Optimal numerical control of single-well to cross-well chaos transition in mechanical systems

Optimal numerical control of single-well to cross-wellchaos transition in mechanical systems

Stefano Lenci a,*, Giuseppe Rega b

a Istituto di Scienza e Tecnica delle Costruzioni, Universit�aa di Ancona, via Brecce Bianche, Monte D’Ago, 60131 Ancona, Italyb Dipartimento di Ingegneria Strutturale e Geotecnica, Universit�aa di Roma ‘‘La Sapienza’’, via A. Gramsci 53, 00197 Roma, Italy

Accepted 27 May 2002

Abstract

The optimal numerical control of nonlinear dynamics and chaos is investigated by means of a technique based on

removal of the relevant homo/heteroclinic bifurcations, to be obtained by modifying the shape of the excitation. To

highlight how the procedure works, the analysis is accomplished by referring to the Duffing equation, although the

method is general and holds, at least in principle, for whatever nonlinear system. Attention is focused on the single-well

to cross-well chaos transition due to a homoclinic bifurcation of an appropriate period 3 saddle [Int. J. Bifur. Chaos 4

(1994) 933]. It is shown how it is possible to eliminate this bifurcation simply by adding a single superharmonic cor-

rection to the basic harmonic excitation. Successively, the problem of the optimal choice of the superharmonic is

addressed and solved numerically. The optimal solutions are determined in the two cases of symmetric (odd) and

asymmetric (even) excitations, and it is shown how they entail practical, though variable, effectiveness of control in

terms of confinement and regularization of system dynamics.

� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

The question of controlling nonlinear dynamics and chaos of various mechanical systems largely benefits from the

knowledge of the main properties of system dynamics as well as of the basic mechanisms involved in its organization

and evolution. As a matter of fact, only the understanding––and the successive detection––of the involved mechanisms

can suggest the most appropriate––and possibly optimal––modifications of the external action applied to the system, or

of the system itself, aimed at achieving control of its dynamics. In other words, although in the past several empirical

methods have been developed and applied with some success to control nonlinear dynamics and chaos (see, for example

[4,19] and the very accurate and updated Ref. [5]), it is believed that any kind of ‘‘optimal’’ control should be based on

the appropriate knowledge of system dynamics, irrespective of the meaning of the underlying optimization.

Contrary to what can be thought at first glance, this aspect is shared also by several methods which apparently do

not require system knowledge, like, for example, the well-known OGY method [17]. In fact, in these methods the

dynamics are usually allowed for a preliminary free evolution after which, based on the analysis of sensors output, an

approximation of a mathematical model of the system or a full reconstruction of its dynamics is performed by means of,

e.g., the delay coordinates embedding theory [1,2]. The control is then applied––and possibly optimized––just exploiting

the dynamical properties of the considered model.

Even in the relatively simple case of a single-d.o.f. systems, nonlinear dynamics are characterized by several items

including local and/or global bifurcations, cascades and other routes to chaos, external and/or parametric resonances,

* Corresponding author. Fax: +390-71-220-4576.

E-mail address: [email protected] (S. Lenci).

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Chaos, Solitons and Fractals 15 (2003) 173–186www.elsevier.com/locate/chaos

mail to: [email protected]

attractors versus unstable solutions, invariant manifolds, basins of attractions. Among others, an important role is

played by the invariant manifolds [7,26,27], whose analysis ‘‘provides a useful stepping stone in the understanding

of the overall system dynamics’’ [9]. As a matter of fact, they constitute the boundaries of basins of attractions, they

are usually the skeleton of chaotic attractors, they are involved in global bifurcations and in other topological

phenomena with meaningful dynamic consequences, their intersection determines, via the Smale–Birkhoff theorem, the

chaotic saddle responsible for sensitivity to initial conditions (SIC), and so on. In particular, the homo/hetero-

clinic bifurcations of selected saddles are seen to be the mechanisms responsible for fractalization of basin bound-

aries with consequent SIC, for appearance/disappearance of chaotic attractors or for their sudden enlargement/

reduction (boundary or interior crisis, respectively [6,18]), for transition from single-well to cross-well chaos [9], and

they are also the triggering phenomena of the safe basin erosion suddenly leading to escape from a potential well

[22,23].

The previous considerations illustrate the usefulness of eliminating, by suitable control techniques, the homoclinic

bifurcations embedded in the dynamics of various mechanical systems and responsible for unwanted events. This

question has been studied in depth by the authors [10–14] in a series of papers aimed at investigating the main features

of a specific control technique based on removal of homoclinic bifurcations by varying the shape of the excitation. In

those works, attention was paid to the homoclinic bifurcation of the relevant main (hilltop) saddle. This permits to

reduce the safe basin erosion and the escape from potential well in softening systems [12], and to shift toward higher

excitation amplitudes the appearance of scattered dynamics, while regularizing them, in hardening systems [13]. Within

the general theoretical framework of the control method, one important aspect of previous works is the analytical

detection of global bifurcations, which have been identified either exactly or approximately by the Melnikov’s method

[7]. This has great practical advantages, because it permits to drive analytically the control procedure, a point which has

been extensively exploited to obtain optimal controls. These were achieved by modifying the shape of a reference

harmonic excitation applied to the system through the proper addition of a certain number of superharmonics [10,12–

14].

Unfortunately, such a possibility of analytical treatment is specific of the hilltop saddle of Hamiltonian systems,

while, to the authors’ knowledge, there are no analytical tools able to detect specific homo/heteroclinic bifurcations of

other saddles which are sometimes seen to play the dominant role in system dynamics (see, e.g., [9]). In such cases, the

lack of mathematical tools requires a fully numerical approach, where the invariant manifolds and their bifurcations are

detected numerically [8,25]. Repeating this analysis for different values of the governing parameters around the bi-

furcation threshold, and for uncontrolled versus controlled systems, it is possible to numerically determine the effects of

control, and, at least in principle, to realize an optimal control, i.e., to provide given results with the minimal effort.

This is the main subject of this paper, where, to the authors’ knowledge, the matter is investigated for the first time.

To focus attention on the main ideas and highlight how the procedure works, we consider an archetypal Duffing

equation known to deserve a great interest in the nonlinear dynamics of several mechanical systems. This permits

having one-dimensional manifolds, which are numerically manageable, whereas the restriction to cubic nonlinearity is

unessential, and is chosen only for illustrative purposes. Furthermore, we still apply a control strategy based on varying

the shape of the excitation, possibly in an optimal way.

In our numerical computations, the saddles are determined by a modified Newton method [16], while the invariant

manifolds are detected by standard numerical algorithms based on forward and backward iterations of the unstable and

stable eigenvectors, respectively [3,8,25]. This analysis is to be repeated for several values of the governing parameters,

which requires lengthy and time consuming numerical computations, even though they can be automatized. Thus, in

order to perform reasonable analyses, only the control based on a single superharmonic added to a basic harmonic

excitation is considered. Other cases involving higher superharmonics and/or, e.g., parametric and external excitations,

could be treated analogously with some extra efforts.

The Duffing equation

€xxþ 0:164 _xx� 0:2xþ x3 ¼ A½sinðtÞ þ c1 sinðnt þ c2Þ�; ð1Þ

previously studied by Ueda et al. [24] and Katz and Dowell [9] in the case of harmonic excitation (c1 ¼ 0), is analyzed.The numbers c1 and c2 are the control parameters. In particular, c1 measures the relative amplitude of the superhar-monic (the physical amplitude is Ac1), while c2 measures the phase difference, or simply the phase. Eq. (1) has nega-tive linear stiffness and two symmetric potential wells (with x being measured from the hilltop saddle), so that it

describes the motions of various mechanical systems, including buckled beams, magnetoelastic pendulum, etc. [15]. It

has very rich nonlinear dynamics, with periodic and/or chaotic attractors [20,26,27]. Furthermore, it allows for both

confined (within the potential wells) and scattered (cross-well) dynamics, which can coexist and compete with each

other.

174 S. Lenci, G. Rega / Chaos, Solitons and Fractals 15 (2003) 173–186

We focus attention on the transitions from single-well to cross-well chaos, one of which is due to a homoclinic

bifurcation of a period 3 saddle, as shown by Ueda et al. [24] and Katz and Dowell [9]. This seems to be the most

dangerous dynamical event in many practical applications, because it leads to inefficiencies of the operational condi-

tions and possibly reduces the system lifetime. Thus, our aim is to eliminate or, better, to shift in parameter space, the

homoclinic bifurcation responsible for appearing of cross-well chaos. This is done by properly choosing the control

parameters c1 and c2.The results of previously quoted works are the starting point for our developments, and are summarized in Section 2

within a preliminary analysis of the system local and global dynamics. In the following Section 3, it is shown by some

examples how it is actually possible to eliminate cross-well chaos by adding a single superharmonic to the basic har-

monic excitation. This section also reports on some basic features of control, in particular the difference between odd

and even superharmonics. In Section 4, which is the main part of the paper, the problem of the optimal choice of c1 andc2 is addressed, the optimal excitations are determined, and their effects on the dynamics are systematically checkednumerically. Finally, some conclusions end the paper (Section 5).

2. Preliminary analysis of local and global dynamics

The unforced version of the investigated Duffing equation (1) has three fixed points, the (hilltop) saddle x2 ¼ 0 (alsonamed D1 in the following) and the two symmetric focuses x1;3 ¼ �0:4472 with small oscillations frequency xr ¼ 0:6324.Since the excitation frequency is x ¼ 1, we are quite far from resonance. Moreover, we are on the right side of the vertexof the well-known V-shaped region of cross-well chaos, which, with a different damping coefficient, is depicted, e.g., in

Fig. 2 of [21]. Thus, for increasing excitation amplitude, cross-well chaos first appears at relatively large values of A. For

the following purposes, we also report the homoclinic bifurcation threshold of the hilltop saddle. We numerically get

Ahom ¼ 0:0765, while the Melnikov method [26,27] gives Ahom ¼ 0:0738, a value which differs of 3.6% due to the highvalue of the damping coefficient.

To give an overview of the main bifurcational events occurring under harmonic (reference) excitation for increasing

excitation amplitude, we report in Fig. 1a the general bifurcation diagram extended up to a quite large value of A, which

highlights some points worthy of attention. First, we note that the period 1 small amplitude oscillations ensuing from

the rest positions x1;3 ¼ �0:4472 at A ¼ 0 are initially stable, then unstable, and successively get back their initialstability for large value of A, where they are still confined and small amplitude period 1 oscillations. Successively, they

coalesce and collide with the hilltop saddle through a reverse pitchfork (PF), symmetry breaking bifurcation leaving a

unique period 1, scattered and small amplitude attractor. It is a ‘‘global stable hilltop oscillation’’ surrounding the

hilltop of the potential, and it was previously observed by Zakrzhevsky [28], which also underlined that it occurs for

damping coefficient large enough. In the right part of the diagram it coexists with another period 1, scattered and large

amplitude attractor, which is of interest in the following analysis.

In the range of instability of period 1 oscillations the scenario is quite involved, entailing sequences of confined/

scattered periodic/chaotic attractors with a large variety of transitions between different regimes. In this paper we focus

attention on the last scattered to confined transition, which occurs for A ¼ Aesc and which is marked by arrows in Fig.1a and enlarged in Fig. 1b. More precisely, we want to apply the control method to reduce the final scattered chaotic

Fig. 1. Bifurcation diagrams of Eq. (1) in the case of harmonic excitation, c1 ¼ 0. (a) General diagram; (b) zoom around the interval ofinterest.

S. Lenci, G. Rega / Chaos, Solitons and Fractals 15 (2003) 173–186 175

region. The other transitions of Fig. 1a, and in particular the other boundaries of the two main scattered chaotic re-

gions, can be dealt with by the same method and ideas illustrated in this work.

The sudden change of Fig. 1b, which for decreasing values of A is a transition from single-well to cross-well dy-

namics, has been studied in depth by Ueda et al. [24] and Katz and Dowell [9] in the case of harmonic excitation. Thus,

we now summarize the part of these papers upon which our work is based on. Katz and Dowell [9] estimated

0:3255 < Aesc < 0:326, while we get Aesc ffi 0:3252, the difference being due to the different calibration of numerical toolsand being obviously negligible.

For A > Aesc there are three attractors, a period 1 (P1) scattered cycle and two confined, one in each potentialwell, chaotic attractors born, via classical Feigenbaum cascades, from confined P1 attractors existing for larger values

of A. Thus, in a small range one has coexistence of periodic scattered and chaotic confined attractors. For A < Aesc,on the other hand, the two confined attractors are substituted by a unique scattered chaotic attractor, coexisting

with the scattered periodic solution which persists and is not affected by the crisis (as confirmed by forthcom-

ing Figs. 5b, 6b and 7b). In this range there are only scattered dynamics, with coexistence of chaotic and periodic

attractors.

According to Ueda et al. [24] and Katz and Dowell [9], the transition at Aesc is due to the homoclinic bifurcation ofperiod 3 saddles (one in each well, herein denoted as D3), called destroyers and embedded in the system dynamics. Thecrisis occurs when the lower stable manifold and the right unstable manifold of D3 touch each other in each of the twowells, as shown in Fig. 2, where only one (D31 ¼ ð�0:2898;�0:3360Þ) of the three points D31, D32 and D33 constituting thesaddle in the left well is reported. Note that the left unstable manifold intersects both insets, but this seems to be ir-

relevant to the considered transition. In terms of global dynamics, the illustrated phenomenon corresponds to a

boundary crisis, where the confined chaotic attractor nearby coinciding with the unstable manifold of D31 touches theboundary of its basin of attraction, as shown in Fig. 3.

Although the homoclinic bifurcation is directly responsible for the sudden change of the chaotic attractor, the to-

pological mechanism of transition from confined to scattered dynamics seems to be slightly more involved. In fact, as

noted by Katz and Dowell [9], it is also connected with the heteroclinic intersection of the unstable manifold of D31 andthe stable manifold of the hilltop saddle D1 separating the basins of the two wells (Fig. 4). Following the two successive(homoclinic and heteroclinic) intersections, the neighborhood of the formerly confined chaotic attractor is mapped first

along the unstable manifold of D31, then along the stable manifold of D1 up to near D1, and finally along the branch of

the unstable manifold of D1 entering the other potential well. It is the triggering of this ‘‘turning around’’ mechanism,initiated by the homoclinic bifurcation of D31, which, according to Katz and Dowell [9], is responsible for the confined toscattered transition.

Below Aesc the lower stable and right unstable manifolds of D3 intersect (see forthcoming Fig. 5a) and the scattereddynamics are definitely established. The competing scattered chaotic and periodic attractors coexist in a large range of

excitation amplitudes, as shown in Fig. 1b. In the following sections we will focus attention to this zone where we

control (eliminate) the robust scattered chaotic attractor by avoiding the homoclinic bifurcation of the destroyer. This is

achieved by adding an appropriate superharmonic to the basic harmonic excitation.

Fig. 2. Stable and unstable manifolds of the point D31 of the saddle D3 at A ¼ 0:3252. (a) Large view, (b) zoom.


3. Numerical control of global dynamics and elimination of scattered chaos

In the framework of the considered control method, in this section we show how it is possible to eliminate the

manifolds intersection by adding to the basic harmonic excitation suitable superharmonic corrections as given by the

forcing terms in Eq. (1). We further show, by preliminary observations to be completed in Section 4, how this permits to

control various aspects of the global nonlinear dynamics of the considered oscillator, such as, for example, elimination

of chaos (regularization of the attractor), of scattered dynamics (confinement in the potential wells), and of sensitivity to

initial conditions (de-fractalization of basin boundaries).

We fix the value A ¼ 0:3246, which is inside the region A < Aesc to be controlled (see Fig. 1b), and consider threedifferent excitations: harmonic (reference), harmonic plus odd superharmonic and harmonic plus even superharmonic.

These three cases are sufficient to illustrate the main features of control we wish to discuss, and they permit to check the

practical performances of the system under controlled excitation.

The ‘‘symmetry’’ of system response in the case of harmonic excitation, which means that if x1ðtÞ is a solution, so isx2ðtÞ ¼ �x1ðt þ T=2Þ (for Eq. (1) T ¼ 2p), is maintained by the odd superharmonic and is lost with the even super-harmonic. Accordingly, the excitation with n ¼ 3 is called ‘‘symmetric’’, while that with n ¼ 2 ‘‘asymmetric’’. Thesymmetry of the excitation has deep consequences on control, as will be discussed in due course.

In the case of harmonic excitation there is intersection of the manifolds of the destroyer, as shown by Fig. 5a. Since

A < Aesc, according to Fig. 1b there are two scattered attractors, one periodic and the other chaotic. The attractor-basin

Fig. 3. The two confined chaotic attractors and their basins at A ¼ 0:326 just above the boundary crisis at A ¼ Aesc ¼ 0:3252.

Fig. 4. The unstable manifold of the saddle D31 and the stable manifold of the hilltop saddle D1 at A ¼ 0:3252.


Fig. 6. A ¼ 0:3246, c1 ¼ 0:6, c2 ¼ p, n ¼ 3 (symmetric excitation). (a) Stable and unstable manifolds of D31; (b) attractor-basin phaseportrait with scattered periodic and two confined chaotic attractors.

Fig. 5. A ¼ 0:3246, c1 ¼ 0:0 (harmonic excitation). (a) Stable and unstable manifolds of D31; (b) attractor-basin phase portrait withscattered periodic and chaotic attractors.

Fig. 7. A ¼ 0:3246, c1 ¼ 0:3, c2 ¼ 3p=2, n ¼ 2 (asymmetric excitation). (a) Stable and unstable manifolds of D31 ; (b) attractor-basinphase portrait with scattered and confined periodic attractors.


phase portrait reported in Fig. 5b confirms this. Of the two attractors in this picture only the chaotic cross-well one was

influenced by the homoclinic bifurcation (see Fig. 1b). Thus, it is expected that elimination of the manifolds intersection

influences only the inner basin, while the periodic attractor and its outer basin will remain almost unchanged even with

the unharmonic excitations.

To illustrate the effects of symmetric excitation, we choose n ¼ 3, c1 ¼ 0:6 and c2 ¼ p. Fig. 6a shows that with thisexcitation we are able to eliminate the manifolds intersection, so that numerical control looks theoretically effective in

this case. To illustrate the practical effects of manifolds detachment, we have reported in Fig. 6b the related attractor-

basin phase portrait. This picture shows that, apart from the scattered periodic attractor uninfluenced by the control,

there are two confined chaotic attractors, one in each potential well. Thus, the cross-well chaos has been eliminated,

though the chaoticity survives to a minor extent. However, the confined chaotic attractors of Fig. 6b are those ap-

pearing at the end of two Feigenbaum cascades––similar to those of Fig. 1b for A > Aesc––and thus it is felt that for adifferent calibration of the excitation, or at least for slightly larger values of A, one should be able to obtain confined

and periodic attractors (see forthcoming Fig. 10b). This is obtained, for example, in the case of asymmetric excitation

(Fig. 7b), while the problem of choosing the optimal excitation, i.e., the optimal combination of parameters c1 and c2will be addressed in Section 4.

The basin boundaries of the two confined attractors are very intertwined and are numerically fractal. This is due to

the fact that the invariant manifolds of the hilltop saddle intersect for this set of values, see Fig. 1a: this determines

mixing of the basins of the in-well attractors, just as it occurs in Fig. 3. From a practical point of view, this implies that

we still have SIC, which indeed can be avoided only by eliminating the homoclinic bifurcation of the hilltop saddle

through the same control method [13].

The case of asymmetric excitation is reported in Fig. 7, corresponding to n ¼ 2, c1 ¼ 0:3 and c2 ¼ 3p=2. Fig. 7ashows that the manifold intersection has been eliminated in this case, too, while Fig. 7b shows that the cross-well chaos

is substituted by a unique period 2 attractor belonging to the left potential well and coexisting with the uninfluenced

scattered periodic attractor. Once again, theoretical effectiveness is accompanied by practical performances, which in

this case consist in confinement and regularization of the dynamics. The P2 attractor of Fig. 7b, as the chaotic attractors

of Fig. 6b, belongs to a Feigenbaum cascade, so that for different values of the parameters we may also get a period 1

confined attractor.

It is worth emphasizing that it is the asymmetry of the excitation which allows for having only one confined at-

tractor. This is not possible with symmetric excitations, for which a confined attractor x1ðtÞ has its specular attractorx2ðtÞ ¼ �x1ðt þ T=2Þ in the other potential well. Furthermore, with symmetric excitation, an attractor xðtÞ which has nospecular counterpart satisfies the condition xðtÞ ¼ �xðt þ T=2Þ (i.e., it is symmetric) which actually ensures that it isscattered and has period 1. This is the case of the external scattered attractor in Figs. 5b and 6b.

The (unique) confined periodic attractor which substitutes the scattered chaotic attractor inherits its basin of at-

traction, which remains almost unchanged. Being not divided in two basins with fractal boundaries, as in the case of

Fig. 6b, there is no SIC. This represents another important practical performance of the asymmetric excitation, which

leads indeed to a very desirable dynamics.

Figs. 5–7 show in a preliminary and not systematic way that the chosen excitations are theoretical performant and

provide valuable practical results, the main ones being regularization and confinement of the dynamics, two control

features previously observed by the authors in other systems [11]. Also the elimination of SIC in the case of asymmetric

excitation is worth of attention. A further understanding can be obtained by comparing the relative amplitudes of the

superharmonic corrections. In the case of symmetric excitation it is relatively large, contrary to the asymmetric exci-

tation where a smaller c1 is used, which however produces better results, in terms of regularization, than in the formercase.

We summarize the previous discussions by saying that symmetric excitations maintain the symmetry of the system

but look quite ‘‘expensive’’, while asymmetric excitations are ‘‘cheaper’’ and more performant, but they lose the

symmetry of the system. Another point related to the nature of the method can be made based on Figs. 6 and 7. We

note that a larger distance between stable and unstable manifolds corresponds to a major regularization, and this agrees

very well with the theoretical background of the control method. This question will be deeply investigated in Section 4,

where the excitation allowing for maximum distance between inset and outset––which, according to this observation,

also allows for maximum regularization of the response––is looked for.

To close this section we remark that Figs. 5–7 numerically show that homoclinic intersection is accompanied by

cross-well dynamics (chaos), while homoclinic nonintersection is accompanied by in-well dynamics. This numerically

confirms the findings of Ueda et al. [24] and Katz and Dowell [9], and extends beyond the case of harmonic excitation

their results concerned with (i) D3 being the destroyer and with (ii) the mechanism illustrated in Section 2 being re-sponsible for spread of the dynamics.


4. Optimal numerical control

In this section we extend the previous results by investigating systematically the effects of the added superharmonics

(i) on the homoclinic bifurcation responsible for the confined to scattered transition, and (ii) on the actual system

response. We are interested in better understanding the main properties of the even (asymmetric) and odd (symmetric)

approaches to control, so that they will be analyzed independently in the following subsections.

Based on the theoretical background of the method and on preliminary results of Section 3, we look for the eu-

ristically optimal combinations of the control parameters c1 and c2 realizing the maximum distance between stable andunstable manifolds for fixed values of the excitation amplitude A. This is done by performing extensive numerical

simulations. For each (c1; c2) we compute the position of the saddle D3 by a modified Newton method [16] and considera window of the phase space of fixed size Dx ¼ 0:05 and D _xx ¼ Dy ¼ 0:025 centered in D31, as shown in Fig. 8. Thispermits to compare results with different values of c1, c2, n and possibly A. In this window we draw the stable and

unstable manifolds of D31, which are computed by a standard algorithm [3,8,25]. The right unstable and low stablemanifolds intersection is relevant for our work (see Section 2), so we focus attention on their distance d, which is

measured along the direction of the unstable eigenvector of D31 (see Fig. 8), and is assumed positive if the manifolds donot intersect (Fig. 8), negative otherwise. Fig. 8 also reports the scale of distances, which in that case is d ¼ þ2:4.The points (c1; c2; d) are successively interpolated and the function d ¼ dðc1; c2) is finally obtained. It is the key point

for the developments in Sections 4.1 and 4.2, where we refer to the critical excitation amplitude Aesc ¼ 0:3252 corre-sponding to the homoclinic bifurcation in the case of harmonic excitation. This is the amplitude where scattered first

chaos appears when decreasing A (see Fig. 1b), so it is the first amplitude which needs control.

4.1. Control with odd superharmonic (n ¼ 3)––symmetric excitation

Let us first consider the case of symmetric excitation (n ¼ 3). The polar coordinates contour plot of the distancebetween stable and unstable manifolds for A ¼ 0:3252 is reported in Fig. 9, where the black lines represent the relevantlevel curves and the concentric circles are the loci of constant c1, which are reported for better understanding.Fig. 9 shows that, roughly speaking, the function dðc1; c2) is a paraboloid centered approximately at (0.30, 0.00).

There is only a small bounded region of homoclinic intersection, depicted in gray in Fig. 9, just on the right of the point

(c1; c2Þ ¼ ð0; 0Þ corresponding to the harmonic excitation. Furthermore, in this region the manifolds intersect onlyslightly, being the maximum negative distance �0.15 (compare this value with that of Fig. 8).In the white region we have detachment of the manifolds, which is even quantitatively conspicuous in the left part of

Fig. 9, so we may conclude that with symmetric excitation the control is theoretically effective in a very large region of the

parameter space. This adds meaningfully to their overall effectiveness in phase space, which is a peculiarity of symmetric

excitations already highlighted in previous authors’ works [11], which justifies the name ‘‘global’’ control used therein.

Indeed, with this excitation the elimination of one homoclinic bifurcation automatically implies, by symmetry consid-

erations, elimination of the symmetric homoclinic bifurcation, which, accordingly, is not dealt with explicitly.

Fig. 8. The distance between stable and unstable manifolds. A ¼ 0:3252, c1 ¼ 0:1, c2 ¼ 3p=2, n ¼ 2.


The circles of Fig. 9 permit understanding the role of the phase c2. It is seen that for every fixed superharmonicrelative amplitude c1, the maximum distance is attained approximately for about the same c2 value (c2 1:08p), whichtherefore represents the euristic optimal choice of the superharmonic phase. The influence of c1, on the other hand, issimpler, because, apart from the ineffective gray region, the distance is an increasing function of c1. We may concludethat in the considered case the optimal excitation has c2 1:08p and the largest possible c1.In turn, the limitations on c1 are only of practical nature, because c1 is an excitation amplitude which is somehow

related to the cost of control, the smaller being c1 the cheaper being the control. Furthermore, as the superharmonicterms are here understood as corrections to the basic harmonic excitation––although generalizations are feasible––, it is

reasonable to practically consider values of c1 smaller than 1, which therefore represents a ‘‘natural’’ upper bound (thisis why Fig. 9 deals only with c16 0:8).The practical performances of the optimal excitation are now investigated in the case c2 ¼ 1:08p and c1 ¼ 0:4, a

value which seems to balance the opposite requirements of cheapness and effectiveness. The related bifurcation diagram

is reported in Fig. 10a and, when compared to that of Fig. 1b, it shows the effectiveness of the considered optimal

excitation. In fact, apart from the overall features of the diagram which are not changed by the added optimal su-

perharmonic, the confined to scattered transition is shifted toward the left, and now occurs for A ffi 0:3242. We havethus numerically proved that the (optimal) elimination of homoclinic bifurcation entails (likely optimal) elimination of

Fig. 9. The contour plot of the function dðc1; c2Þ in polar coordinates for Aesc ¼ 0:3252 and n ¼ 3.

Fig. 10. Bifurcation diagrams of Eq. (1) in the case of symmetric optimal excitation n ¼ 3, c2 ¼ 1:08p and (a) c1 ¼ 0:4, (b) c1 ¼ 0:8.


the scattered chaotic attractor, a property which characterizes the excitation amplitude interval [0.3242, 0.3252] called

the ‘‘practical saved region’’ [11].

In this region, however, the chaoticity has not been eliminated at all, as the confined attractors are both chaotic,

although to a lesser extent than with the scattered attractor. To improve these results, a larger superharmonic amplitude

is used, according to previous observations. For example, by considering c1 ¼ 0:8 the bifurcation diagram of Fig. 10b isobtained. In this case, not only the confined to scattered transition is further shifted (now, it occurs for A ffi 0:3220), butalso the periodic attractors of the Feigenbaum cascade enter the practical saved region, so that, at least just below Aesc,we have confinement and regularization of the dynamics, thus exhibiting very valuable practical performances of the

control.

It is worth noticing that the width of the practical saved region is proportional to the control parameter c1. In-terestingly enough, it is seen that it is more than linear, as the doubling of c1 permits to enlarge the saved region of afactor 3.2, as it passes from a width DA ffi 0:0010 to 0.0032. This partially compensates for the major cost of highercontrol amplitudes.

There is only one qualitative difference between the diagrams of Fig. 10a and b. In fact, in the former, the disap-

pearing of the confined (chaotic) attractor corresponds to the appearance of the scattered chaotic attractor, whose

skeleton is the chaotic saddle born at the homoclinic bifurcation of the hilltop saddle (which occurs for a lower value of

A, see Fig. 1a). Thus, in this case, scattering is still accompanied by increasing of chaoticity, as in Fig. 1b. In the latter

case, on the other hand, at the end of the cascade the chaotic saddle does not become immediately attractive, but there is

a well-defined interval where the unique attractor is the outer period 1 oscillation existing for all of the considered

values of A. Thus, in this circumstance scattering is accompanied by an overall regularization of the dynamics. Only for

lower values of A the chaotic saddle becomes attractive, likely due to another reverse boundary crisis. This transition,

however, is not related to control and is not investigated for being of minor interest to the purposes of this paper. We

only note that other, not reported, numerical simulations show that the occurrence of such a range of regularized

dynamics is a robust feature for increasing values of c1.

4.2. Control with even superharmonic (n ¼ 2)––asymmetric excitation

The case of asymmetric excitation (n ¼ 2) is considered in this section following the same guidelines as in Section 4.1.However, there is a basic difference which needs to be mentioned from the very beginning. In fact, the symmetry of the

response is lost, so that the control of one homoclinic bifurcation does not necessarily entail control of its symmetric––

and, indeed, it does not. However, there is no need to consider the right and left homoclinic bifurcations independently,

because if x1ðtÞ is a solution of (1) with n ¼ 2 and with given c�1 and c�2, then x2ðtÞ ¼ �x1ðt þ pÞ is a solution of (1) withc1 ¼ �c�1 and c2 ¼ c�2 or, more appropriately, with a phase shift of the superharmonic correction of a half period, c1 ¼ c�1and c2 ¼ c�2 þ p. Note that this is the counterpart of the symmetry property of odd superharmonics.From a practical point of view, this property is very important because it means that if for (c1; c2) the manifolds of,

say, the left saddle have a positive distance d left, then the manifolds of the right saddle for (c1; c2 þ p) also have a positivedistance dright, and the larger is d left, the larger is d right. If we agree to measure left and right distances on (stroboscopic)Poincar�ee sections shifted of half-period along the time, then we may conclude that the distance of the manifolds in theright potential well is just the same as that of manifolds in the left potential well computed with a half-period shift of the

superharmonic phase, namely

drightðc1; c2Þ ¼ d leftðc1; c2 þ pÞ: ð2Þ

We will show in the following (see forthcoming Fig. 11) that d leftðc1; c2Þ �d leftðc1; c2 þ pÞ, so that d right �d left. Thisimplies that, contrary to the case of symmetric excitations, it is no longer possible to eliminate both homoclinic bi-

furcations, but rather one (i) must choose which homoclinic bifurcation is to be controlled, and (ii) must accept a worse

behaviour of the uncontrolled bifurcation (in fact, positive distances in, say, the left well imply negative distances in the

right well). This theoretical behaviour is characteristic of asymmetric excitations, as noted in other authors’ works [11],

and, although appearing somewhat unpleasant, it is compensated by a major effectiveness of this superharmonic in

detaching the invariant manifolds, and in its consequent practical performances (see the following).

According to Eq. (2), we can limit the study to one homoclinic bifurcation, and we consider the saddle

D31 ¼ ð�0:2898;�0:3360Þ of Fig. 2b belonging to the left potential well. Thus, we are controlling the left homoclinicbifurcation or, generally speaking, the left potential well. Fig. 11 reports the polar coordinates contour plot of the

function d leftðc1; c2Þ ¼ dðc1; c2Þ. As in Fig. 9, the thick line d ¼ 0 represents the locus of parameters (c1; c2) givinghomoclinic tangency, in the gray region there is homoclinic intersection, while in the complementary white region the

even superharmonic is able to detach the manifolds.


The considered region of parameter space is divided in two, approximately equal, parts, one where the control is

theoretically effective (d > 0) and the other where it is ineffective (d < 0). This resembles the behaviour of the evensuperharmonic in phase space, where the control is effective in eliminating one homoclinic bifurcation and ineffective for

the other. This is another characteristic of this kind of excitation, which justifies the name ‘‘one-side’’ control used

elsewhere [11].

Fig. 11 shows that the optimal superharmonic phase is c2 1:65p, which is the best choice irrespective of the su-perharmonic relative amplitude c1, so that the left optimal excitation providing the maximum distance between left

manifolds has c2 1:65p and the largest possible c1. As in the previous section, the limitations on c1 are only ofpractical nature and related to the cost of control.

By comparing Figs. 9 and 11 it is possible to observe the major theoretical effectiveness of the asymmetric excitation.

For example, we have dðc1 ¼ 0:2, c2 ¼ 1:65p, n ¼ 2Þ 4:9 versus dðc1 ¼ 0:2, c2 ¼ 1:08p, n ¼ 3Þ 0:34 and dðc1 ¼ 0:4,c2 ¼ 1:65p, n ¼ 2Þ 11:2 versus dðc1 ¼ 0:4, c2 ¼ 1:08p, n ¼ 3Þ 0:79, showing that asymmetric excitations are ap-proximately 14 times more efficient than symmetric excitations. Thus, we have remarkable distances also with low

values of c1 and this explains why Fig. 11 reports only the region c16 0:4.The high theoretical effectiveness is the other distinguished feature of the considered excitation, which was previously

observed in other works where the control is applied––with different purposes––to the homoclinic bifurcation of the

hilltop saddle [10–12,14]. This property and the possibility of controlling only one part of the space, versus minor

effectiveness and control of whole space typical of symmetric excitations, highlight the complementary and not com-

peting nature of ‘‘global’’ and ‘‘one-side’’ controls. This distinction is important from an application point of view,

because it suggests the appropriate use of one or the other optimal excitation depending on the pursued results in

practical applications.

Based on relation (2), we can observe that the distance of the manifolds of the right saddle is just that of Fig. 11

rotated of p around the point c1 ¼ 0. Thus, the right optimal excitation for controlling the right potential well is

characterized by c2 0:65p and the largest possible c1, and the considerations made for the left optimal excita-tion apply. As previously anticipated, in the first approximation this rotation entails changing the sign of the dis-

tance, namely, the right distance is simply obtained by inverting the gray with the white region without further

modifications.

As in the previous section, the practical performances of asymmetric left optimal excitations c2 ¼ 1:65p are detectedby systematic numerical investigations aimed at confirming the theoretical effectiveness shown in Fig. 11. The bifur-

cation diagrams corresponding to c1 ¼ 0:20 and 0.40 are depicted in Fig. 12a and b, respectively, and are now discussed.The lost of symmetric behaviour, which is the main characteristic of asymmetric excitations, is well recognizable in

Fig. 12, which shows how the confined attractor in the right (upper) uncontrolled potential well is shifted toward higher

values of A, while the confined cascade of the left (lower) controlled potential well is shifted toward lower values of A.

Fig. 11. The contour plot of the function dðc1; c2Þ in polar coordinates for Aesc ¼ 0:3252 and n ¼ 2.


This agrees with theoretical predictions, which suggest an improvement in the controlled well and a worsening in the

uncontrolled well.

The second important point is the numerical confirmation that high theoretical effectiveness provides strong

practical effectiveness in avoiding scattered chaotic dynamics. In fact, the confined to scattered transition is shifted to

A ffi 0:3183 for c1 ¼ 0:2 and to A ffi 0:3102 for c1 ¼ 0:4, and the related widths of the practical saved region areDA ffi 0:0069 and DA ffi 0:0150, respectively. This latter value, in particular, can be compared with the valueDA ffi 0:0010 observed in the case of symmetric excitation for the same superharmonic amplitude c1 ¼ 0:4, whichconfirms also practically that asymmetric excitations are 14 times more effective than symmetric excitations.Previous considerations concern the confinement of dynamics. The practical effectiveness of the asymmetric exci-

tations, on the other hand, also improves with respect to the regularization features. In fact, the practical saved regions

in Fig. 12 are seen to be mainly characterized by periodic attractors, contrary to what happens in the case of symmetric

excitations (see Fig. 10).

One more valuable characteristic associated with asymmetric excitations can be deduced from Fig. 12. The presence

of a single attractor in the practical saved region implies that there are only two basins of attraction with smooth

boundary, similarly to what occurs in Fig. 7b. In particular, there is no sensitivity to initial conditions, contrary to what

happens with symmetric excitations where the two confined competing attractors have fractal basin boundaries (see Fig.

6b), and there is chaotic transient and sensitivity to initial conditions. Thus, asymmetric excitations are more perfor-

mant also with respect to this dynamical aspect.

As a final check, we have reported in Fig. 13 the bifurcation diagrams for the right optimal control, corresponding to

c2 ¼ 0:65p and to the same superharmonic excitations c1 ¼ 0:2 and 0.4 used in Fig. 12. According to theoretical pre-dictions, we have exactly a ‘‘specular’’ behaviour, with the confined attractor in the saved region belonging to the

Fig. 12. Bifurcation diagrams of Eq. (1) in the case of asymmetric left optimal excitation n ¼ 2, c2 ¼ 1:65p and (a) c1 ¼ 0:2, (b)c1 ¼ 0:4.

Fig. 13. Bifurcation diagrams of Eq. (1) in the case of asymmetric right optimal excitation n ¼ 2, c2 ¼ 0:65p and (a) c1 ¼ 0:2, (b)c1 ¼ 0:4.


controlled (now the right) potential well, all the other considerations being unchanged with respect to the left control, in

particular, the width of the practical saved regions.

5. Conclusions

The problem of controlling nonlinear dynamics and chaos in mechanical systems by properly modifying the shape of

the excitation has been addressed. This is a very general control approach, which was previously developed by the

authors, and which has been applied here to the Duffing oscillator in order to numerically highlight its effectiveness,

although it may actually be employed for controlling many other oscillators. For the same reason, attention has been

focused on excitations made of a basic harmonic term plus a single controlling superharmonic. Extension to more

involved excitations is straightforward and does not modify the main features of control.

The undesired dynamical phenomenon to be controlled (eliminated or, better, shifted in parameter space) is the

homoclinic bifurcation responsible for transition from single-well (confined) to cross-well (scattered) chaos. This event

cannot be detected analytically, as the involved saddle is not hilltop, so that the problem has been approached in a

completely numerical way, this constituting the main novelty of the present paper.

It has been shown by preliminary examples how adding a superharmonic leads first to the elimination of the ho-

moclinic bifurcation and then, as a consequence, to the elimination of the unwanted scattered chaotic attractor. Suc-

cessively, the problem has been investigated systematically, and the optimal controlled excitations able to furnish

maximum distances between the stable and unstable manifolds meaningful for transition have been obtained numer-

ically. Two classes of excitations, namely symmetric and asymmetric, have been investigated separately and their

specific, complementary, characteristics have been studied in depth and compared with each other.

The practical effectiveness of the optimal excitations has also been investigated. It has been shown how, in various

circumstances and to a different extent, the control excitations allow for: (i) confinement of the dynamics, i.e., elimi-

nation of scattered attractor; (ii) regularization, i.e., substitution of chaotic with periodic attractor; and (iii) elimination

of sensitivity to initial conditions and chaotic transient, by de-fractalization of basin boundaries.

In conclusion, the present paper shows the theoretical and practical effectiveness of the control method proposed by

the authors also in the case where the homoclinic bifurcation must be detected numerically. This extends the results of

previous works well beyond the cases where the homoclinic bifurcation is detected by the Melnikov’s method, and

further confirms the generality of the control method also highlighted in [14] in a nearly completely theoretical

framework.

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Optimal numerical control of single-well to cross-well chaos transition in mechanical systems

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