Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator

Chaos, Solitons and Fractals 21 (2004) 1031–1046

www.elsevier.com/locate/chaos

Global optimal control and system-dependent solutionsin the hardening Helmholtz–Duffing oscillator

Stefano Lenci a,*, Giuseppe Rega b

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

Accepted 22 July 2003

Abstract

A method for controlling nonlinear dynamics and chaos based on avoiding homo/heteroclinic bifurcations is applied

to the hardening Helmholtz–Duffing oscillator, which is an archetype of a class of asymmetric oscillators having distinct

homoclinic bifurcations occurring for different values of the parameters. The Melnikov’s method is applied to ana-

lytically detect these bifurcations, and, in the spirit of the control method developed by the authors, the results are used

to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter

space. The two main novelties with respect to previous authors’ works consist in the occurrence (i) of a more involved

control scenario, and (ii) of system-dependent optimal solutions. In particular, the possibility of having global control

‘‘with’’ or ‘‘without symmetrization’’, related to the optimization of the critical amplitudes versus the critical gains, is

illustrated. In the former case, two sub-cases are found, namely, ‘‘pursued’’ and ‘‘achieved’’ symmetrization, which take

care of the possibly simultaneous occurrence of right and left homoclinic bifurcations under optimal excitation.

� 2004 Published by Elsevier Ltd.

1. Introduction

This paper deals with a method for controlling nonlinear dynamics and chaos of various mechanical systems pre-

viously developed by the authors in a series of works [9,10,13]. The idea is quite simple in principle, and is based on the

elimination (or better shifting in parameter space) of the homo/heteroclinic bifurcations embedded in system dynamics,

which are the topological events triggering undesired dynamical phenomena [7,17], such as chaotic transient, fractal

basin boundaries, sensitivity to initial conditions [4,16,27,28], cross-well chaos [6,21], erosion of safe basin [25], escape

from potential well [13,24], boundary or interior crisis [3], etc.

The method consists in choosing the shape of the periodic excitation which permits to avoid, in an optimal manner,

the transverse intersection of the stable and unstable manifolds of the relevant (usually hilltop) saddle. It is developed in

various sequential steps: (i) detection of the homo/heteroclinic bifurcation, which is accomplished by the Melnikov’s

method and its generalizations in the case of quasi-Hamiltonian systems, or numerically otherwise [12]; (ii) study of the

dependence of the bifurcation on the shape of the excitation; (iii) formulation of the mathematical problem of opti-

mization, which, for a given excitation frequency, consists in determining the theoretical optimal excitation which

maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical

amplitude for homo/heteroclinic bifurcation; (iv) solution of the previous optimization problems, which is a mathe-

matical task sometimes complemented by the requirement of satisfying a physical admissibility constraint. Of course, a

* Corresponding author. Tel.: +390-71-220-4452; fax: +390-71-220-4576.

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

0960-0779/$ - see front matter � 2004 Published by Elsevier Ltd.

doi:10.1016/S0960-0779(03)00387-4

mail to: [email protected]

1032 S. Lenci, G. Rega / Chaos, Solitons and Fractals 21 (2004) 1031–1046

further step consists in checking the practical effectiveness of the method, to be investigated via numerical simulations

aimed at verifying theoretical predictions and the control-induced improvements on system response [10,11,13].

The idea of eliminating homoclinic bifurcations of the hilltop saddle by varying the shape of the excitation has been

first proposed by Shaw [20] for the inverted pendulum with lateral barriers, a piece-wise linear mechanical system. The

author considered a very simple case, an excitation constituted by a basic harmonic sinðxtÞ plus the first odd super-

harmonic sinð3xtÞ, and computed the solution easily. This work has been extended in [9] by considering infinite su-

perharmonics, thus allowing for the most general shape of the excitation. It has been shown that the critical excitation

amplitude for homoclinic bifurcation can be doubled, at least in principle. Detailed numerical simulations confirming

the effectiveness of the method are given in [10], which also reports on the questions related to the simultaneous

presence of two homoclinic bifurcations, leading to different control strategies (called ‘‘one-side’’ and ‘‘global’’ control),

which were first introduced in [8,9].

The method has been successively applied to smooth, strongly nonlinear mechanical systems. In [13] the asymmetric

single-well Helmholtz oscillator––exhibiting only quadratic nonlinearities––has been considered. A detailed theoretical

investigation has been performed, and several kinds of optimal solutions (reduced, constrained and reduced/con-

strained), required for having physically admissible solutions, have been obtained explicitly. Extensive numerical

simulations verifying the theoretical predictions and illustrating the practical performances, in particular the reduction

of the safe basin erosion, are also reported in [13].

The symmetric double-well Duffing oscillator––which exhibits only cubic nonlinearities––has been investigated in

[11], a work which extends the previous one in various points. From a control point of view, the presence of two

homoclinic bifurcations permits to consider also the ‘‘global’’ control (contrary to Helmholtz, where only ‘‘one-side’’

control applies), and to compare it with the ‘‘one-side’’ control. In turn, from a mechanical point of view, Duffing is a

hardening system (contrary to the softening Helmholtz), and this has strong practical consequences. For example, the

unwanted phenomena to be controlled by elimination of homoclinic bifurcations is no longer the escape but rather the

appearance of scattered (cross-well) dynamics, a question which has been addressed in [11].

A generalization of the previous works is made in this paper, where we consider the asymmetric double-well

Helmholtz–Duffing oscillator––with quadratic and cubic nonlinearities––which is an archetype of the class of systems

having more than one homo/heteroclinic bifurcation occurring for different values of the excitation amplitude.

Accordingly, if, in the context of a global control, one wishes to control simultaneously the two critical events, one has

to deal with two different expressions of the Melnikov’s distance: this is a difficult task which entails a major mathe-

matical complicatedness with respect to the cases analyzed previously, and which constitutes one main element of

novelty of the present work. It leads to global control ‘‘with’’ or ‘‘without symmetrization’’, which is related to the

optimization of either the excitation amplitudes or of the gains, and to the ‘‘pursued’’ and ‘‘achieved’’ symmetrization,

which is concerned with the possibility to have two simultaneous homoclinic bifurcations depending on the system

properties. The one-side control, on the other hand, remains unchanged because it involves only a single homoclinic

bifurcation, irrespective of the behaviour of the other.

Another characterizing element of the paper consists in highlighting how the optimal solutions obtained in the case

of global control do depend on the system properties, contrary to the generic––namely system-independent––solutions

ensuing from one-side controls, as well as from global control of symmetric systems (see also [14]).

Strictly speaking, the results obtained in the present paper are related only to the hardening Helmholtz–Duffing

oscillator. However, it is felt that the main ideas and phenomena encountered in this case are sufficiently general and

common to the whole class of oscillators having distinct homoclinic bifurcations. Other specific systems could be

studied following the guidelines of the present work, with only minor differences, possibly of technical nature.

A detailed analysis is performed along the main steps of the method previously illustrated. More precisely, in Section

2 the Melnikov analysis permitting a theoretical prediction of global bifurcations is performed, and the dependence on

the shape of the excitation is determined. The optimal problems required to detect the best shape of the excitation are

also stated in Section 2, whereas they are analyzed and numerically solved in Section 3 paying attention to the system-

dependent solutions which characterize various cases of global control. The regions of pursued and achieved sym-

metrization in parameter space are also determined in Section 3.2. Finally, some conclusions end the paper (Section 4).

2. Global bifurcations and control strategy

Let us consider the hardening Helmholtz–Duffing equation

€xþ ed _x� rx� 3

2ðr� 1Þx2 þ 2x3 ¼ ecðxtÞ; ð1aÞ

Fig.

S. Lenci, G. Rega / Chaos, Solitons and Fractals 21 (2004) 1031–1046 1033

cðsÞ ¼X1j¼1

cj sinðjsþWjÞ ¼ c1X1j¼1

cjc1

sinðjsþWjÞ ð1bÞ

in which ed is the viscous damping, ecðxtÞ the generic T -periodic ðT ¼ 2p=xÞ external excitation and e is a dimensionless

smallness parameter introduced to emphasize the smallness of damping and excitation. We use the expression (1b) of

the excitation because we assign c1 the role of amplitude, while the dimensionless parameters cj=c1 govern the shape of

the excitation, namely, they measure the superharmonic corrections to the basic harmonic excitation.

Eq. (1a) describes the single mode dynamics of one dimensional structural systems with initial curvature, such as

shallow arches [15]. The system asymmetry is measured by the parameter r, which is assumed greater than 1 without

loss of generality. For example, for the arch whose buckling was experimentally checked in [2] we have r ¼ 1:014–1:395,while for the experimental arch whose dynamics has been recently studied by Alaggio and Benedettini [1] we have

r ¼ 1:098. For r ¼ 1:2, the associated potential V ðxÞ ¼ �rx2=2� ðr� 1Þx3=2þ x4=2 and the unperturbed phase space

are depicted in Fig. 1(a) and (b), respectively.

An expression similar to (1) governs, among others, the single mode dynamics of buckled beams with lateral loads

[5,19], and, more generally, under various kinds of imperfections. In addition to previous mechanical applications, Eq.

(1) is the archetype of the twin-well asymmetric oscillators and exhibits very rich nonlinear dynamics. Contrary to the

Duffing equation (which (1) is reduced to in the case r ¼ 1), extensively studied from a theoretical, numerical,

experimental and control point of view [11,16,22,23,26,28], Eq. (1) was not deeply investigated in the past. However, it

shares several properties of its symmetric version, like, e.g., the ‘‘robust’’, cross-well (scattered) chaotic attractor

occurring for large values of the excitation amplitude. This is an important effect, usually considered more undesired

than the in-well (confined) chaotic attractors appearing at the end of some Feigenbaum cascades and rapidly disap-

pearing by boundary crises. It is triggered by the two homoclinic bifurcations of the unique hilltop saddle, and its

shifting towards higher excitation amplitudes constitutes a very appealing practical performance of the proposed

method in the general case of hardening oscillators [11,14].

The unperturbed undamped dynamics is characterized by the presence of two centers x1 and x3 and a unique (hilltop)

saddle x2 ¼ 0, having two homoclinic loops surrounding the two potential wells (see Fig. 1(b)), which can be expressed

in the form:

xr;lhomðtÞ ¼2r

�ðrþ 1Þ coshðtffiffiffir

pÞ � ðr� 1Þ : ð2Þ

The presence of two simultaneous asymmetric homoclinic orbits has very important consequences in terms of control,

as will be shown in the following.

The homoclinic loops separate the (left and right) in-well periodic oscillations from the large amplitude scattered

swayings. When the perturbations are added, they split into distinct stable and unstable manifolds, which may or may

not intersect depending on the relative magnitude of c1 and d. There exist critical values corresponding to homoclinic

bifurcations, which can be analytically computed by the Melnikov’s method. This theory proves that the first order

approximation (in e) of the signed distance between stable and unstable manifolds is given by [4,28]

1. (a) The potential V ðxÞ and (b) the unperturbed phase space of the hardening Helmholtz–Duffing equation (1) for r ¼ 1:2.


Mr;lðmÞ ¼Z 1

�1_xr;lhomðtÞ½�d _xr;lhomðtÞ þ cðxt þ mÞ�dt ð3Þ

for the right and left manifolds, respectively. After some computations, (3) can be rewritten in the form

Mr;lðmÞ ¼ �df r;lðrÞ 1

"� c1ch;r;l1;cr ðxÞ

hr;lðmÞ#; ð4aÞ

f r;lðrÞ ¼ffiffiffir

p

12ð3r2 þ 2rþ 3Þ � ðrþ 1Þ2ðr� 1Þ

4arctanðr�1=2Þ; ð4bÞ

ch;r;l1;cr ðxÞ ¼ df r;lðrÞ2px

sinh xpffiffir

p� �

sinh xffiffir

p arccos � 1�r1þr

� �h i > 0; ð4cÞ

hr;lðmÞ ¼X1j¼1

hr;lj cosðjmþWjÞ; hr;lj ¼cjc1jsinh xpffiffi

rp

� �sinh jxpffiffi

rp

� � sinh jxffiffir


� �h isinh xffiffi

rp arccos � 1�r

1þr

� �h i : ð4d;eÞ

Note that hr;l1 ¼ 1 and that hr;lðmÞ are 2p-periodic and have zero mean value. The parameters hrj and hlj, j > 1, measure

the effects of the superharmonic corrections on the Melnikov’s functions MrðmÞ and MlðmÞ, respectively. They con-

tribute to the ‘‘amplitude free’’ oscillating parts hr;lðmÞ of the distances between the manifolds.

As it will be shown in the following Section 2.1, the curves ch;r;l1;cr ðxÞ represent the loci of right/left homoclinic

bifurcations in the reference case of harmonic excitation, and are depicted in Fig. 2 for r ¼ 1:2. Contrary to the Duffing

equation [11], however, they no longer coincide (Fig. 2). This fact represents the source of the strong differences (in

terms of control) between symmetric and asymmetric hardening two well oscillators.

As in the case of Duffing equation [11], the presence of two homoclinic orbits permits to choose among different

control strategies, some of which are here more involved because ch;r1;cr and ch;l1;cr are now distinct. Indeed, we can control,

i.e. eliminate, only the right (left) homoclinic bifurcation, irrespective of what happens in the left (right) potential well,

or we can try to control simultaneously the right and the left homoclinic bifurcations. This distinction is very important

from the application point of view, and it concerns, roughly speaking, the control of one well versus the whole phase

space. The two approaches have been discussed by the authors in other works [8,10,11], where it has been shown that

the former (‘‘one-side’’ control) provides large gains, while the latter (‘‘global’’ control) provides lower gains. Their

different features have been shown in [10], where it is also numerically confirmed that, at least for the inverted pen-

dulum, the global and one-side controls are actually complementary rather than competing.

In turn, the asymmetry of the Helmholtz–Duffing equation has deep effects in the ‘‘global control’’ strategies, which

disappear in the case of symmetric systems and which therefore have not been studied previously. For this reason, we

will now investigate separately four different cases, paying less attention to the one-side controls (which are identical to

Fig. 2. The curves ch;r1;cr, ch;l1;cr and cr1;cr, c

l1;cr for M

r ¼ M l ¼ 0:5 and r ¼ 1:2.


those considered in [11]) and much more attention to the global controls, which are studied here for the first time in the

case of asymmetric systems.

2.1. ‘‘One-side’’ control on the right well (right control)

The Melnikov’s theory guarantees that, for e sufficiently small, there is homoclinic intersection of the right stable and

unstable manifolds if and only if MrðmÞ has a simple zero for some m, i.e., if and only if

hrðmÞ ¼ �ch;r1;crðxÞ

c1; ð5Þ

for some m in ½0; 2p�. Since the right hand side is a negative number, this is possible if and only if

minm2½0;2p�

fhrðmÞg < �ch;r1;crðxÞ

c1; ð6Þ

namely, iff

c1 >ch;r1;crðxÞM r

¼def cr1;crðxÞ; ð7Þ

where

M r ¼ � minm2½0;2p�

fhrðmÞg ¼ maxm2½0;2p�

f�hrðmÞg ð8Þ

is a positive number which accounts for the shape of the excitation. In the space of governing parameters ðx; c1Þ, thecurve cr1;crðxÞ (corresponding to a generic excitation and illustrated in Fig. 2 for r ¼ 1:2 and M r ¼ 0:5) separates thezone where homoclinic intersections do not occur (below the critical curve) from that where homoclinic intersections do

occur (above the critical curve).

In the reference case of harmonic excitation, which is usually employed to analyze the basic nonlinear forced

dynamics, we have hrðmÞ ¼ cosðmþW1Þ andM r ¼ 1 so that cr1;cr ¼ ch;r1;cr. This actually shows that the curve ch;r1;cr, which is

illustrated in Fig. 2 for r ¼ 1:2, is the critical curve in the reference case of harmonic excitation.

The control method aims at reducing the region of homoclinic intersection of Fig. 2 by varying the shape of the

excitation. Since only M r depends on the shape in (7), this entails decreasing M r, the smaller being M r the smaller being

the upper region in the parameter space. To measure the increment of critical threshold with respect to the reference

harmonic excitation, we introduce the gain, defined as the ratio between the critical amplitudes of the unharmonic and

harmonic excitations (see Eq. (7)),

G ¼ Gr ¼defcr1;crch;r1;cr

¼ 1

M r: ð9Þ

The zone above ch;r1;cr and below cr1;cr, where there is homoclinic intersection with harmonic excitation and no intersection

with unharmonic excitation, is the region where the control is expected to be effective, at least from a theoretical point

of view, and it is called saved region (Fig. 2).

It is now clear that the optimal shape is that which permits enlarging as much as possible the saved region. This

entails solving the following optimization problem:

Maximizing G ¼ Gr by varying the Fourier coefficients hrj and Wj; j ¼ 2; 3; . . . ; of hrðmÞ; ð10Þ

which is exactly the problem encountered and solved in [11]. After having obtained the best hrj (see, for example, the

forthcoming Table 1), we can get the Fourier coefficients of the optimal excitation by simply inverting relations (4e).

2.2. ‘‘One-side’’ control on the left well (left control)

This case is analogous of that of Section 2.1, with the difference that the homoclinic intersection of the left stable and

unstable manifolds is given by the condition MlðmÞ ¼ 0 (instead of MrðmÞ ¼ 0) for some m 2 ½0; 2p�. This occurs if andonly if


hlðmÞ ¼ch;l1;crðxÞ

c1; ð11Þ

for some m in ½0; 2p�. Since the right hand side is a positive number, this is possible if and only if

maxm2½0;2p�

fhlðmÞg >ch;l1;crðxÞ

c1; ð12Þ

namely, iff

c1 >ch;l1;crðxÞM l

¼def cl1;crðxÞ; ð13Þ

where

M l ¼ maxm2½0;2p�

fhlðmÞg ð14Þ

is a positive number which accounts for the shape of the excitation. The critical curves cl1;crðxÞ and ch;l1;crðxÞ, corre-sponding to the generic and harmonic excitation, respectively, are reported in Fig. 2 for r ¼ 1:2 and M l ¼ 0:5. Similarly

to (9), the gain is now defined as

G ¼ Gl ¼defcl1;crch;l1;cr

¼ 1

M lð15Þ

and the optimization problem is still (10) with the obvious substitution of the apex r with l.

Remark 1. The two optimization problems are (apparently) different, because in one case the functional to be optimized

is related to the minimum of the admissible functions hrðmÞ (right control), while in the other case it is related to the

maximum of the admissible functions hlðmÞ (left control). In spite of this, they are mathematically equivalent, and it is

possible to show that they provide the same optimal gain, and that the functions where this gain is attained are related

by the condition hrj ¼ ð�1Þðjþ1Þhlj.

2.3. ‘‘Global’’ control without ‘‘symmetrization’’

If we want to control simultaneously the right and the left homoclinic tangencies, we have to increase simultaneously

the right and the left gains, Gr ¼ 1=M r and Gl ¼ 1=M l. However, this point needs some care, because, for fixed r and x,M r and M l are not independent. In fact, since we want to realize a simultaneous control with a single excitation, from

(4e) we have

hljhrj

¼sinh xffiffi

rp arccos 1�r

1þr

� �h isinh jxffiffi

rp arccos 1�r

1þr

� �h i sinh jxffiffir



rp arccos � 1�r

1þr

� �h i ¼ bðj;x; rÞ 2�0; 1�; ð16Þ

so that hrðmÞ and hlðmÞ, and consequently the numbers M r ¼ �minfhrðmÞg ¼ maxf�hrðmÞg and M l ¼ maxfhlðmÞg, arestrictly related. The function bðj;x; rÞ has values in ]0,1], and is depicted in Fig. 3 for j ¼ 2 and 3. Apart from the

uninteresting case x ¼ 0, one has that hrðmÞ and hlðmÞ coincide only for r ¼ 1, namely, for symmetric systems.

Mathematically, increasing Gr and Gl simultaneously entails to increase their minimum value, namely:

Maximizing G ¼ minfGr;Glg by varying the Fourier coefficients hrj and Wj; j ¼ 2; 3; . . . ;

of hrðmÞ; the Fourier coefficients of hlðmÞ being given by ð16Þ: ð17Þ

In (17) we can alternatively choose to vary the Fourier coefficients of hlðmÞ, those of hrðmÞ being in this case computed

by inverting (16).

With the present approach we increase the critical threshold of both the right and left homoclinic bifurcations, even

if the proportional increments (i.e., the gains) with respect to the harmonic case may be different in general. However,

all the numerical solutions of (17) which will be obtained in Section 3 will satisfy the condition Gr ¼ Gl ¼ G, namely,

�minfhrðmÞg ¼ maxfhlðmÞg. In such a case, the problem (17) can be rewritten in the following form:

Fig. 3. The functions bð2;x;rÞ and bð3;x;rÞ relating the Fourier coefficients of hrðmÞ and hlðmÞ.


Maximizing Grð¼ GlÞ by varying the Fourier coefficients hrj and Wj; j ¼ 2; 3; . . . ; of hrðmÞ; under theconstraint max

m2½0;2p�fhlðmÞg ¼ � min

m2½0;2p�fhrðmÞg;where the Fourier coefficients of hlðmÞ are given by ð16Þ; ð18Þ

which shows that (18) is actually a constrained version of (10). Accordingly, the optimal gain is lesser than that of (10),

as it will be confirmed numerically in Section 3. The counterpart of this reduction is the possibility to control the whole

phase space.

Apart from the reduction of the optimal gain, there is an important novelty in (18). Indeed, while the solution of

(10) does not depend on the system, the solution of (18) does, because of the constraint (16) which depends on x and

r. This fact constitutes the most important distinction between the two approaches in terms of a unifying control

perspective, and it has also some consequences from a practical point of view. In fact, the solution of the opti-

mal problem for one-side control is generic, in the sense that it can be computed once and then applied to any system:

it is sufficient to know the relations between the Fourier coefficients of hðmÞ and those of the excitation, which follow

from the Melnikov analysis. In contrast, in the case of global control without symmetrization, the solution strictly

depends on the considered system, because the relation between hrðmÞ and hlðmÞ does, and a generalization seems not

possible.

2.4. ‘‘Global’’ control with ‘‘symmetrization’’

In the previous strategy the right and the left homoclinic bifurcation thresholds are cr1;cr ¼ Gch;r1;cr and cl1;cr ¼ Gch;l1;cr

(ch;r1;cr and ch;l1;cr are given by (4c)), respectively, and they are distinct (see forthcoming Fig. 5(a), third column), except for

the special case ch;r1;cr ¼ ch;l1;cr, which however occurs for a single value of x, see Fig. 2, and is thus out of interest. The

difference due to the asymmetry of the system (which disappears for r ¼ 1, see Fig. 5(b), third column) is thus

maintained by the controlled excitation. However, another strategy can be followed. In fact, instead of increasing the

lowest gain as done in Section 2.3 it is possible to increment the lowest of the two values cr1;cr ¼ Grch;r1;cr and

cl1;cr ¼ Glch;l1;cr––where Gr and Gl are given by (9) and (15), respectively––corresponding to a generic excitation, even at

the price of reducing the highest. In this case, the aim is that of increasing the threshold of the first critical event, thus

improving the worst performance of the system while still keeping an overall control. From this point of view, the

critical threshold is c1;cr ¼ minfcr1;cr; cl1;crg. The associated optimization problem is then:

Maximizing c1;cr ¼ minfGrch;r1;cr;Glch;l1;crg by varying the Fourier coefficients hrj and Wj; j ¼ 2; 3; . . . ;

of hrðmÞ; the Fourier coefficients of hlðmÞ being given by ð16Þ: ð19Þ

As in the case of Section 2.3, the solution of (19) depends on the system. Here, however, the things seem to be slightly

more involved, since the solution behaves differently according to whether ch;l1;cr is close to or different from ch;r1;cr. This

distinction is conjectured from a theoretical point of view, and is confirmed by the forthcoming numerical results

(Section 3). To qualitatively locate the regions of different behaviour (closeness versus remoteness of ch;l1;cr and ch;r1;cr) in the

parameter space ðx; rÞ, we have depicted in Fig. 4 the contour plot of the ratio

Fig. 4. The contour plot of the function aðx;rÞ.


aðx; rÞ ¼ch;l1;cr

ch;r1;cr

¼ f lðrÞf rðrÞ

sinh xffiffir

p arccos þ 1�r1þr


rp arccos � 1�r

1þr

� �h i > 0: ð20Þ

The exact boundary between the regions will be determined in Section 3.2.

Let us first consider the case of ch;l1;cr � ch;r1;cr, which is characteristic of this kind of approach. In this case, in fact, the

increment of ch;l1;cr and the (possible) decrement of ch;r1;cr are comparable, and, similarly to the case of Section 2.3, the

numerical results suggest that the solution is characterized by the condition

Grch;r1;cr ¼ Glch;l1;cr; ð21Þ

i.e., by cr1;cr ¼ cl1;cr. This implies that, contrary to the case of harmonic excitation, the right and left homoclinic bifur-

cations occur simultaneously, so that the involved features of the system dynamics have been ‘‘symmetrized’’ (Fig. 5(a),

fourth column). The present approach is called ‘‘global control with symmetrization’’.

The class of excitations satisfying (21) is characterized, in term of the functions hrðmÞ and hlðmÞ, by the following

relation:

maxm2½0;2p�

fhlðmÞg ¼ �aðx; rÞ minm2½0;2p�

fhrðmÞg; ð22Þ

where aðx; rÞ is given by (20) and is identically equal to 1 for symmetric systems ðr ¼ 1Þ.On the basis of (22) and noting that maximizing Grch;r1;cr entails maximizing Gr since ch;r1;cr does not depend on the shape

of the excitation, it is possible to rewrite the mathematical problem (19) in the form:

Maximizing Gr by varying the Fourier coefficients hrj and Wj; j ¼ 2; 3; . . . ; of hrðmÞ; under the constraint

maxm2½0;2p�

fhlðmÞg ¼ �aðx; rÞ minm2½0;2p�

fhrðmÞg;where the Fourier coefficients of hlðmÞ are given by ð16Þ; ð23Þ

which permits a direct comparison with (18). In fact, one difference is the factor aðx; rÞ which appears in the constraint

of (23) and which is 1 in (18), while another is that (18) seems to be valid, i.e., equivalent to (17), for all values of r, while(23) is correct only when ch;l1;cr � ch;r1;cr. Apart from these points, the two optimization problems are similar, so that (23)

has the same general properties as (18). In particular, it is strictly dependent on the system and provides a theoretical

optimal gain lesser than that of the one-side control, even if in this case measuring the gain requires some care. In fact,

to quantitatively measure the increment, it is again possible to define the gain as the ratio between the critical threshold

c1;cr ¼ cr1;cr ¼ cl1;cr with the actual excitation and the first critical threshold ch1;cr ¼ minfch;r1;cr; ch;l1;crg with the harmonic

excitation. Since ch;r1;cr ¼ c1;crMr and ch;l1;cr ¼ c1;crM

l, we have that ch1;cr ¼ c1;cr minfM r;M lg. This finally proves that

G ¼c1;crch1;cr

¼ 1

minfM r;M lg ¼ maxfGr;Glg: ð24Þ

(a)

(b)

Fig. 5. A summary chart of the various control approaches for hardening systems. (a) asymmetric; (b) symmetric.


Let us now consider the case of ch;l1;cr very different from ch;r1;cr (largely asymmetric systems or high excitation fre-

quency). To fix ideas, we suppose that ch;l1;cr � ch;r1;cr, which occurs in the upper left region of Fig. 4 and, more generally,

for large values of r (because ch;l1;cr=ch;r1;cr asymptotically behaves like 32=ð15r2Þ as r ! 1). In this case, due to the large

difference between ch;l1;cr and ch;r1;cr, the procedure exhausts all resources in increasing cl1;cr, the possible associated lowering

of cr1;cr being unessential because, being ch;r1;cr very large, cr1;cr ¼ Grch;r1;cr remains always greater than cl1;cr even if Gr is much

lower than 1 (Fig. 5(a), fifth column). Thus, in this case, the left critical threshold is increased up to its maximum extent,

and the same gain and hlðmÞ are obtained as in the (left) one-side control (Fig. 5(a), first column) where only one

homoclinic bifurcation is increased. Indeed, for large r, one can think of the right potential well in Fig. 1(a) being so

deep and the right center being so far from the hilltop saddle, that the whole potential in the neighbourhood of the latter

is actually similar to that of the Helmholtz oscillator [13], to which Eq. (1) reduces for r ! 1 after a proper rescaling.

The simplified problem (23) no longer holds in this case.

Since the threshold for right homoclinic bifurcation remains greater than that for left bifurcation, the system has not

actually been symmetrized in the sense previously mentioned. Thus, strictly speaking, this is a case of global control

without symmetrization, which is however a distinction that refers to a dynamical point of view. In contrast, from a

control point of view, this case is similar to that ch;l1;cr � ch;r1;cr because the same (universal) control strategy is adopted,

irrespective of the solution of the problem which is an ‘‘a posteriori’’ result. In this respect, one can indeed consider the

solution as that which realizes the maximum symmetrization allowed for by the specific characteristics of the system and

of the excitation. Looking for a different label of the two situations, one could name the case ch;l1;cr � ch;r1;cr as ‘‘global

control with achieved symmetrization’’ and the other as ‘‘global control with pursued symmetrization’’ (Fig. 5(a)).

To close this section, we remind that, as in the previous cases, after having determined the best hrjs and Wjs, the

Fourier coefficients cj of the optimal excitation can be computed by inverting the relation (4e). The constraint (16)

between the hrj and the hlj guarantees that there is no ambiguity in the construction of the optimal excitation.

Remark 2. The global control of the symmetric system (Duffing oscillator, see [11]) can be obtained as limit case of the

global controls discussed in the previous section for r ! 1. In this case one has aðx; r ¼ 1Þ ¼ 1 so that problems (17)

and (19) coincide. This means the distinction between global control with and without symmetrization no longer

holds in the symmetric case, and there is a unique, simpler, system-independent global control. Furthermore, also


bðj;x; r ¼ 1Þ ¼ 1, so that hrðmÞ ¼ hlðmÞ in (17) and (19) (and in the simplified versions (18) and (23)), and this finally

proves that, as expected, they are identical to the global control problem of the Duffing oscillator [11].

Remark 3. The various control approaches are summarized (and compared) in the schematic diagram of Fig. 5(a),

while for the sake of completeness and comparison the simpler case of symmetric hardening systems (Duffing) is re-

ported in Fig. 5(b). Fig. 5 illustrates the main characteristics of the various cases, and it has been drawn recalling that

the mathematical optimal gain of the one side controls is 2, while the mathematical optimal gain of global control of

symmetric systems is 4=p ffi 1:2732 [11].

3. Optimal control

In the previous section it has been shown that, depending on the control strategy, the problem of choosing the best

excitation mathematically reduces to the optimization problems (10), (17) and (19). The former, which is the unique to

possess system-independent solutions, has been discussed in detail and solved in various forms elsewhere [13], see also

[14]. In particular, the mathematical solution has been first obtained, which is physically inadmissible. To overcome this

drawback, reduced solutions with a finite number of superharmonics and constrained solutions with a constraint

guaranteeing physical admissibility are then obtained. Finally, the mixed case of reduced and constrained solutions,

deserving practical interest, is also addressed.

The reduced solutions in the case of right control are given in Table 1, which is a copy of Table 1 of [13] herein

reported for easiness of reading, being repeatedly referred to in the following. The reduced solutions in the case of left

control, on the other hand, are just those of Table 1 with the even coefficients hj changed of sign (see Remark 1).

3.1. System-dependent solutions

The global control of asymmetric systems is slightly less general than the one-side control, because the solution is

system-dependent. In this respect, it is worth pointing out that all the results of the present section refer to the

asymmetric Helmholtz–Duffing oscillator (1), whereas they are not directly valid for other asymmetric mechanical

systems, although these are likely to exhibit the same main characteristics and phenomena. In spite of this, the present

case has some worthing characteristics, such as the questions related to the global control with and without symme-

trization, which are typical of asymmetric systems and are herein investigated for the first time.

The global control of asymmetric systems requires solving problems (17) (global control without symmetrization) or

(19) (global control with symmetrization). However, these problems are mathematically more difficult than the global

control problem of symmetric system, basically due to the relation (16) between hrðmÞ and hlðmÞ, and we have not got

the exact solutions in the general case with no constraint of physical admissibility. A major source of difficulty is due to

the superharmonics being no longer in phase, contrary to what happens for the Duffing oscillator [11], so that the

number of unknowns in the optimization problems becomes double. Accordingly, only numerical solutions with a finite

number of unbounded superharmonics are determined, using the algorithm of Nelder and Mead [18] properly modified.

Initially, the behaviour for increasing number of superharmonics has been determined. We choose r ¼ 1:2, which is

a reasonable value for real shallow arches (see Section 2 and Figs. 1 and 2). For this value, the Helmholtz–Duffing

oscillator has two stable equilibrium positions xl ¼ �0:7032 and xr ¼ 0:8532, and the frequencies of small oscillations

around these points are xl ¼ 1:4795 and xr ¼ 1:6297. In order to investigate a case sufficiently close to both resonances,

Table 1

The numerical results of various optimization problems with increasing finite number of superharmonics in the case of one-side control

N GN h2 h3 h4 h5 h6 h7 h8 h9

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

6 1.8476 0.807624 0.567084 0.334898 0.153043 0.042422

7 1.8794 0.842528 0.635867 0.422667 0.237873 0.103775 0.027323

8 1.9000 0.872790 0.706011 0.527198 0.355109 0.205035 0.091669 0.024474

9 1.9130 0.877014 0.705931 0.518632 0.341954 0.195616 0.091497 0.031316 0.005929

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2 1 1 1 1 1 1 1 1

Table 2

The numerical results of various optimization problems with increasing unbounded finite number of superharmonics in the case of

global control of asymmetric systems without symmetrization

N c1;cr GN h2 W2 h3 W3 h4 W4 h5 W5

2 0.9151 1 0.000000 0.000000 Right

0.7753 0.000000 0.000000 Left

3 1.0463 1.1433 )0.008976 )0.095646 )0.202643 )0.004106 Right

0.8864 )0.006980 )0.095646 )0.121802 )0.004106 Left

4 1.0604 1.1587 )0.040962 )0.062851 )0.240144 0.004503 )0.047464 0.074554 Right

0.8983 )0.031851 )0.062851 )0.144342 0.004503 )0.022050 0.074554 Left

5 1.0747 1.1744 )0.046380 )0.423565 )0.283030 )0.112952 0.003180 )0.007272 0.092714 )0.033888 Right

0.9105 )0.036063 )0.423565 )0.170119 )0.112952 0.001477 )0.007272 0.033289 )0.033888 Left

x ¼ 1:55, r ¼ 1:2, W1 ¼ 0 (without loss of generality). Note that ch;r1;cr ¼ 0:9151 and ch;l1;cr ¼ 0:7753.

Table 3

The numerical results of various optimization problems with increasing unbounded finite number of superharmonics in the case of

global control of asymmetric systems with (achieved) symmetrization

N c1;cr GN h2 W2 h3 W3 h4 W4 h5 W5

2 0.8364 0.9139 )0.094529 0.097428 Right

1.0788 )0.073503 0.097428 Left

3 0.9573 1.0460 )0.155196 0.000000 )0.205347 0.000000 Right

1.2347 )0.120676 0.000000 )0.123427 0.000000 Left

4 0.9628 1.0521 )0.140583 )0.024134 )0.207741 0.005656 0.028273 )0.159942 Right

1.2418 )0.109314 )0.024134 )0.124866 0.005656 0.013134 )0.159942 Left

5 1.0001 1.0929 )0.156225 )0.022596 )0.259358 0.006480 0.057252 )0.183003 0.071511 0.009883 Right

1.2900 )0.121476 )0.022596 )0.155891 0.006480 0.026597 )0.183003 0.025676 0.009883 Left

x ¼ 1:55, r ¼ 1:2,W1 ¼ 0 (without loss of generality). Note that ch1;cr ¼ minfch;r1;cr; ch;l1;crg ¼ minf0:9151; 0:7753g ¼ 0:7753 and that, being

the actual gain the maximum between Gr and Gl, in this case G ¼ Gl.


we consider x ¼ 1:55. For these values of r and x, one has ch;r1;cr ¼ 0:9151 and ch;l1;cr ¼ 0:7753. The numerical results are

reported in Tables 2 and 3 for the global control without and with symmetrization, respectively.

The specific characteristics of the two approaches are well recognizable in the two tables. In particular, in the case of

control without symmetrization (Table 2) the left and right gains are seen to be equal (and optimal, in each restricted

class of excitations), so that the amplitude thresholds for the right and left homoclinic bifurcations remain distinct also

with the optimal excitation. The equal relative increment of the right and left critical amplitudes is obtained at the price

of not very high gains, at least if compared with those of Table 3. They may be satisfactory in all applications where we

need only a minor reduction of the ‘‘cross-well chaotic’’ region.

On the other hand, for the control with (achieved) symmetrization (Table 3) in a case of intermediate x, it can be

seen that the two critical amplitudes are equal with the optimal excitation, and accordingly Gr 6¼ Gl. This permits to

increase the lower critical threshold, even accepting that the other critical amplitude may reduce (see the case N ¼ 2 of

Table 3; this phenomenon is more evident in the forthcoming Table 5). This approach spends less in optimization, and

consequently it permits to obtain larger gains on the worst side of the phase space (in the case of Table 3, it is G ¼ Gl),

more than one and a half those of Table 2. Furthermore, the larger is the number N of superharmonics, the larger is the

difference between the gains provided by the two different approaches.

The absence of the analytical solution prevents from having an appropriate upper bound to be used as a reference

measure for the reduction of theoretical performances obtainable with the two approaches when considering a finite

instead of infinite number of superharmonics. However, we can compare the gains of Tables 2 and 3 with the corre-

sponding ones of Table 1, which are optimal gains in the case of reduced one-side controls and therefore represent

numerical upper bounds. This permits to estimate the reduction of gain due to the constraint (16) occurring in the case

of global control. From this comparison a strong reduction is seen to occur when passing from one-side to global

control, similarly to what happens in the case of global control of the symmetric system [11]. This decrement is more

(less) marked without (with) symmetrization and it is intrinsic to the nature of the global control.

The influence of the asymmetry parameter r on the optimal excitation is investigated in Tables 4 and 5, which are

summarized in Fig. 6, for the case N ¼ 3 and x ¼ 1:55. By considering the limit case r ¼ 1 reported in the first row, the

latter also permits the comparison between the effects of global control on symmetric and asymmetric systems.

Table 4

Influence of asymmetry parameter r: Numerical results of various optimization problems with finite unbounded number of super-

harmonics in the case of global control without symmetrization

r c1;cr ch1;cr G3 h2 W2 h3 W3

1.0 0.9090 0.7872 1.1547 0 0 )0.166667 0 Right

0.9090 0.7872 0 0 )0.166667 0 Left

1.1 0.9745 0.8466 1.1510 0.000000 0.000000 )0.191396 0.000000 Right

0.8986 0.7807 0.000000 0.000000 )0.144819 0.000000 Left

1.2 1.0463 0.9151 1.1433 )0.008976 )0.095646 )0.202643 )0.004106 Right

0.8864 0.7753 )0.006980 )0.095646 )0.121802 )0.004106 Left

1.4 1.2116 1.0776 1.1244 )0.030938 )0.433606 )0.231417 )0.057337 Right

0.8629 0.7674 )0.020243 )0.433606 )0.097694 )0.057337 Left

1.7 1.5331 1.3823 1.1091 )0.066329 )0.423600 )0.238232 )0.111547 Right

0.8451 0.7620 )0.036670 )0.423600 )0.070723 )0.111547 Left

2.0 1.9386 1.7617 1.1004 )0.078432 )0.398051 )0.249788 )0.119979 Right

0.8393 0.7628 )0.039150 )0.398051 )0.059433 )0.119979 Left

2.5 2.8110 2.5726 1.0927 )0.098548 )0.423781 )0.250444 )0.154411 Right

0.8455 0.7738 )0.045016 )0.423781 )0.048508 )0.154411 Left

3.0 3.5975 3.6304 1.0901 )0.103320 )0.392954 )0.250992 )0.149112 Right

0.8648 0.7933 )0.045613 )0.392954 )0.044271 )0.149112 Left

4.0 7.1953 6.6089 1.0887 )0.104349 )0.420943 )0.256306 )0.159839 Right

0.9226 0.8474 )0.046139 )0.420943 )0.043660 )0.159839 Left

x ¼ 1:55, N ¼ 3, W1 ¼ 0 (without loss of generality).

Table 5

Influence of asymmetry parameter r: Numerical results of various optimization problems with finite unbounded number of super-

harmonics in the case of global control with symmetrization

r c1;cr ch1;cr G3 h2 W2 h3 W3

1.0 0.9090 0.7872 1.1547 0 0 )0.166667 0 Right

0.7872 1.1547 0 0 )0.166667 0 Left

1.1 0.9353 0.8466 1.1047 )0.074261 0.000000 )0.189087 0.000000 Right

0.7807 1.1980 )0.064682 0.000000 )0.143072 0.000000 Left

1.2 0.9573 0.9151 1.0460 )0.155196 0.000000 )0.205347 0.000000 Right

0.7753 1.2347 )0.120676 0.000000 )0.123427 0.000000 Left

1.4 0.9883 1.0776 0.9172 )0.293358 0.004946 )0.203205 )0.011549 Right

0.7674 1.2878 )0.191944 0.004946 )0.085784 )0.011549 Left

1.7 1.0291 1.3823 0.7444 )0.475486 0.014708 )0.133773 )0.104380 Right

0.7620 1.3504 )0.262872 0.014708 )0.039713 )0.104380 Left

2.0 1.0734 1.7617 0.6093 )0.644803 0.000867 )0.003750 )0.353494 Right

0.7628 1.4072 )0.321855 0.000867 )0.000892 )0.353494 Left

2.5 1.1630 2.5726 0.4521 )0.893145 0.021414 0.320768 0.163502 Right

0.7738 1.5030 )0.407982 0.021414 0.062129 0.163502 Left

3.0 1.2701 3.6304 0.3498 )1.138719 0.000270 0.719791 )0.026388 Right

0.7933 1.6010 )0.502715 0.000270 0.126959 )0.026388 Left

3.1 1.2862 3.8743 0.3324 )1.187869 0.000322 0.820670 )0.003042 Right

0.7979 1.6119 )0.522866 0.000322 0.143244 )0.003042 Left

3.2 1.2985 4.1294 0.3144 )1.239677 0.001525 0.940469 )0.006260 Right

0.8028 1.6175 )0.544570 0.001525 0.162776 )0.006260 Left

3.2381 1.3020 4.2297 0.3078 )1.259000 0.000000 0.989453 0.000000 Right

1.3020 0.8047 1.6180 )0.552756 0.000000 0.170789 0.000000 Left

4.0 2.0318 6.6089 0.3074 )1.250129 0.000000 1.002606 0.000000 Right

1.3712 0.8474 1.6180 )0.552756 0.000000 0.170789 0.000000 Left

x ¼ 1:55, N ¼ 3, W1 ¼ 0 (without loss of generality). Note that, being the actual gain the maximum between Gr and Gl, in this case

G ¼ Gl.


It is clear from Fig. 6 that in the cases with and without symmetrization the asymmetry has different effects: there is a

strong increment of the gain in the former case and a slight decrement in the latter. Thus, the asymmetry of the system

Fig. 6. The reduced optimal gain G3 in the case of global control of asymmetric systems with and without symmetrization for

x ¼ 1:55. The case r ¼ 1 corresponds to the global control of symmetric systems [11]. The limiting constant value of G3 for one-side

control is also reported.


has a favourable effect on control in the first approach, while it is unfavourable if one is aimed at controlling the system

by the global control without symmetrization.

Focusing attention on the global control with symmetrization, which is theoretically more performant, we see that

the gain increases rapidly and reaches the numerical upper bound G3 ¼ 1:6180 (see Table 1) approximately for

r ¼ 3:2381. This value represents the transition from ‘‘achieved’’ to ‘‘pursued’’ symmetrization, and will be discussed in

detail in the following section. Furthermore, also the coefficients of the hlðmÞ tend to the coefficients h2 ¼ �0:552756(see Remark 1), h3 ¼ 0:170789, W2 ¼ W3 ¼ 0 of the optimal solution of Table 1, and this shows that for r > 3:2381 the

global control with symmetrization basically reduces to the one-side control of the left homoclinic bifurcation. This

agrees with the discussion of Section 2, because for r > 3:2381 (largely asymmetric systems) ch;l1;cr is so lower than ch;r1;cr

(see the third column of Table 5) that all resources are used to increase cl1;cr up to its maximum value given by the one-

side control, whereas cr1;cr remains larger than cl1;cr (see second column of Table 5) though being strongly reduced with

respect to the harmonic case (see third column of Table 5).

The behaviour of the solution illustrated in the previous comments is felt to be qualitatively correct also for opti-

mization problems with larger number of superharmonics, the only differences being of quantitative type. For example,

the gain will be certainly larger than those of Fig. 6, and the transition to the flat plateau will occur for a different value

of r, however maintaining the same qualitative aspect of the diagram.

3.2. On the transition from ‘‘achieved’’ to ‘‘pursued’’ global control with symmetrization

Fig. 6 of previous section highlights the occurrence of a well defined threshold (r ¼ 3:2381 in that particular case)

above which the right and left homoclinic bifurcations are so far apart that the optimization spends all resources in

increasing the lower bifurcation, the strong reduction of the higher one being irrelevant. This threshold marks the

transition from ‘‘achieved’’ to ‘‘pursued’’ symmetrization.

For practical purposes, it is worth determining the overall boundary between these different regions in the parameter

space ðx; rÞ, which permits to better characterize the expected theoretical performances of control. This question is

investigated in the present section by referring again to only numerical solutions with a finite number of unbounded

superharmonics. The extent of the investigated regions depends on the number N of superharmonics, the larger being Nthe larger being the possibility to increase the lower bifurcation and reduce the gap between the lower and the higher

ch1;cr. Thus, the ‘‘achieved’’ region enlarges and the ‘‘pursued’’ region reduces with increasing N .

Let us consider the N -reduced left-side optimal solution �hlðmÞ (see Table 1 and Remark 1) and the associated optimal

gain GN given by 1=maxf�hlðmÞg and reported in the second column of Table 1. By means of (16), the corresponding�hrðmÞ is given by

�hrðmÞ ¼ cosðmÞ þXNj¼2

�hljbðj;x; rÞ cosðjmÞ ð25Þ

Fig. 7. The regions of ‘‘achieved’’ and ‘‘pursued’’ global control with symmetrization in the parameter space ðx; rÞ.


and it is easy to compute Gr

N ¼ Gr

N ðx; rÞ ¼ �1=minf�hrðmÞg. With the associated excitation, the left homoclinic

bifurcation occurs for cl1;cr ¼ GNch;l1;cr, while the right homoclinic bifurcation occurs for cr1;cr ¼ G

r

Nch;r1;cr. Since c

l1;cr cannot

be further increased, one has ‘‘pursued’’ symmetrization as long as cl1;cr < cr1;cr, namely (remember Eq. (20)),

Gr

N ðx; rÞGNaðx; rÞ

> 1: ð26Þ

In the space ðx; rÞ, Eq. (26) represents an implicit inequality defining a ‘‘pursued’’ region. It is the upper-left region

depicted in Fig. 7 for N ¼ 3 and 5. For x ¼ 1:55 and N ¼ 3 the boundary is at r ¼ 3:2381, which is just the case of Fig. 6.

Let us now consider the N -reduced right-side optimal solution �hrðmÞ and the associated optimal gain GN given by

�1=minf�hrðmÞg. By means of (16), the corresponding �hlðmÞ is

�hlðmÞ ¼ cosðmÞ þXNj¼2

�hrjbðj;x; rÞ cosðjmÞ ð27Þ

and it is easy to compute Gl

N ¼ Gl

N ðx; rÞ ¼ 1=maxf�hlðmÞg. With the associated excitation, the left homoclinic bifur-

cation occurs for cl1;cr ¼ Gl

Nch;l1;cr, while the right homoclinic bifurcation occurs for cr1;cr ¼ GNc

h;r1;cr. Since cr1;cr cannot be

further increased, one has ‘‘pursued’’ symmetrization as long as cr1;cr < cl1;cr, namely,

Gl

N ðx; rÞaðx; rÞGN

> 1: ð28Þ

In the space ðx; rÞ, Eq. (28) represents an implicit inequality defining another ‘‘pursued’’ region. It is the lower-right

region depicted in Fig. 7 for N ¼ 3 and 5.

In the limit case N ! 1 one has �hrj ! 1 and G1 ¼ 2 (Table 1). The function �hlðmÞ is

�hlðmÞ ¼ cosðmÞ þX1j¼2

bðj;x; rÞ cosðjmÞ ð29Þ

and its maximum is

Ml ¼ 1þ

X1j¼2

bðj;x; rÞ ¼X1j¼1

bðj;x; rÞ; ð30Þ

(it is an upper bound attained for m ¼ 0), which is a rapidly convergent series. Thus, the ‘‘pursued’’ region in the limit

case of infinite superharmonics is implicitly defined by the inequality

aðx; rÞ2P1

j¼1 bðj;x; rÞ> 1 ð31Þ

and is also depicted in Fig. 7. It represents the accumulation set for the ‘‘pursued’’ regions for N ! 1.


4. Conclusions

An extensive analysis has been performed to illustrate the behaviour of a method for controlling nonlinear dynamics

and chaos in the case of hardening Helmholtz–Duffing equation having two asymmetric homoclinic bifurcations. This

represents an archetype of a more general class of oscillators having distinct homoclinic bifurcations occurring for

different values of the parameters, in particular of the excitation amplitude.

These systems are characterized by a behaviour of the solution of the optimization problems which is system-

dependent in the case of global control, contrary to what happens for one-side controls and for the global control of

symmetric (Duffing) system, where the solutions are system-independent [14]. This entails a more difficult mathematical

problem, which was solved numerically in the case of finite number of superharmonics (reduced solution). The

extension to reduced/constrained solutions seems straightforward, and has not been performed only to limit the length

of the work. The extension to constrained solutions with infinite terms appears instead impracticable, at least at this

stage of the research. However, it deserves mainly a theoretical interest, because the reduced solutions produce gains,

i.e. theoretical performances, which are already quite large and satisfactory from an application point of view.

It has been shown that the considered system has special characteristics in terms of global control. In fact, two

different approaches have been followed, based on increasing of either gains (global control without symmetrization) or

excitation amplitudes (global control with symmetrization). In the latter case the situation is even more involved,

because the symmetrization of system dynamics can be actually ‘‘achieved’’ or simply ‘‘pursued’’. This is a peculiarity of

the solution, and not of the control strategy, and depends on the asymmetry of the system and on the excitation

frequency.

Finally, the regions of ‘‘pursued’’ and ‘‘achieved’’ global control with symmetrization in the parameter space have

been determined, too. The pursued region is constituted by two different zones, one for highly asymmetric systems and

another for large excitation frequencies. The achieved regions enlarge by increasing the number of superharmonics.

References

[1] Alaggio R, Benedettini F. The use of experimental tests in the formulation of analytical models for the finite forced dynamics of

planar arches. In: Proc. DECT’01 2001 ASME Design Eng Tech Conf, Pittsburgh, Pennsylvania, USA, 9–12 September 2001 (CD

ROM DETC2001/VIB21613).

[2] Gjelsvik A, Bodner SR. The energy criterion and snap buckling of arches. ASCE J Eng Mech Div 1962;(October):89–134.

[3] Grebogi C, Ott E, Yorke JA. Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 1983;7:181–200.

[4] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer-

Verlag; 1983.

[5] Higuchi K, Dowell EH. Effect of constant transverse force on chaotic oscillations of sinusoidally excited buckled beam. In:

Schiehlen W, editor. Proc. of IUTAM Symposium, Nonlinear dynamics in engineering systems. Stuttgart, Germany 21–25 August

1989. Berlin: Springer-Verlag; 1989. p. 99–106.

[6] Katz A, Dowell EH. From single well chaos to cross well chaos: a detailed explanation in terms of manifold intersections. Int J Bif

Chaos 1994;4:933–41.

[7] Kovacic G, Wiggins S. Orbits homoclinic to resonance, with an application to chaos in a model of the forced and damped sine-

Gordon equation. Physica D 1992;57:185–225.

[8] Lenci S, Rega G. Global chaos control in a periodically forced oscillator. In: Bajaj AK, Namachchivaya NS, Franchek MA,

editors. Proc ASME Int Mech Eng Congress. Nonlinear Dynamics and Control, DE-vol. 91. 17–22 Nov. 1996, Atlanta, GA.

p. 111–6.

[9] Lenci S, Rega G. A procedure for reducing the chaotic response region in an impact mechanical system. Nonlinear Dyn

1998;15:391–409.

[10] Lenci S, Rega G. Controlling nonlinear dynamics in a two-well impact system. Parts I and II. Int J Bif Chaos 1998;8:2387–424.

[11] Lenci S, Rega G. Optimal control of nonregular dynamics in a Duffing oscillator. Nonlinear Dyn 2003;33:71–86.

[12] Lenci S, Rega G. Optimal numerical control of single-well to cross-well chaos transition in mechanical systems. Chaos, Solitons &

Fractals 2003;15:173–86.

[13] Lenci S, Rega G. Optimal control of homoclinic bifurcation: theoretical treatment and practical reduction of safe basin erosion in

the Helmholtz oscillator. J Vib Control 2003;9:281–316.

[14] Lenci S, Rega G. A unified control framework of the nonregular dynamics of mechanical oscillators. J Sound Vibr, in press.

[15] Mettler E. Dynamic buckling. In: Fl€ugge W, editor. Handbook of engineering mechanics. New York: Mc Graw-Hill; 1962.

[16] Moon FC. Chaotic and fractal dynamics. An introduction for applied scientists and engineers. New York: Wiley; 1992.

[17] Moon FC, Cusumano J, Holmes PJ. Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum. Physica D

1987;24:383–90.

[18] Nelder JA, Mead R. A simplex method for function minimization. Comput J 1964;7:308–13.

[19] Pinto OC, Gonc�alves PB. Non-linear control of buckled beams under step loading. Mech Syst Sign Proc 2000;14:967–85.


[20] Shaw SW. The suppression of chaos in periodically forced oscillators. In: Schiehlen W, editor. Proc of IUTAM Symposium,

Stuttgart, Germany, 21–25 August 1989. Nonlinear Dynamics in Engineering Systems. Berlin: Springer-Verlag; 1989. p. 289–96.

[21] Szemplinska-Stupnicka W. Cross-well chaos and escape phenomena in driven oscillators. Nonlinear Dyn 1992;3:225–43.

[22] Szemplinska-Stupnicka W. The analytical predictive criteria for chaos and escape in nonlinear oscillators: a survey. Nonlinear

Dyn 1995;7:129–47.

[23] Thompson JMT, Stewart HB. Nonlinear dynamics and chaos. New York: Wiley; 1986.

[24] Thompson JMT. Chaotic phenomena triggering the escape from a potential well. Proc R Soc Lond A 1989;421:195–225.

[25] Thompson JMT, McRobie FA. Indeterminate bifurcations and the global dynamics of driven oscillators. In: Kreuzer E, Schmidt

G, editors. Proc First Eur Nonlinear Oscillations Conf, 16–20 August. Hamburg, Berlin: Akademie Verlag; 1993. p. 107–28.

[26] Tseng W-Y, Dugundji J. Nonlinear vibrations of a buckled beam under harmonic excitation. ASME J Appl Mech 1971;38:467–76.

[27] Wiggins S. Global bifurcation and chaos: analytical methods. New York, Heidelberg, Berlin: Springer-Verlag; 1988.

[28] Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. New York, Heidelberg, Berlin: Springer-Verlag; 1990.

Documents

Global optimal control and system-dependent solutions in the hardening Helmholtz–Duffing oscillator