Optimal Control of Nonregular Dynamics in a Duffing Oscillator

Nonlinear Dynamics 33: 71–86, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

Optimal Control of Nonregular Dynamics in a Duffing Oscillator

STEFANO LENCIIstituto di Scienza e Tecnica delle Costruzioni, Università Politecnica delle Marche, via Brecce Bianche,I-60131 Ancona, Italy; E-mail: [email protected]

GIUSEPPE REGADipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma ‘La Sapienza’, via A. Gramsci 53,I-00197 Roma, Italy; E-mail: [email protected]

(Received: 11 July 2002; accepted: 27 May 2003)

Abstract. A method for controlling nonlinear dynamics and chaos previously developed by the authors is appliedto the classical Duffing oscillator. The method, which consists in choosing the best shape of external periodicexcitations permitting to avoid the transverse intersection of the stable and unstable manifolds of the hilltopsaddle, is first illustrated and then applied by using the Melnikov method for analytically detecting homoclinicbifurcations. Attention is focused on optimal excitations with a finite number of superharmonics, because theyare theoretically performant and easy to reproduce. Extensive numerical investigations aimed at confirming thetheoretical predictions and checking the effectiveness of the method are performed. In particular, the eliminationof the homoclinic tangency and the regularization of fractal basins of attraction are numerically verified. Thereduction of the erosion of the basins of attraction is also investigated in detail, and the paper ends with a study ofthe effects of control on delaying cross-well chaotic attractors.

Keywords: Periodic excitation, one-side and global optimal control, homoclinic bifurcations, Duffing oscillator,basin erosion, cross-well chaos.

1. Introduction

The hardening twin-well Duffing equation is one of the most widely investigated equationsin the field of applied nonlinear dynamics [1–6], and it describes the single-mode nonlineardynamics of buckled beams [7], of magnetoelastic pendulum [2] and of many others mech-anical systems and structures. In addition to these mechanical applications, it is the archetypeof twin-well symmetric oscillators and exhibits very rich nonlinear dynamics. Among othersphenomena, we mention

1. the existence and possible competition of confined (in-well) and scattered (cross-well)attractors;

2. the existence and possible competition of periodic and chaotic attractors;3. the occurrence of fractal basin boundaries responsible for chaotic transient and sensitivity

to initial conditions.

Scattered chaotic dynamics and fractal basin boundaries, in particular, are usually unwanteddynamical events whose elimination (or shift in the parameter space) represents a very desir-able goal.

All of the previous properties are related to the intersection of the stable and unstablemanifolds of the hilltop saddle, and are triggered by its homoclinic bifurcation. Based on these

72 S. Lenci and G. Rega

considerations, this paper deals with the problem of eliminating the homoclinic bifurcations ofthe hilltop saddle of the hardening two-well Duffing oscillator by means of a general controlmethod recently developed by the authors [8–11].

The method was first formulated for optimal control of chaos in a discontinuous mechan-ical system [8], and then generalized to continuous nonlinear oscillators and systematicallyapplied for controlling nonregular dynamics in a Helmholtz oscillator [11]. As regards theDuffing oscillator, this method adds to other chaos control approaches developed and appliedin the past. Moon et al. [12], after a preliminary summary of various control methods, experi-mentally control (regularize) the nonlinear dynamics of a magnetoelastic pendulum (a modelof a buckled beam) by using a minor variant of the occasional proportional feedback control,a method developed by Hunt [13] which in turn is a smart, semi-empirical variant of thefamous OGY method [14]. Pinto and Gonçalves still refer to the buckled beam, and controlits nonlinear dynamics (i) by varying the axial load by means of a classical optimized statefeedback control [15], and (ii) by applying concentrated moments at suitable points along thebeam axis [16]. Osipov et al. [17] apply the method of feedback impulsive chaos suppressionto the Duffing oscillator with positive instead of negative linear stiffness.

A comparison of various control methods applied to the Duffing equation is performedin [18, 19]. Sifakis and Elliott [19] consider four different control techniques. (i) Open loop-periodic perturbation method, which, similarly to the method investigated in the present work,consists in adding a periodic perturbation to the system excitation. However, in [19] thisperturbation is choosen empirically by a trial-and-error procedure, while in our work it isoptimally determined on the basis of a deep theoretical analysis based on system dynamicalproperties. (ii) Continuous delayed feedback method, developed by Pyragas [20], which con-sists in applying a force proportional to the difference between the current state of the systemand the state at some time earlier. (iii) The Hunt method [13] and (iv) the OGY method [14].Agrawal et al. [18], on the other hand, consider more classical control techniques and applythe optimal polynomial control and the robust sliding mode control.

The main difference between our method and other methods is that while other approachesare concerned with stabilization (or creation) of a single, given, orbit, our approach is aimedat providing an overall control of the dynamics, since it is based on the elimination of one ofthe underlying causes of complex and undesired behaviour. Thus, one must be aware of whatkind of control is actually needed for the application at hand, which may also depend on therelevant cost of implementation, and choose the control strategy accordingly.

The present method consists in identifying the shape of the periodic excitation whichpermits to avoid, in an optimal manner, the transverse intersection of the stable and unstablemanifolds of the hilltop saddle. It is developed in various sequential steps: (i) detection ofthe homoclinic bifurcation, which is accomplished by the Melnikov’s method; (ii) study ofthe dependence of the homoclinic bifurcation on the shape of the excitation; (iii) formulationand resolution of the mathematical problem of optimization which consists in determiningthe (optimal) theoretical excitation which maximizes the distance between stable and unstablemanifolds for a fixed excitation amplitude or, equivalently, the critical amplitude for occur-rence of homoclinic bifurcation; (iv) numerical implementation of the optimal excitation,which is required to confirm the theoretical predictions and to check the feasibility and theactual performances of the technique.

The first three steps, which are reported in Section 2, are mainly theoretic, and followthe same guidelines as other authors’ works [8, 10, 11]. The last point, on the other hand,is considered in Section 3, where extensive numerical simulations aimed at verifying the

Optimal Control of Nonregular Dynamics in a Duffing Oscillator 73

effectiveness of the control procedure are performed. The following points are consideredspecifically: (i) numerical verification of the Melnikov prediction of the homoclinic bifurca-tions (Section 3.1); (ii) analysis of the regularization of fractal basin boundaries, which is thefirst predicted effect of homoclinic bifurcation elimination (Section 3.2); (iii) reduction of thefractal erosion of basins of attraction of confined attractors (Section 3.3); (iv) effect of controlon cross-well chaotic dynamics, to be investigated by the analyses of the relevant bifurcationdiagrams (Section 3.4). Finally, some conclusions end the paper (Section 4).

2. Theoretical Aspects of the Control Method

Let us consider the dimensionless Duffing equation

x + εδx − x

2+ x3

2= εγ (ωt) = εγ1

∞∑j=1

γj

γ1sin(jωt + j), (1)

which is the version used in [3, 4, 12, 21] and which is used here to refer to the results of theseworks. Other versions are commonly found in the literature (for example, x+εδx−x+2x3 =εγ (ωt) is used in [10] and ¨x + εδ ˙x − x + x3 = εγ (ωt) in [22, 23, 19]), but all of them areequivalent up to a rescaling of space and time.

In Equation (1) εδ is the damping coefficient, which is assumed equal to 0.1 in all forth-coming numerical simulations (just as done in [3, 4]), and εγ (ωt) is the generic 2π/ω-periodicexternal excitation. We use the right hand side expression of γ (ωt) because we assign εγ1 therole of amplitude, while the remaining dimensionless parameters γj/γ1 govern the shape ofthe excitation. In practice, they measure the superharmonic corrections to the basic harmonicexcitation, which is commonly used in the applied nonlinear dynamics literature and is hereconsidered as a reference to measure the improvements provided by the control method.

The dimensionless parameter ε is introduced to emphasize the smallness of damping andexcitation, which indeed are considered as perturbations of the conservative case ε = 0.The case of parametric and external excitations acting simultaneously on the system can betreated by considering an ‘equivalent external excitation’, just as done in [11] in the case ofthe Helmholtz oscillator.

The potential V (x) = −x2/4 + x4/8 associated to Equation (1) and the unperturbed phasespace are depicted in Figure 1. This picture clearly shows the two homoclinic loops of thehilltop saddle x1 = 0, which are analytically given by

xhoml,r (t) = ∓

√2

cosh(t/√

2), (2)

while the stable centers are x0,2 = ∓1.When the perturbations are added, the stable and unstable manifolds of x1 split, and for

sufficiently large values of the excitation amplitude they intersect. The first excitation givingtangencies of manifolds corresponds to the homoclinic bifurcation [1] and can be computedanalytically by the Melnikov’s method [6]. The application of this method is standard, and theMelnikov’s function representing the first order approximation (in ε) of the signed distancebetween stable and unstable manifolds is given by

Ml,r(m) =∞∫

−∞xhom

l,r (t)[−δxhoml,r (t) + γ (ωt + m)] dt. (3)


Figure 1. The potential (a) and the unperturbed phase space (b) of the Duffing equation.

The argument m of the Melnikov functions is actually given by m = ωt0 + φ0 [6], where(t0,φ0) is a parametrization of the two-dimensional (in the three-dimensional space (x, x,t))stable and unstable manifolds. In particular, t0 is a time-like parameter along the unperturbedhomoclinic loop on a given, fixed Poincaré section, and φ0 is a phase between two consecutivestroboscopic Poincaré sections. Thus, fixing φ0 means measuring the manifolds distance ona fixed Poincaré section (as done, for example, in Figures 4 and 5), while fixing t0 meansmeasuring the manifolds distance along the time for a fixed point of the unperturbed phasespace.

The integrals in Equation (3) can be computed analytically, and the results are written inthe following form:

Ml,r(m) = −2√

2δ

3

[1 ∓ γ1

γ h1,cr(ω)

h(m)

], γ h

1,cr(ω) = δ√

2

3

cosh(

ωπ√2

)ωπ

,

h(m) =∞∑

j=1

hj cos(jωt + j), hj = γj

γ1

j cosh(

ωπ√2

)cosh

(jωπ√

2

) . (4)

Note that h1 = 1, that h(m) is 2π -periodic and has zero mean value, and that the effectsof superharmonic corrections in the Melnikov function are governed by the parameters hj ,j ≥ 2.

Expression (4) shows that the distance is made of a constant part plus an oscillating part, thefirst being proportional to the damping and the second to the excitation amplitude. Accordingto the Melnikov’s theory, we have homoclinic intersection of the left (right) manifolds if thereexists an m ∈ [0, 2π ] such that Ml,r(m) have a simple zero, respectively. It is not difficult toshow that the equation Ml(m) = 0 has solution, namely, there is homoclinic intersection on theleft part of the phase space, if and only if

γ1 >γ h

1,cr(ω)

Ml

def= γ l1,cr(ω), Ml def= max

m∈[0,2π]{h(m)}. (5)

Analogously, the equation Mr(m) = 0 has solution, namely, there is homoclinic intersectionon the right part of the phase space, if and only if

γ1 >γ h

1,cr(ω)

Mr

def= γ r1,cr(ω), Mr def= − min

m∈[0,2π]{h(m)}. (6)


Figure 2. The curves γ h1,cr(ω), and γ

l,r1,cr(ω) for Gl,r = 4/π (mathematical solution of global control) and

Gl,r = 2 (mathematical solution of one-side control).

For a given, generic, excitation, the curves γl,r1,cr(ω) represent the loci of the left and right

homoclinic bifurcations, respectively, in the frequency/amplitude parameter space (ω, γ1),and separate the zone where homoclinic intersections do not occur (below the critical curve)from that where homoclinic intersections do occur (above the critical curve) (Figure 2). Notethat in general γ l

1,cr(ω) and γ r1,cr(ω) are distinct, and coincide only for the class of excitations

satisfying max{h(m)} = − min{h(m)}. For example, this occurs for the harmonic excita-tion, for which h(m) = cos(ωt + 1) and Ml = Mr = 1. This further shows that thecurve γ h

1,cr(ω), which is also reported in Figure 2, represents the (coinciding) left and righthomoclinic bifurcations with harmonic excitation.

The strip above γ h1,cr(ω) and below γ

l,r1,cr(ω)) is called saved (i.e., controlled) region in

Figure 2, and it represents the zone where the unharmonic excitation is theoretically effective.Its (maximum) enlargement constitutes the objective of the control method.

To quantitatively measure the increment of the critical thresholds of the unharmonic ex-citations with respect to the harmonic reference case we introduce the gains, which are theratios

Gl def= γ l1,cr(ω)

γ h1,cr(ω)

= 1

Ml, Gr def= γ r

1,cr(ω)

γ h1,cr(ω)

= 1

Mr. (7)

Let us first consider the right homoclinic bifurcation. The central idea of the control methodis to reduce the region of homoclinic intersection by varying the shape of the excitation. Thisentails increasing γ r

1,cr as much as possible by varying the Fourier coefficients γj and j ofγ (ωt), and mathematically requires solving the following optimization problem:

Maximizing Gr by varying hj and j , j ≥ 2, of h(m). (8)

Once the coefficients hj and j solution of (8) have been determined, the optimal excitationpermitting the maximum increase of the critical threshold for the right homoclinic bifurcationcan be obtained by inverting (4)4, i.e.,

εγ ropt(ωt) = εγ1

∞∑j=1

hj

cosh(

jωπ√2

)j cosh

(ωπ√

2

) sin(jωt + j). (9)


Table 1. The reduced optimal solutions in the case of right one-sidecontrol.

N GrN h2 h3 h4 h5

2 1.4142 0.353553

3 1.6180 0.552756 0.170789

4 1.7321 0.673525 0.333274 0.096175

5 1.8019 0.751654 0.462136 0.215156 0.059632

. . . . . . . . . . . . . . . . . .

∞ 2 1 1 1 1

This is called ‘optimal right excitation’, and the method based on this excitation is called‘one-side (right) control’ [9].

The case of left homoclinic bifurcation is similar, with the unique difference that in (8)one must maximize Gl instead of Gr and, accordingly, the approach is called ‘one-side (left)control’ [9]. Since − min{h(m)} = max{−h(m)}, these two problems are mathematicallyequivalent.

Apart from the solution of the optimization problems, to be discussed in the following,the peculiarity of ‘one-side’ controls is that only one homoclinic bifurcation is considered,irrespective of what happens to the other. Thus, generally speaking, in this case one is ableto control only one part of the phase space, while having to accept a possible worseningon the other part. It seems therefore natural to develop also another approach, based on thesimultaneous increasing of the two critical thresholds for homoclinic bifurcations. Since

γ l1,cr = γ r

1,cr ⇔ Gl = Gr ⇔ max{h(m)} = − min{h(m)}, (10)

this entails solving

Maximizing Gdef= (Gl = Gr) by varying hj and j , j ≥ 2, of h(m),

under the constraint − min{h(m)} = max{h(m)}. (11)

The control based on the solution of (11) is called ‘global control’ [9] and its main feature isthat of controlling the whole phase space. The ‘optimal global excitation’ is still given by (9),where now hj and j are solutions of (11).

Problem (11) is clearly a constrained version of (8). Accordingly, the optimal gain is lesser,so that the global control is theoretically less performant then the one-side controls, but, atleast in principle, it permits controlling the whole phase space. These considerations showthat the two approaches are complementary and not competing.

The mathematical solutions of problems (8) and (11) are Gr = 2, hj = 1, j = 0 (whichcorresponds to the constant function −1/2 plus a Dirac delta function of amplitude π at m =0), and G = 4/π , hj = sin(jπ/2)/j , j = 0 (this instead corresponds to the function h(m) =(π/4)sign[cos(m)]), respectively [8]. Unfortunately, they are physically inadmissible, becausethe Fourier series of the corresponding excitations γ (ωt) would be divergent, as clearly seenby (9), so that other solutions are sought. Among various possibilities, in this work we considerthe (optimal) solutions obtained with a finite number N of superharmonics, while the case ofa constraint added to the optimization problem to guarantee the physical admissibility of themathematical solution is investigated in [10].


Table 2. The reduced optimal solutions in the case of global control.

N GN h3 h5 h7 h9

3 1.1547 -0.166667

5 1.2071 -0.232259 0.060987

7 1.2310 -0.264943 0.100220 -0.028897

9 1.2440 -0.284314 0.125257 -0.053460 0.016365

. . . . . . . . . . . . . . . . . .

∞ 1.2732 -0.333333 0.200000 -0.142857 0.111111

The considered reduced solutions are still characterized by the condition j = 0, andtheir superharmonic coefficients are reported in Tables 1 and 2 for increasing number N ofemployed superharmonics. Note that the optimal solutions of the global control, which isaimed at being ‘symmetric’, have no even superharmonics.

These solutions are interesting because they are easy to reproduce in experiments, ap-plications, and in the following numerical simulations (Section 3). Moreover, they providegains which are quite close to the upper bounds 2 and 4/π corresponding to the physicallyinadmissible mathematical solutions, which are reported in the last row of Tables 1 and 2,respectively, for the sake of comparison.

3. Numerical Simulations

This section is devoted to the numerical simulation of the theoretical performances and ofthe effectiveness of the control method previously illustrated. A classical fourth order Runge–Kutta’s method is employed to integrate Equation (1), with time steps equal to 1/200 of theperiod of the excitation.

Based on the observation that cross-well chaos is a really unwanted phenomenon, weinvestigate the parameter region where it occurs for the lowest values of the excitation. Ithas been shown by Szemplinska-Stupnicka and co-workers [3, 4] that in the case of refer-ence harmonic excitation, this happens at the vertex of a V-shaped region in the excitationfrequency-amplitude parameter space, which is reported in Figure 3. The tip of this regionoccurs on the left, i.e., for lower frequencies, of the principal external resonance ω = 1.

In the case of Equation (1) for εδ = 0.10 (the value used here and in [3] and [4]), thevertex of the V-region occurs for ω = 0.78 ÷ 0.79 (Figure 3). This point is characterized by acodimension 2 event (the simultaneous occurrence of the final saddle node bifurcation of thenonresonant cycle and of the final boundary crisis of the complete period doubling cascadeoriginating from the resonant oscillation, see Figure 3), and may involve some degeneracies.Thus, in order not to consider pathological effects, but still remaining close to the vertex, thevalue ω = 0.8 is used in the following. From Equation (4)2 we have that the homoclinic bifurc-ation in the case of harmonic excitation is εγ h

1,cr = 0.0570, while by inverting Equation (4)4

we obtain the following Fourier coefficients of the reduced optimal excitations to be used inthe following: γ3/γ1 = −1.8884 (global control with 1 superharmonic); γ3/γ1 = −2.6317and γ5/γ1 = 14.4963 (global control with 2 superharmonics); γ2/γ1 = 1.0170 (right one-sidecontrol with 1 superharmonic).

Since |γj/γ1| > 1, the control is practically not suitable. However, (i) this does not happenfor every value of the excitation frequency, and (ii) in any case, we can consider another type


Figure 3. The V-shaped region of cross-well chaos in the case of harmonic excitation (see [21]).

of optimal excitations where, roughly speaking, the solution is sought in the class of functionssatisfying |γj/γ1| < const. These are the bounded (possibly reduced) optimal excitationswhich have been obtained in a very general context in [10]. They entail a practically acceptableenergy of the excitation, but are more involved, and their investigation is out of the scope ofthe present paper.

3.1. CONTROL OF HOMOCLINIC BIFURCATIONS

The Melnikov method is a perturbative technique which gives accurate results only for ε suf-ficiently small. Accordingly, some numerical simulations are performed to confirm Melnikovtheoretical predictions on homoclinic bifurcations. This will enlighten us on the mechanismof elimination of the homoclinic bifurcation accomplished by the optimal excitations.

We first deal with global control, and consider εγ1 = εγ1,cr|N=3 = 1.1547∗εγ h1,cr = 0.0659

corresponding to the homoclinic bifurcation threshold in the case of a single superharmonic(Table 2). Thus, with this excitation we should theoretically have homoclinic tangency, andthis is confirmed by the saddle-manifolds phase portrait of Figure 4b. In the case of harmonicexcitation, on the other hand, the homoclinic bifurcation occurs for a lower amplitude, andaccordingly Figure 4a shows homoclinic intersection in this case. Finally, in the case of op-timal excitation with two superharmonics (N = 5, see Table 2), the homoclinic bifurcationoccurs for εγ1,cr|N=5 = 1.2071 ∗ εγ h

1,cr = 0.0688, and in fact the manifolds in Figure 4c donot intersect.

Figure 4 further shows how the control excitations topologically avoid the manifold inter-sections. Focusing attention on the right neighborhood of the saddle, it is possible to see thatthe control excitation reverses the peak of the (right) stable manifold intersecting the (right)unstable manifold (compare Figures 4a and 4b), and this is done in an optimal way – i.e., thereversal is maximized – by the optimal excitation.

The ‘symmetric’ behaviour of Figures 4a and 4b is lost in the case of one-side (say,right) control, where the right and left homoclinic bifurcations occur for different excitationamplitudes, εγ r

1,cr|N=2 = 1.4142 ∗ εγ h1,cr = 0.0807 and εγ l

1,cr|N=2 = 0.7388 ∗ εγ h1,cr = 0.0421,

respectively. These values are computed as follows: the optimal superharmonic coefficient inthe case N = 2 is h2 = √

2/4 = 0.353553 (Table 1), so that the optimal h(m) is given by


Figure 4. The saddle-manifolds phase portraits for εγ1 = 0.0659, ω = 0.80, and for (a) harmonic excitation,(b) global optimal excitation with one superharmonic, and (c) global optimal excitation with two superharmonics.

h(m) = cos(m)+ (√

2/4) cos(2m). The maximun and the minimum of this function are givenby 1 + √

2/4 and −√2/2, respectively, so that Gr = −1/ min{h(m)} = √

2 = 1.4142(Table 1) and Gl = 1/ max{h(m)} = 4/(4 + √

2) = 0.7388. Note that only the right(controlled) homoclinic bifurcation is increased, while the left (uncontrolled) homoclinic bi-furcation is lowered (gain lesser than one). This represents a characterizing aspect of theone-side control, which is aimed at controlling only one bifurcation, even accepting a worsebehaviour of the other.

Figures 5a and 5b report the saddle-manifolds phase portraits for εγ1 = 0.0421 and εγ1 =0.0807, respectively. According to theoretical predictions, we have homoclinic tangency onthe left and homoclinic detachment on the right part in Figure 5a, while we have homoclinicintersection on the left and homoclinic tangency on the right in Figure 5b. Thus, we canconclude that the Melnikov predictions are numerically confirmed.

3.2. REGULARIZING FRACTAL BASIN BOUNDARIES

The regularization of fractal basin boundaries is an important expected result of the proposedmethod, and is investigated in this section for εγ1 = 0.0650. It is worth to remark that onlyfractality between different wells can be eliminated by controlling the homoclinic bifurcationof the hilltop saddle, while the in-well fractality between coexisting confined attractors (e.g.,resonant and non-resonant oscillations) is likely due to other homoclinic bifurcations, and, atleast in principle, can be controlled by the same method applied to the involved – non-hilltop– saddle [24].

The considered excitation amplitude is larger than εγ h1,cr = 0.0570, and accordingly we

observe in Figure 6a fractal basin boundaries between the right and left attractors under har-


Figure 5. The saddle-manifolds phase portraits for right one-side optimal excitation, ω = 0.80, and for thetheoretical values of (a) left, εγ1 = 0.0421, and (b) right, εγ1 = 0.0807, homoclinic bifurcations.

Figure 6. Basins of attractions for εγ1 = 0.0650, ω = 0.80 and for (a) harmonic excitation, (b) global optimalexcitation with one superharmonic, and (c) right one-side optimal excitation with one superharmonic.

monic excitation. The (left-light grays/right-dark grays) fractality is modest, due to closenessof the considered excitation to the homoclinic bifurcation threshold, and the behaviour is‘symmetric’ according to the nature of the excitation.

The ‘symmetry’ is maintained by the global control (Figure 6b), which indeed entailsno fractal basin boundaries, according to the fact that the homoclinic bifurcation for thisexcitation occurs for a (slightly) larger value of the amplitude (εγ1,cr|N=3 = 0.0659).

In the case of (right) one-side control the ‘symmetry’ is lost. In fact, for εγ1 = 0.065 wehave homoclinic intersection of the left manifolds (being εγ l

1,cr = 0.0421) and no intersectionof the right manifolds (being εγ r

1,cr = 0.0807). Thus, as expected, we observe in Figure 6c(left-light gray/right-dark grays) fractal basins of attraction on the left, and regular basins


Figure 7. The basins erosion curves for ω = 0.80 and for (a) harmonic excitation, (b) global optimal excitationwith one superharmonic, and (c) right one-side optimal excitation with one superharmonic.

on the right. Furthermore, the (left-light gray/right-dark grays) fractality is rather extendedon the left due to the considerable distance from the left homoclinic bifurcation threshold inparameter space.

Figure 6 permits to compare the different behaviours of global and one-side controls. Infact, the former permits a limited reduction of fractality (related to the low theoretical gain) onthe whole phase space, while the latter gives a strong reduction of fractality in the controlledpotential well (related to the high theoretical gain), but extended fractality in the uncontrolledwell, namely, a localized control in the phase space. The different extension of the fractalzones with harmonic and one-side optimal excitations is due to the different distance from therelevant homoclinic bifurcations triggering the fractalization.

3.3. REDUCING BASIN EROSION

In the previous section it has been shown how the controlled excitations are able to regularize(left-light grays/right-dark grays) fractal basin boundaries, a question which is now invest-igated in greater detail. In fact, it is well known that for growing values of the excitationamplitude, fractal tongues penetrate the basins of the confined attractors and destroy theirintegrity, a phenomenon which has been called ‘Basin Erosion’ and which has been deeplyinvestigated in the past [22, 23]. This erosion first leads to extended fractality of the wholephase space, namely, to strong sensitivity to initial conditions, and then facilitates the finalcrisis leading to the appearance of the scattered chaotic attractor. Since this phenomenon istriggered by the homoclinic bifurcation of the hilltop saddle, it is expected that it can bereduced by optimal excitations.

To have a quantitative measure of the reduction of basins erosion, we employ the IntegrityFactor (IF), which is defined in the Appendix and therein compared with another index ofbasins erosion. The erosion can be highlighted by drawing the IF versus the excitation amp-litude, while the reduction of this erosion can be detected by comparing the erosion curves forthe three considered excitations: harmonic (reference), global control with one superharmonicand right one-side control with one superharmonic. This is done in Figure 7, which clearlyshows the sharp fall of the erosion curves (this justifies the name ‘Dover Cliff’ erosion used,e.g., in [22]) and which permits to determine the influence of control.

The first aspect to be underlined is that Figure 7 clearly shows how the global optimalexcitation (curve (b)) is able to shift toward larger amplitudes the erosion curve with re-


spect to the harmonic excitation (curve (a)), namely, it is effective in reducing the erosion.Furthermore, the starting points of erosion are just after, and in good agreement with, therespective homoclinic bifurcation thresholds (marked by vertical segments), so confirmingthat the homoclinic bifurcation is the necessary pre-requisite for erosion, though this is alsogoverned by other and more complex nonlinear phenomena [22].

The controlled excitation profile is sharper than that of the reference excitation, and afterthe fall the IF is smaller. This agrees with similar characteristics observed in the case of theHelmholtz oscillator [11], and proves that there is a well defined interval (in between thevertical segments of curves (a) and (b)) where the control is effective, which has been calledpractical saved region in [11].

The case of (right) one-side control has different properties. In fact, due to the asymmetryof this excitation, the left and right wells have a different behaviour. The erosion curve of theleft uncontrolled potential well is denoted as (c)l , and, according to the theoretical predictions,it is lower than curve (a), namely, there is a strong fractalization in the uncontrolled potentialwell, and even a strong reduction of the extent of the basin of the left attractor(s) (Figure 6c).In the right controlled potential well, on the other hand, basically there is no erosion at all(curve (c)r ), in very good agreement with the theoretical predictions.

The one-side erosion curves end at εγ1∼= 0.066, where the last attractor belonging to

the left potential well disappears by a saddle-node bifurcation, as shown in the forthcomingFigure 8c. For εγ1 > 0.066 (and until εγ1 < 0.0951, see Figure 8c) there are only confinedattractors in the controlled potential well, so that the question of cross-well fractalization ofthe basins makes no more sense.

We close this section by noting that the curve (c)r of Figure 7 does not fall down. Thisagrees with the fact that the right homoclinic bifurcation occurs at εγ r

1,cr = 0.0807, a valuewell above the point where the curve disappears, which, accordingly, is not reported in Fig-ure 7.

3.4. DELAYING CROSS-WELL CHAOTIC ATTRACTOR

The final numerical check concerns the effects on the scattered chaotic attractor. In thisrespect, it is worth to remark that there is no precise theoretical expectation (as for the regular-ization of fractal basin boundaries), because the homoclinic bifurcation of the hilltop saddledoes not directly influence the attractors scenario. Nevertheless, previous numerical investiga-tions made on a different (non-smooth versus smooth) two well system [9] highlighted, amongother aspects, the practical coincidence of the homoclinic bifurcation threshold and of thecritical excitation entailing scattered steady chaos. So, it seems worth investigating this aspectfor the Duffing oscillator, too, and verifying whether, incidentally, the optimal excitations areable to shift the first appearance of cross-well chaotic attractor.

The bifurcation diagram of the reference harmonic excitation is reported in Figure 8a,which clearly shows the two well known S-shaped paths of the non-resonant (higher branch)and resonant (lower branch), left and right, confined oscillations. The resonant cycle finallyundergoes a period doubling cascade which first leads to a confined chaotic attractor, existingin a very narrow range of excitation amplitudes and named ‘structurally unstable’ in [21], andthen to the scattered, robust chaotic attractor.

Confirming previous results (see, e.g., [21] and Figure 3), no strict connection is observedin this case – contrary to the system in [9] – between the homoclinic bifurcation thresholdand the cross-well chaos threshold, whose value εγ1 = 0.0878 is greater than εγ h

1,cr = 0.0570


Figure 8. The bifurcation diagrams for ω = 0.80 and for (a) harmonic excitation, (b) global optimal excitationwith one superharmonic and (c) right one-side optimal excitation with one superharmonic.

(Figure 8a). In terms of control, one could thus conjecture that, very roughly speaking, anincrease of the homoclinic bifurcation threshold of at least 0.0878/0.0570 = 1.54 is requiredto possibly eliminate (or, better, delay the occurrence of) the scattered chaotic attractor. Thissuggests using at least the one-side control with two superharmonics, which is the minimaloptimal excitation satisfying GN ≥ 1.54, as shown in Tables 1 and 2.

The ineffectiveness of global control is highlighted by the bifurcation diagram of Figure 8b,which shows that indeed scattering is anticipated by the controlled excitation. However, this isnot actually a drawback of the method, because the control excitations have been (optimally)designed to eliminate the mixing due to the homoclinic bifurcation of the hilltop saddle (andthey do succeed in this respect, as shown in the previous sections), and not to delay thescattering transition, which is not theoretically connected with that homoclinic bifurcation.Whether the scattering transition delay were the objective, one should proceed in a differentway, still applying the proposed method, as done, for example, in [24].

The performance is very different in the case of one-side control, as shown by the bi-furcation diagram reported in Figure 8c. The asymmetry of this excitation is clearly reflectedin the asymmetry of the system response, and now the two S-paths are distinct: the one inthe controlled potential well occurs for amplitudes higher than those corresponding to thepath in the uncontrolled well. The resonant cycle of the former reaches the scattered chaoticattractor, whereas the resonant cycle of the latter ends at εγ1 = 0.0604 after a classical perioddoubling cascade. Both non-resonant branches of the two S-paths disappear by saddle-nodebifurcations, at εγ1

∼= 0.075 and εγ1∼= 0.066, respectively.

In the range 0.066 < εγ1 < 0.0951, all attractors are confined and belong to the controlledpotential well. Thus, the mechanism of control consists in capturing the dynamics in the


controlled well by destroying the attractor in the uncontrolled well. This phenomenon hasbeen called confinement in [9, 24].

Contrary to the case of global control and in spite of being G2 = 1.41 < 1.54, thecontrolled excitation is able to delay the appearance of scattered chaotic attractor, thus ex-hibiting partially unexpected resources. Thus, in the range 0.0878 < εγ1 < 0.0951 notonly the dynamics have been confined, but they have also been regularized (see also [24]),as the cross-well chaos has been substituted by confined periodic attractors. Confinement andregularization are very interesting characteristics of one-side control.

4. Conclusions and Further Developments

The problem of controlling nonlinear dynamics and chaos in the Duffing oscillator by a shrewdchoice of the shape of the external periodic excitation, has been addressed.

After having developed the theoretical aspects of control, which are similar to those en-countered in other cases [8, 10, 11], attention has been focused on the numerical simulation ofthe system response under harmonic and controlled excitations, the former being consideredas a reference to measure the performances of control. Four points have been highlightedspecifically:

1. the numerical reliability of the Melnikov predictions of homoclinic bifurcations, whichcan be usefully applied in spite of their perturbative nature;

2. the control induced regularization of fractal basin boundaries, which confirms the firstpredicted effect of elimination of the homoclinic bifurcation;

3. the reduction of fractal erosion of the basins of attraction of confined attractors, which isvery useful for reducing the sensitivity to initial conditions;

4. the effect of control on the scattered chaotic attractor, which has been investigated throughthe analysis of the relevant bifurcation diagrams, showing the effectiveness of the one-sidecontrol in increasing the in-well to cross-well transition threshold.

In all numerical simulations attention has been paid to the analysis of one-side versusglobal controls, and it has been numerically confirmed that the two approaches are comple-mentary with each other, the former providing low gain but control of the whole phase space,the latter high gain but control of only a part of the phase space.

Summarizing, we can say that this work has shown that the theoretical predictions of themethod are numerically confirmed, and that the control procedure works effectively. Further-more, it has been shown that, in some cases, the control performances extend also beyond thetheoretical expectations, permitting, for example, the delay of the in-well to cross-well chaostransition in the case of one-side control.

This paper extends other authors’ works aimed at showing how it is possible to simplycontrol the nonlinear dynamics of various mechanical systems by an appropriate choice ofthe shape of the excitation. In this respect, it represents a further step toward a full under-standing of the control procedure, which should be accomplished by considering also morecomplex systems, e.g., with more degrees of freedom. Furthermore, it is believed that physicalexperimentations would certainly increase the knowledge and reliability of the method, andconstitute a possible development of this work.


Appendix: The Integrity Factor

The Integrity Factor (IF), which is a quantitative measure of the basins erosion, is definedin the following way. Let us fix the phase space region where all of the relevant phenomenaare localized. In the present work, we consider the window −1.8 ≤ x ≤ 1.8, −1.0 ≤ x ≤1.0, which surrounds the two potential wells (Figure 1b). For a given value of the excitationamplitude εγ1 we construct the attractor-basin-phase portrait and we draw the largest circleswhich contain only points of the right (respectively, left) basins of attraction. Clearly, thelarger are these circles, the larger is the cross-well integrity of the basin, the smaller is thecross-well fractality and the related sensitivity to initial conditions. The IF is the normalizedradius of this circle, namely, the ratio between the radius of the circle for the given εγ1 andthe radius of the corresponding circle for εγ1 = 0. Examples of these circles are reported inFigure 6. Note that the circles (partly) cover the basins of all attractors belonging to the samepotential well, so that the IF is actually a property of the well rather than of the basins.

The present approach is different from that used in [22], where the erosion is measuredby considering the ‘Safe Basin of Attraction’, defined as the set of all initial conditions that,roughly, do not change well in time, and the Global Integrity Measure is defined as the nor-malized area of this set. Although different, the two approaches are similar in some respect:indeed, both of them do not distinguish between different in-well attractors, and both giveerosion curves which are very similar to each other and likely equivalent from a practicalpoint of view (compare our Figure 7 with figure 4 in [22]).

The main difference, on the other hand, is that in our approach the integrity is measuredon the steady dynamics, irrespective of the trajectory possibly visiting the other well duringthe transient, while the authors of [22] take into account both the cross-well transient and thesteady dynamics, by including only the initial conditions leading to in-well transient. Thus,the latter approach is more conservative from an engineering viewpoint, and more appropriatein the applications where the transient dynamics are important. However, it is worth to remarkthat referring to steady dynamics seems to be more consistent with the features of the presentcontrol method and more suitable for checking its performances. Indeed, the elimination ofthe invariant manifolds intersection directly influences the basins of attraction while it onlyindirectly influences the safe basins of attraction as defined in [22]: these are not actuallybasins of attraction in the dynamical systems sense (although they are numerically close tothem) and, accordingly, they are not bounded by stable manifolds.

Another basic difference is related to the unlike measure of integrity, namely, (largest)circle of initial conditions leading to in-well attractors versus safe basins of attraction. In thisrespect, it is the present approach to be topologically more conservative, as in the Lansburyet al. [22] approach also fractal tongues of initial conditions, which give sensitivity to initialconditions and are practically unwanted, contribute to the integrity measure, while they areneglected in the present approach.

Acknowledgement

The authors wish to thank the COFIN research program of MIUR, Italy.


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Optimal Control of Nonregular Dynamics in a Duffing Oscillator