (left) homoclinic orbit of the underlying integrable two-well Duffing oscillator (*δ*=*F*=*η*=0); and *B*(*m*,*n*) is the Euler beta function. It has been demonstrated (Chacón 2001*a*) that, in general, a second harmonic excitation can reliably play an enhancer or inhibitor role *solely* from adjusting its initial phase. The Melnikov function *M*^{+}(*t*_{0}) will be used here to illustrate the approach to the enhancement of chaos. Indeed, consider that, in the absence of any second parametric excitation (*C*=0), the associated Melnikov function changes sign at some *t*_{0}, i.e. |*D*|≤*A*. If one now lets the second excitation act on the system such that *C*≤*A*−|*D*|, this relationship represents a sufficient condition for *M*^{+}(*t*_{0}) to change sign at some *t*_{0}. Thus, a necessary condition for *M*^{+}(*t*_{0}) to always have the same sign (*M*^{+}(*t*_{0})<0) is *C*>*A*−*D*≡*C*_{min}. It was mentioned above (Chacón 1999) that a sufficient condition for *C*>*C*_{min} to also be a sufficient condition for inhibiting chaos is *Ω*=*pω* (subharmonic resonance condition), *C*≤*C*_{max}≡*A*/*p*^{2}, *p* an integer, and that and −*C*_{min,max}sin (*Ωt*_{0}+*Θ*) are *in opposition*. This condition yields the optimal suppressory values . It was demonstrated (Chacón 2001*a*) that imposing to be *in phase* with −*C*_{min,max}sin (*Ωt*_{0}+*Θ*) is a *sufficient* condition for *M*^{+}(*t*_{0}) to change sign at some *t*_{0}. This condition provides the optimal enhancer values of the initial phase, , in the sense that *M*^{+}(*t*_{0}) presents its highest maximum at , i.e. one obtains the maximum gap from the homoclinic tangency condition. Now, the following remarks are in order.

First, for a given homoclinic orbit forming (part of) a separatrix, one has in general (i.e. for any Melnikov function (2.4))(2.16)for each resonance order.

Second, for *C*=*C*_{min}, there always exists a *maximum-range* interval(2.17)of permitted initial phases for enhancement of chaos in the sense that, for values of *Θ* belonging to that interval, the maxima of *M*^{+}(*t*_{0}) are higher than those of . Similarly, for *C*=*C*_{max}, there always exists a *different* maximum-range interval(2.18)of allowed initial phases for enhancement of chaos, and also(2.19)which is a consequence of the dissipation. It must be emphasized that the definition of is general, i.e. it refers to any resonance and any Melnikov function (2.4).

Third, for general separatrices, i.e. those formed by several homoclinic and (or) heteroclinic loops, the above scenario of control of chaos holds for *each* homoclinic (heteroclinic) orbit. However, it is common to find that the different homoclinic (heteroclinic) orbits of a given separatrix yield *distinct* values. This is a consequence of the survival of the symmetries existing in the absence of the second excitation. Thus, the actual scenario is usually more complicated. For instance, let and be the optimal values associated with the right and left homoclinic orbits, respectively, of a typical separatrix with a ‘figure-of-eight’ loop, as in the two-well Duffing oscillator (2.13). One then obtains that the best chance for enhancing chaos should now be at See Chacón (2001*a*) for more details.

### (c) Further developments

The case of subharmonic resonance between the chaos-inducing and chaos-controlling frequencies has been briefly discussed earlier. However, a number of theoretical (Salerno 1991; Salerno & Samuelsen 1994), numerical (Braiman & Goldhirsch 1991) and experimental (Uchida *et al*. 1998) studies show that chaos can be reliably controlled by other non-subharmonic resonances. The work of Chacón (2001*b*) presents a Melnikov method-based approach concerning reduction of homoclinic and heteroclinic instabilities for the family of systems (1.1), where the harmonic excitations verify an ultrasubharmonic resonance condition: *Ω*/*ω*=*p*/*q*, *q*>1 (*p*≠*q*), *p*, *q* positive integers and *Ω*(*ω*) the chaos-suppressing (inducing) frequency. Such general results can be used to approach the case of *incommensurate* chaos-suppressing excitations by means of a series of ever better rational approximations, which are the successive convergents of the infinite continued fraction associated with the irrational ratio *Ω*/*ω*. This procedure has been much employed in characterizing strange non-chaotic attractors in quasi-periodically forced systems as well as in studying phase-locking phenomena in both Hamiltonian and dissipative systems. To illustrate the method, one intentionally chooses the golden section , since it is the irrational number which is the worst approximated by rational numbers (in the sense of the size of the denominator). As is well known, the golden section can be approximated by the sequence of rational numbers (*Ω*/*ω*)_{i}=*F*_{i−1}/*F*_{i}, where *F*_{i}=1,1,2,3,5,8, … are the Fibonacci numbers, such that . For each (*Ω*/*ω*)_{i}, one replaces each quasi-periodically excited system(2.20)by the respective periodically excited system(2.21)giving a sequence of periodically excited systems whose associated frequencies satisfy an ultrasubharmonic resonance condition. The work of Chacón & Martínez (2002) applied this approach to the problem of the reduction of chaotic escape from a potential well using the simple model(2.22)where *βηx*^{2}sin (*Ωt*+*Θ*) is the escape-suppressing excitation. They found that, for irrational escape-suppressing frequencies, the effective escape-reducing initial phases are found to lie close to the *accumulation* points of the set of suitable initial phases that are associated with the complete series of convergents up to the convergent giving the chosen rational approximation.

A Melnikov method-based approach (Chacón 2002) was presented concerning the *relative effectiveness* of harmonic excitations in suppressing homoclinic (and heteroclinic) chaos of the family (1.1) for the main resonance between the chaos-inducing and chaos-suppressing excitations. A criterion based on the area in the suppressory amplitude/initial phase parameter plane, where suppression of homoclinic chaos is guaranteed, was deduced and shown to be useful in choosing the most suitable of the possible chaos-suppressing excitations. Additionally, the choice of the most suitable chaos-suppressing excitation was shown to exhibit *sensitivity* to the particular initial chaotic state.

The work of Chacón *et al*. (2003) presents general findings concerning control of chaos for the family(2.23)where *U*(*x*) is a general potential; , a generic dissipative force; , a general multiple chaos-inducing excitation; and , an as yet undetermined suitable multiple chaos-controlling excitation, with *F*_{ch,i}(*t*) and *F*_{co,j}(*t*) being harmonic functions of common frequency *ω* and initial phases 0 (*i*=1, …, *N*) and *φ*_{j} (*j*=1, …, *M*). The effectiveness of this approach, in suppressing spatio-temporal chaos of chains of identical chaotic-coupled oscillators, was demonstrated through the example of coupled Duffing oscillators, where coherent oscillations were achieved under *localized* control.

The work of Chacón *et al*. (2002) studied the robustness of the suppression of bidirectional chaotic escape of a harmonically driven oscillator from a quartic potential well by the application of weak parametric excitations. It was numerically shown that Melnikov method-based theoretical predictions also work for harmonic escape-inducing excitations in the presence of external noise and for chaotic escape-inducing excitations having a sharp Fourier component with a sufficiently high power.

The method proposed in the work of Lenci & Rega (2003) consists of choosing the shape of external and/or parametric periodic excitations, which permits one to avoid, in an optimal manner, a homoclinic bifurcation. They numerically investigated the effectiveness of the control method with respect to the basin erosion and escape phenomena of a perturbed Helmholtz oscillator.

## 3. Some applications

### (a) Taming chaotic escape from a potential well

The work of Chacón *et al*. (1996, 1997, 2001*a*,*b*) and Balibrea *et al*. (1998) applies the above-mentioned Melnikov method-based approach to the problem of chaotic escape from a potential well. This is a general and ubiquitous phenomenon in science. Indeed, one finds it in very distinct contexts: the capsizing of a boat subjected to trains of regular waves (Thompson 1989), the stochastic escape of a trapped ion induced by a resonant laser field (Chacón & Cirac 1995) and the escape of stars from a stellar system (Contopoulos *et al*. 1993) are some important examples. Remarkably, such complex escape phenomena can often be well described by a low-dimensional system of differential equations. The case considered by Chacón and co-workers is that where escape is induced by an external periodic excitation added to the 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 both dissipative and Hamiltonian systems. In particular, Chacón *et al*. (1996) studied the simplest model for a universal chaotic escape situation(3.1)where *βηx*^{2}sin (*Ωt*+*Θ*) and *βηx*sin (*Ωt*+*Θ*) are the (independently considered) escape-suppressing parametric excitations. It was demonstrated that the parametric excitation of the linear (quadratic) term suppresses chaotic escape more efficiently than that of the quadratic (linear) term for small (large) driving periods of the primary chaos-inducing excitation. Chacón *et al*. (1997) studied the inhibition of chaotic escape of a driven oscillator from the cubic potential well that typically models a metastable system close to a fold(3.2)where and are the (independently considered) linear and nonlinear damping terms, respectively. They demonstrated that the effectiveness of a parametric excitation at suppressing chaotic escape from such a cubic potential well diminishes as the system approaches a period-1 *parametric resonance*, and that, for linear damping, the parametric excitation inhibits chaotic escape more efficiently than for nonlinear damping. The role of a nonlinear damping term, proportional to the *n*th power of the velocity, on the escape-inhibition scenario is considered in the work of Chacón *et al*. (2001*a*,*b*),(3.3)where *ηx*^{2}cos (*Ωt*+*Θ*) is the escape-suppressing parametric excitation. In this case, the effectiveness of the parametric excitation of the quadratic potential well in inhibiting chaotic escape diminishes as the system approaches either a period-1 or a period-2 parametric resonance. In addition, the effectiveness of the parametric excitation in the presence of the nonlinear dissipative force is less than that for a linear dissipative force.

### (b) Taming chaotic solitons in Frenkel–Kontorova chains

Control of chaos in spatially extended systems is one of the most important and challenging problems in the field of nonlinear dynamics. Instances of possible applications include the stabilization of superconducting Josephson-junction arrays (Barone & Paterno 1982), periodic patterns in optical turbulence and semiconductor laser arrays (Schöll 2001), to cite just a few. Martínez & Chacón (2004) presented a Melnikov method-based general theoretical approach to control chaotic *solitons* in damped, noisy and driven Frenkel–Kontorova chains. Specifically, they studied the model(3.4)where *βF* cos (*Ωt*+*φ*) is the chaos-suppressing excitation and *ξ*(*t*), a bounded noise term. They obtained an effective equation of motion governing the dynamics of the soliton centre of mass for which they deduced Melnikov method-based predictions concerning the regions in the control parameter space where homoclinic bifurcations are frustrated. Numerical simulations indicated that such theoretical predictions could be reliably applied to the original Frenkel–Kontorova chains, even for the case of *localized* application of the soliton-taming excitations. It is worth mentioning that the same effectiveness of such a localized control in suppressing spatio-temporal chaos of chains of identical chaotic-coupled oscillators was demonstrated through the example of coupled Duffing oscillators (Chacón *et al*. 2003).

### (c) Taming *chaotic-charg*ed particles in the field of an electrostatic wave packet

The interaction of charged particles with an electrostatic wave packet is a basic and challenging problem appearing in diverse fundamental fields, such as astrophysics, plasma physics and condensed matter physics. While the Hamiltonian approach to this problem is suitable in many physical contexts, the consideration of dissipative forces seems appropriate in diverse phenomena, such as the stochastic heating in the dynamics of charged particles interacting with plasma oscillations. In any case, stochastic (chaotic) dynamics already appears (can appear) when the wave packet solely consists of two electrostatic plane waves. Such a non-regular behaviour of the charged particles may yield undesirable effects on a number of technological devices, such as the destruction of magnetic surfaces in tokamaks. Thus, apart from its general intrinsic interest, the problem of regularization of the dissipative dynamics of charged particles in an electrostatic wave packet by a small-amplitude uncorrelated wave (which is added to the initial wave packet) is especially relevant in plasma physics. Chacón (2004) considered the simplest model equation to examine this problem:(3.5)where *E*_{c}sin (*k*_{c}*x*−*ω*_{c}*t*) and *E*_{s}sin (*k*_{s}*x*−*ω*_{s}*t*) are the chaos-inducing and chaos-suppressing waves, respectively. In a reference frame moving along the main wave *E*_{0}sin (*k*_{0}*x*−*ω*_{0}*t*), (3.5) transforms into a perturbed pendulum equation, which is capable of being studied by means of Melnikov's method. Two suppressory mechanisms were identified. One mechanism requires chaos-inducing and chaos-suppressing waves to have both commensurate wavelengths and commensurate relative phase velocities, while the other allows chaos to be tamed when these quantities are incommensurate.

## 4. Conclusions and open problems

The present review summarizes some of the main results and applications of a preliminary theoretical approach to control chaos in dissipative, non-autonomous dynamical systems, capable of being studied by Melnikov's method, by means of periodic excitations. Diverse extensions and applications of the current theory remain to be developed. Among them:

to obtain the boundaries of the regularization regions in the control parameter space for the case of a general resonance (not just the main) between the involved excitations,

to extend the theoretical approach to (some family of) multidimensional systems capable of being studied by (some generalized version of) Melnikov's method,

to develop a multiharmonic control theory beyond the main resonance case,

to extend the theoretical approach for the case of periodic excitations to the case of random excitations,

to obtain analytical approximations of the system's solutions when the conditions for suppressing chaos are satisfied,

to extend the current theory described for harmonic excitations to the case of general periodic excitations (both chaos-inducing and chaos-controlling). In particular, the

*waveform effect*should be taken into account in the control scenario,to extend the current theory to the case where the chaos-controlling excitation is a parametric excitation of the amplitude of the chaos-inducing excitation, as well as to the case where it is a parametric excitation of the frequency of the chaos-inducing excitation,

to apply the current theory to ratchet systems to improve the directed energy transport, and

to apply the current theory to control chaotic population oscillations between two coupled Bose–Einstein condensates with time-dependent asymmetric potential and damping.

## Acknowledgments

The author acknowledges the financial support from the Spanish MCyT and the European Regional Development Fund (FEDER) program through FIS2004-02475 project.

## Footnotes

One contribution of 15 to a Theme Issue ‘Exploiting chaotic properties of dynamical systems for their control’.

- © 2006 The Royal Society