parametrically excited liehnard systems

7/31/2019 Parametrically Excited LieHnard Systems

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International Journal of Non-Linear Mechanics 35 (2000) 239}262

Modulated motion and in"nite-period homoclinic bifurcationfor parametrically excited LieHnard systems

Attilio Maccari

Technical Institute **G. Cardano++, Piazza della Resistenza 1, 00015 Monterotondo, Rome, Italy

Received 26 August 1998; received in revised form 5 February 1999

Abstract

A parametrically excited LieHnard system is investigated by an asymptotic perturbation method based on Fourier

expansion and time rescaling. Two coupled equations for the amplitude and the phase of solutions are derived. Their

"xed points correspond to limit cycles for the LieHnard system and we determine stability of steady-state response as well

as response-parametric excitation and response-frequency curves. We use the PoincareH }Bendixson theorem, the Dulac's

criterion and energy considerations to study existence and characteristics of limit cycles of the two coupled equations.

A limit cycle corresponds to a modulated motion in the LieHnard system. We show that modulated motion can also be

obtained for very low values of the parametric excitation and construct an approximate analytic solution. Moreover, we

observe an unusual in"nite-period homoclinic bifurcation, because in certain cases due to the symmetry of the two

coupled equations two stable limit cycles approach a saddle point and merge to form a greater stable limit cycle.

Subsequently, this limit cycle and another unstable limit cycle coalesce and annihilate through a saddle-node bifurcation.

Comparison with the solution obtained by the numerical integration con"rms the validity of our analysis. 1999

Elsevier Science Ltd. All rights reserved.

Keywords: Parametric excitation; LieHnard systems; Asymptotic analysis; In"nite-period bifurcation

1. Introduction

It is well known that many mechanical systems and oscillating circuits can be modeled by the LieHnard

system [1}3]

X$ (t)#f(X(t))"g(X(t))XQ (t), (1)

where the dot denotes di!erentiation with respect to the time and the arbitrary functions f(x) and g(x) are

supposed to be analytic. Obviously, Eq. (1) can be interpreted mechanically as the equation of motion for

a unit mass subject to a non-linear damping force g(x)x and a non-linear restoring force!f(x).We restrict our study to the following particular case of Eq. (1):

X$ (t)#

X(t)#f

X(t)#f

X(t)"g

XQ (t)#g

X(t)XQ (t)#g

X(t)XQ (t) (2a)

0020-7462/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 1 2- 8


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corresponding to the choice

f(x)"

x#f

x#f

x, g(x)"g#g

x#g

x. (2b)

Eqs. (2a) and (2b) can also be considered a generalized Van der Pol}Du$ng system, because it includes as

particular cases the Van der Pol oscillator ( f

,f

, g"0 and g

"!g

O0) and the Du$ng equation

( f"g

"g

"0 and g

,fO0). Through a second-order perturbation analysis we have investigated Eqs.

(2a) and (2b) in Ref. [4] and the existence of one or two limit cycles has been demonstrated. Moreover,a resonant external excitation has been added to Eqs. (2a) and (2b) and by a lower-order perturbation

analysis we have derived a su$cient condition to obtain a two-period quasiperiodic motion, when a second

low-frequency appears, in addition to the forcing frequency. Finally, analytic approximate solutions have

been checked by numerical integration.

In this paper we consider the LieHnard oscillator in the case of parametric excitation. In particular, we

introduce a principal parametric excitation 2fX(t) cos(2t) into Eqs. (2a) and (2b) and obtain

X$ (t)#

X(t)#f

X(t)#f

X(t)"g

XQ (t)#g

X(t)XQ (t)#g

X(t)XQ (t)#2fX(t) cos(2t). (3)

Principal parametric resonance or

-subharmonic external resonance have been studied in many papers and

systems similar to Eq. (3) have been considered in Refs. [5,6].

In Section 2 we use the asymptotic perturbation (AP) method [7,8] and calculate an approximate analytic

solution. We derive a model system of two coupled di!erential equations in the phase and amplitude of

solutions. Stable (unstable)"xed points of the model system correspond to stable (unstable) limit cycles of the

parametrically excited LieHnard system (3). Subsequently, we compare frequency-response and parametric

excitation-response curves with results of the numerical integration.

In Section 3 we perform a global analysis of the model system by the PoincareH }Bendixson theorem, the

Dulac's criterion and energy considerations. We derive several conditions in order to exclude or permit the

existence of limit cycles, corresponding to modulated motions for the LieHnard systems (3). Moreover, we

study the modulated motion that appears for very low values of the parametric excitation. In this case the

two-period quasiperiodic motion is characterized by a slight modulation of the fundamental oscillation, with

a modulation amplitude proportional to the magnitude of the parametric excitation.In Section 4 we demonstrate the presence of an unusual in"nite-period homoclinic bifurcation. In certain

cases the symmetry of the model system gives rise to the appearance of two stable limit cycles, a saddle point

and a surrounding unstable limit cycle. If we increase the parametric excitation, the two limit cycles approach

the saddle point and subsequently merge to form a greater stable limit cycle. Finally, with a further increase

of the parametric excitation this limit cycle and the unstable limit cycle coalesce and disappear through

a saddle-node bifurcation.

Note that in the usual homoclinic bifurcation a limit cycle approaches a saddle point and at the bifurcation

point the limit cycle and a branch of both the stable and unstable manifolds of the saddle point coincide,

forming a homoclinic connection, which is essentially a cycle limit with in"nite period. As the control

parameter increases, the limit cycle suddenly vanishes (`blue sky catastrophea). On the contrary, in the model

system of the parametrically excited LieHnard oscillator we observe two limit cycles, which at the same timetouch the saddle point, merge and give rise to a great limit cycle surrounding the saddle point.

In the last section, we brie#y recapitulate the most important results and indicate some possible

generalizations of the present study.

2. Periodic solutions and their stability

The AP method [9] can be considered as a technique that links together the most useful characteristics of

the harmonic balancing and the multiple scales method [10,11], but its origin can be found in the theory of

240 A. Maccari / International Journal of Non-Linear Mechanics 35 (2000) 239 }262


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weakly non-linear partial di!erential equations [12,13]. We introduce a detuning parameter de"ned by"# and in order to balance the e!ect of the non-linearity and damping with the parametric

excitation we scale the parametric excitation coe$cient f and the dissipative coe$cient g

as f, g

.

No conditions are imposed on the coe$cients f

,f

, g

, g

, which can be of order 1.

Eq. (3) yields

X$

(t)#

X(t)#2X(t)#

X(t)#fX

(t)#fX

(t)

"g

XQ (t)#g

X(t)XQ (t)#g

X(t)XQ (t)#2fX(t) cos(2t). (4)

We now introduce the slow time

"Ot, (5)

where q is a rational positive number, which will be "xed afterwards. The value ofq "xes the magnitude order

of the temporal asymptotic limit in such a way that the non-linear e!ects become consistent and not

negligible. If tPR, then P0, when assumes a "nite value.If we take "0 in Eq. (4) and neglect non-linear terms, we see that it admits simple harmonic solutions

X(t)"A exp(!it)#c.c., where A is a constant depending on initial conditions and c.c. stands for complexconjugate. We expect that non-linear e!ects will induce a modulation of the amplitude A and the appearance

of higher harmonics. By means of the rescaled time (5) we can describe the slow modulation due to the

non-linear terms.

The assumed solution X(t) of Eq. (4) can be expressed by means of a power series in the expansion

parameter , we formally write

X(t)">

L\

ALL(, ) exp(!int), (6)

with L""n" for nO0, and

"r is a positive number, which will be "xed later on; being the assumed

solution X(t) real, we have

L(, )"H

\L(, ). (7)

The assumed solution (6) can be written more explicitly

X(t)"P

(; )#(

(; ) exp(!it)#

(; ) exp(!2it)

#

(; ) exp(!3it)#

(; ) exp(!4it)#c.c.)#O() (8)

and we see that it can be considered a combination of the various harmonics with coe$cients depending on

and .

We suppose that the functions L(, )'s can be expanded in power series of , i.e.

L(; )"

G

GGL

(). (9)

We have assumed in Eq. (9) that the limit of the L(; )'s for P0 exists and is "nite. In the following, for

simplicity we use the abbreviations L"

Lfor nO1 and

" for n"1.

Note that the introduction of the slow time (5) implies that

d

dt(

Lexp(!int))"!inL#O

dL

d exp(!int). (10)

A. Maccari / International Journal of Non-Linear Mechanics 35 (2000) 239 }262 241


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In order to determine the coe$cients L(; ), we insert the assumed solution (8) into Eq. (4) and obtain

various equations for each harmonic n and for a "xed order of approximation on the perturbation parameter

.For n"0, we get

P#

2f

""#O(, P>)"0. (11)

A correct balance of terms shows that r"2 and then we derive the following relation:

"!

2f

("")#O(). (12)

For n"2, taking into account Eq. (10), we have

!3#f

"!ig

#O(, >O) (13)

and then

"f#ig

3#O(). (14)

For n"1, Eq. (4) yields

2idd

>O!2!2f

(#

H)!3f

""!ig

!ig(

#

H)!ig

""#fH#O(, >O)"0. (15)

if q"2, the "rst term has the same magnitude order of all the other terms.

As we can see from Eqs. (12) and (14), we can derive a di!erential equation for the evolution of the complex

amplitude ,dd"(

#i

)#(

#i

)""#iH (16)

with

"

g

2,

"!, "

f

2, (17)

"

g

2!

gf

2,

"

1

g

6!

3f

2#

5f

3. (18)Substituting the polar form

()"() exp(i()), (19)

into Eq. (16), and separating real and imaginary parts, we arrive at the following model system:

dd"

#

# sin2, (20)

d

d"

#

# cos2. (21)



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Taking into account Eqs. (12), (14) and (19), the approximate solution of Eq. (4) can be written as a sum of

a contribution of order and a contribution of order

X(t)"X

(t)#X

(t)#O(),

X

(t)"2(t) cos(t!(t)),

X

(t)"!2f

(t)#

2f

3(t) cos(2t!2(t))#

2g

3(t) sin(2t!2(t)). (22)

The validity of the approximate solution should be expected to be restricted on bounded intervals of the

-variable and then on time-scale t"O(1/). If one wishes to construct approximate solutions on largerintervals such that "O(1/) then the higher terms will in general a!ect the solution and must be included.Moreover, the approximate solution (22) will be within O() of the true solution on bounded intervals of the-variable, and if the solution is periodic, for all t.

By means of the variable change

tP1

""t, P

, (23)

which implies

P"",

P"

"

, P"

"

, (24)

we can always set "$1,

"$1 and then the model system (20), (21) has only three independent

parameters

,

, .Phase-locked periodic solutions of the LieHnard system (4) correspond to "xed points of the model system

(20), (21), i.e. to the solutions of the equations d/d"d

/d"0.Note that the model system (20), (21) is invariant under the transformation

(, )P(, #), (25)

and hence possesses the corresponding symmetry. Thus if there is an equilibrium point at (

,

), then there

is also one at (

, #). In order to simplify the following discussion, we consider only half of the system. If

we state that the model system contains an equilibrium point, then it actually contains two equilibria points,

the other one being located at the symmetrical position under the transformation (25).

The trivial solution ("0) is possible but also a steady-state "nite-amplitude response (

,

) exists and is

given by

r"!(

#

)$(1#

, r"

, (26a)

"

1

2arctan

#

r

#

r, (26b)

where

"(1#

)!(!

). (27)



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Fig. 1. Response-parametric excitation curve, i.e. amplitude of the response () as function of the excitation ( f). Solid lines stand forstable and dashed lines for unstable solutions while boxes correspond to the numerical exact solution.

The standard linearization method permits the computation of the eigenvalues of the Jacobian matrix

relative to each equilibrium point. They are

"

#2

r#(1!4

r(

#

r),

"

#2

r!(1!4

r(

#

r). (28)

We can determine the parametric excitation-response curve because as the amplitude of the parametric

excitation is varied the stability of the solution is determined by Eq. (28). Moreover, we have checked the

validity of our analysis by numerical integration of Eq. (4) with the standard "fth-order

Runge}Kutta}Fehlberg method.In Fig. 1 we show the results of the stability analysis for a typical case. A jump phenomenon is clearly

observable because for increasing values of the parametric excitation we observe a discontinue transition

from the trivial solution to a "nite steady-state periodic response. Moreover, we observe that this stable

periodic solution comes from a saddle-node fold (or cyclic fold) bifurcation. A fold bifurcation corresponds to

a vertical tangency in the parametric excitation-response space, where the derivative of the response with

respect to the control parameter is in"nite.

Iff(f

, only the stable trivial solution is possible. When f(f(f

, there are three possible solutions: the

stable trivial solution and two non-trivial solutions, the larger of which is stable. When f"f

a subcritical

pitchfork bifurcation occurs and then if f'f

, there are two possible solutions: the unstable trivial solution



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Fig. 2. Frequency-response curve, i.e. the response () as function of the detuning (). Solid lines stand for stable and dashed lines forunstable solutions.

and a non-trivial solution, which is stable. We see that the parametric resonance is not excited when f(f

,

while for f(f(f

we observe the subcritical instability because the parametric resonance may or may not

be excited, depending on the initial conditions. Finally, it is always excited if f'f

.

If we want to observe the jump phenomenon we must vary ffrom a point below f

and the response will be

trivial until f reaches f

. A small increase in f beyond f

causes an upward jump from zero to point B. As

f increases further, B continues to increase slowly towards point C. Obviously, if one starts at point C and

decreases gradually the parametric excitation, the response will slowly decrease until point A is reached.

Finally, a further decrease determine a downward jump to zero and the response becomes trivial. The

frequency-response curve is

"

r$f

4!1!r!2

r. (29)

Eq. (29) presents a threshold level fA

for the activation of the parametric resonance, given by

fA"2((1#r#2

r). (30)

In Fig. 2 we show a typical frequency-response curve. Depending on the value of , one, two or threesolutions are possible. If(

, only the trivial solution is possible, which is stable. If

((

, there are



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Fig. 3. Phase space diagram (X(t),>(t)) with>(t)"XQ (t) with f"1.0, f"!0.11, g"0.01, g"1.0, g"!1.0, "0.01, f"0.01.Circles are the numerical solution and crosses represent the approximate solution.

two possible solutions: a non-trivial solution, which is stable and the trivial solution, which is unstable. If

'

, there are three possible solutions: the trivial solution, which is stable, and two non-trivial solutions,

the larger of which is stable. The subcritical instability appears because in the latter case the response

depends on the initial conditions and the system tends asymptotically to the trivial solution or to the

steady-state periodic response. If'!

, only the stable trivial solution is possible.

In order to verify if the AP method is a valid tool to approximate solutions of non-linear oscillators with

small dissipation coe$cients, we compare the analytic approximate solution with the numerical integration

in the following two examples.We have chosen the following set of parameters:

f"1.0, f

"!0.11, g

"!0.01, g

"1.0, g

"!1.0, f"0.01, "0.01. (31)

We expect the appearance of a stable limit cycle with amplitude "0.085 (see Eq. (25)). In Fig. 3 we show in

the phase space (X(t),>(t)"XQ (t)) a comparison between the analytic approximate solution (22) and the

numerical solution: circles represent the numerical solution and crosses represent the approximate solution.

Since the solution repeats itself one cycle after another, we have represented only a cycle. The agreement of

the results appears to be excellent, because the maximum di!erence is 3;10\ and the medium di!erence is1;10\, i.e. of order as expected.



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Fig. 4. Phase space diagram (X(t),>(t)) with >(t)"XQ (t) with f"1.0, f

"!0.11, g

"0.01, g

"1.0, g

"!1.0, "0.01, f"0.1.

Circles are the numerical solution and crosses represent the approximate solution.

In Fig. 4, we have increased the parametric excitation of an order of magnitude ( f"0.1). According to the

model system (20), (21) and the Eq. (25), in this case we have a stable limit cycle with amplitude "0.165.

The maximum di!erence between the approximate solution (22) and the numerical solution is now 0.04 and

the medium di!erence is 0.02, i.e. of order as expected.

3. Global analysis of the model system

In this section we perform a global analysis of the model system (20), (21) and determine several conditions

for the existence of limit cycles. They correspond to two-period quasiperiodic solutions of the LieHnard system

(4).

A well-known theorem about the unforced LieHnard equation states that this system has a unique, stable

limit cycle under appropriate hypotheses on f(x) and g(x) [14,15].

If f(x) and g(x) satisfy the following conditions:

(i) f (x) and g(x) are continuously di!erentiable for all x;

(ii) f(x)"!f(!x) for all x, i.e. f(x) is an odd function;



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(iii) g(x)"g(!x) for all x, i.e. g(x) is an even function;

(iv) f(x)'0 for x'0;

(v) the odd function G(x)"V

g(u) du has exactly one positive zero at x"x

, is positive for 0(x(x

, is

negative and non-increasing for x'x

, and G(x)P!Ras xPR;

then the system has a unique, stable limit cycle surrounding the origin in the phase plane. The assumptions

(iii) and (v) on g(x) imply that the damping is negative at small "x" and positive at large "x". The assumptions

(ii) and (iv) on f(x) imply that the restoring force acts like an ordinary spring, and then tends to reduce anydisplacement. Since large oscillations are damped down and small oscillations are pumped up, we expect that

the system tends to settle into a self-sustained oscillation of some intermediate amplitude.

By inspection of Eq. (4) for f"0, we can apply the theorem for f"g

"0, f

'0, g

'0, g

(0 because

the function

G(x)"g

x#g

x

3, (32)

where

x"

!3g

g

(33)

satis"es all the given conditions.

However, if we consider the model systems (20), (21) for "f"0, we conclude that a stable steady-stateresponse is possible if

'0 and

(0, i.e. if

g'0, g

(g

f

. (34)

Note that these conditions are less restrictive than those requested by the theorem, but obviously are only

local results and not global properties of the LieHnard system (4).

If the conditions (34) are veri"ed, we obtain a stable equilibrium point which corresponds to a stable limit

cycle for Eq. (4) and its approximate expression is given by Eq. (22), with

(t)"#"!

"constant, (t)"(#

#

)t. (35)

We now consider the most general case fO0. At "rst we demonstrate that if the parametric excitation

amplitude is very low, the limit cycle (35) transforms into a modulated motion. We set "!#

and Eqs.

(20) and (21) yield

dd"!2

#

#sin(2)#3

##

# sin(2), (36)

dd"

#

## cos2#2

##

. (37)

For very low values of the parametric excitation and of , the solution can be expanded in a power series in and . We can retain only the leading terms in Eqs. (36) and (37) and then obtain

dd"!2

#

#sin(2), (38)

d

d"

#

#

. (39)



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Eqs. (38) and (39) can be easily resolved and we get

()") #

, (40)

(t)"

exp(!2

t)

##2

sin(2) t#2

)!2) cos(2) t#2

)#(!2

sin(2

)#2 cos(2

)) exp(!2

t)

4(#) ) ,

(41)

where

)"#

#

(42)

and

, "

!

#depend on the initial conditions. Note that

;

#for the validity of the approxima-

tion.

The resulting asymptotic motion is given by Eq. (40) and

(t)"##

#2 sin(2) t#2)!2) cos(2) t#2)

4(#) )

. (43)

The approximate solution for the LieHnard system (4) can be obtained if we insert the expression for (t), (t)into Eq. (22). The resulting motion is then characterized by a fundamental oscillation with frequency slowlymodulated in amplitude with frequency ) .

In Fig. 5 we observe a modulated motion characterized by a very low value of the parametric excitation

and an amplitude given by Eq. (43).

In Fig. 6 we show a comparison between the approximate solution (22) and the numerical solution of the

same motion. We represent the associated map of the non-autonomous Eq. (4), obtained with the values

(X(0),>(0)), (X(),>()),(X(2),>(2)),2, where is the period of the parametric excitation. Crosses

represent the approximate solution and circles represent the numerical solution. The closed curves reveal

that the motion is quasi-periodic, because of the presence of the frequency (42). The agreement of the results is

excellent, because the maximum di!erence is 0.0031 and the medium di!erence is 0.0014, i.e. of order asexpected.

We now consider some methods in order to exclude the existence of closed orbits and set "$1,

"$1 as in the previous section.

The Dulac's criterion is based on Green's theorem [16]. Let XQ"F(X) be a continuously di!erentiable

vector "eld de"ned on a simply connected subset R of the plane. Moreover, we suppose that there exists

a continuously di!erentiable real-valued function D(X) such that ' (DXQ ) has one sign throughout R. Thenthere are no closed orbits lying entirely in R.

We rewrite the model system in the form

dR

d"2

R#2

R#2R sin2, (44a)

d

d"

#

R# cos2, (44b)

where R". Therefore, we choose D"1 and being X"(R,) obtain from Eqs. (44a) and (44b)

(DXQ )"2#4

R. (45)



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Fig. 5. Representation in the X}> space of an orbit with f"1.0, f

"0.56, g

"0.02, g

"2.0, g

"!1.0, f"0.004, "!0.02.

We conclude that closed orbits do not exist for (i) "

"1 in all the plane, (ii)

"

"!1 in all the

plane and for (iii) (0 in the circular region de"ned by

(!

2

"1

(2, (46)

because in these cases ' (DXQ ) has always one sign. Obviously, in the third case we do not exclude thepossibility of closed orbits which extend from region (46) on the outside in such a way that ' (DXQ ) changesits sign.

More stringent results can be obtained if we consider the energy-like function

E"#

2# cos2 (47)

and suppose that there is a periodic solution X(t) of period . After one cycle, and return to their startingvalues, and therefore E"0 around any closed orbit.

On the other hand,

dE

d"2(

#

)(

#

# cos2)"2(

#

)

d

d(48a)



13/24


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We suppose that

(i) R is a closed, bounded subset of the plane;

(ii) XQ"F(X) is a continuously di!erentiable vector "eld on a open set containing R;

(iii) R does not contain any "xed points;

(iv) there exists a trajectory C that starts in R and stays in R for all future time, i.e. that is con"ned in R; then

either C is a closed orbit, or it spirals towards a closed orbit as tPR. In either case, we conclude that

R contains a closed orbit.It is easy to verify that the model system (20), (21) satisfy conditions (i) and (ii). For the condition (iv) we use

the standard trick to construct a trapping region R, i.e. a closed connected set such that the vector "eld points

inward everywhere on the boundary of R. Then all trajectories in R are con"ned. Moreover, if according to

condition (iii) we can also arrange that there are no "xed points in R, then the PoincareH }Bendixson theorem

ensures that R contains a closed orbit.

We seek two concentric circles with radii K

and +

, such that d/d(0 on the outer circle and d/d'0in the inner circle. Then the annulus 0(

K))

+will be our desired trapping region.

IfK

and +

can be found then the PoincareH }Bendixson theorem will imply the existence of a closed orbit.

To "nd K

, we require d/d'0 for all . Since cos *!1, a su$cient condition for K

is

K

(#

K!"")'0. (50)

Hence any

K(

!"" with

"!1 (51)

will be useful.

By a similar argument the #ow is inward on the outer circle if

+'

#"" with

"!1. (52)

Therefore a stable closed orbit exists for "!1 and it lies somewhere in the annulus

(!""(((

#"". (53)

Note that we require also "1'"" to ensure that

K,

+'0.

From Eq. (21) we can see that there are no "xed points in the annulus and then condition (iii) of the

PoincareH }Bendixson theorem is satis"ed if

#

!

(!1 or#

!

'1, (54)

when is in region (53).We now demonstrate the uniqueness of the limit cycle for 0(""(

. Suppose that there were two di!erent

rotations. One of the rotations would have to lie strictly above the other because trajectories cannot cross.

Let

() and

() denote the upper and lower rotations, where

()'

() for all . The existence of two

such rotations leads to a contradiction, because if we consider the energy-like function (47), then the change

in energy E must vanish, after one circuit around any rotation. Hence any rotation must satisfy

0"E"L

2(1!) d. (55)



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Fig. 7. A two-period quasi-periodic motion for the LieHnard system with "0.01, "!1, "!0.02, "1, "0.005.

If we consider the two rotations

(),

() we have

L

(!

)[1!(

#

)] d"0. (56)

The quantity (!

) is strictly positive and the other is strictly negative because from Eq. (53)

(1!!

)(!1#2""(0 (57)

Then the integral (56) is strictly negative for 0(""(

and we reach a contradiction and so Eq. (55) cannot

hold for both rotations. This contradiction proves that the rotation is unique for 0(""(, as claimed.We can summarise all the results coming from the global analysis of the model system (20), (21) in the

following statement:

(i) if "

"1 or

"

"!1, then there are no closed orbits and then no two-period quasiperiodic

motion for the LieHnard oscillator (4);

(ii) if"1,

"!1, there are no closed orbits lying entirely in the region (1 or in the region '1 and

moreover at least a stable closed orbit exists in the annulus

(1!""(((1#"", (58)

if 0(""(1 and condition (54) is veri"ed in annulus (58). For 0(""(

the closed orbit is unique.



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Fig. 8. Phase space diagram (X" cos , >" sin ) of the limit cycle of the model system corresponding to the motion of Fig. 7.

(iii) if"!1,

"1 then there are no closed orbits lying entirely in the region (1 or in the region

'1 and moreover at least an unstable closed orbit exists in annulus (58) if 0(""(1 and condition(54). For 0(""(

the closed orbit is unique.

In Fig. 7 we show a two-period quasi-periodic motion for the LieHnard system (4) with the coe$cients

f"1.0, f

"0.56, g

"0.02, g

"1.0, g

"!1.0, "0.02, f"0.01.

The corresponding limit cycle for the model system (20), (21) is shown in Fig. 8. Note that the limit cycle is

in annulus (58) which through the transformation (23), (24) correspond to 0.071((0.122.

4. In5nite-period symmetric homoclinic bifurcation

In Figs. 9}13 we show the occurrence of an in"nite-period homoclinic bifurcation. We keep the following

set of parameters:

f"1.0, f

"2.56, g

"!0.03, g

"!2.0, g

"1.0, "0.02 (59)

and use f("2) as control parameter.



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Fig. 9. Phase space diagram (X" cos , >" sin ) for the limit cycle of the model system (20), (21) corresponding tof"0.040,

"0.01,

"0.02,

"!1.5,

"1.5.

The parameters of the model system (20), (21) are

"!0.01,

"!0.02,

"1.5,

"!1.5. (60)

We observe the following scenario:

(i) for !0.045(f(0 the origin is a stable node and an unstable limit cycle exists for the model

system (20), (21); the existence of the unstable limit cycle can be expected for very low value of the

parametric excitation by the PoincareH }Bendixson theorem or numerically checked if we change the

signs of

,

,

,

, in the model system (20), (21), because in this case the limit cycle becomes stable(Fig. 9);

(ii) for f"!0.045 the origin loses its stability, becomes a saddle point through a supercritical pitchfork

bifurcation and two stable nodes arise; after this point, the asymptotic motion is governed by the

steady-state response given by the non-trivial "xed points (26) of the system (20), (21);(iii) forf"!0.056 the two equilibrium points become attractive spirals (foci) (Fig. 10a), while the asymptotic

motion corresponds to a periodic behavior for the LieHnard oscillator (Fig. 10b).

(iv) for f"!0.059 the two equilibrium points become repulsive spirals and through a secondary Hopf

bifurcation two limit cycles appear (Fig. 11a); the frequency is easily deduced from the linearization of the

model system (20), (21) around the equilibrium points (

,

) furnished by Eqs. (26a) and (26b) and is

&"2

(

#(

!

)0.027, (61)

while for the LieH nard oscillator we observe a two-period quasi-periodic motion with the carrier

frequency 1 and the modulation frequency (61) (Fig. 11b);



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Fig. 10. (a) Phase space diagram (X" cos , >" sin ) for an attractive focus of the model system (20), (21) corresponding tof"!0.056, "!0.01,

"!0.02,

"1.5,

"!1.5. It is shown the presence of the other attractive focus. (b) A periodic

motion for the LieHnard oscillator corresponding to the motion of Fig. 10a.



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Fig. 11. (a) Phase space diagram (X" cos , >" sin ) for a limit cycle arising from a secondary Hopf bifurcation of the modelsystem (20), (21) corresponding to f"!0.059,

"!0.01,

"!0.02,

"1.5,

"!1.5. (b) A two-period quasi-periodic

motion for the LieHnard oscillator corresponding to the motion of Fig. 11a.



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Fig. 12 (a,b).

Fig. 12. (a) Phase space diagram (X" cos , >" sin ) for the in"nite-period homoclinic bifurcation of the model system (20), (21)corresponding to f"!0.060,

"!0.01,

"!0.02,

"1.5,

"!1.5. (b) A two-period quasi-periodic motion for the LieHnard

oscillator corresponding to the motion of Fig. 12a. (c) (opposite) Fourier spectrum of the motion of Fig. 12b. No windows were used in

computing the spectrum. Three hundred and ten lines of resolution in a 3.2 Hz baseband were used. The spectrum contains sidebands

uniformly spaced around the carrier frequencies 1, 2 and 3 owing to the presence of the modulation frequency (61).

(d) (opposite) Fourier spectrum of the motion of Fig. 12b. Three hundred and ten lines of resolution in a 0.32 Hz baseband were used.

No windows were used in computing the spectrum.



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Fig. 12 (c,d). Caption on facing page.



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Fig. 13. (a) Phase space diagram (X" cos , >" sin ) for the limit cycle of the model system (20), (21) corresponding tof"!0.061,

"!0.01,

"!0.02,

"1.5,

"!1.5. (b) A two-period quasi-periodic motion for the LieHnard oscillator

corresponding to the motion of Fig. 13a.



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(v) if fcontinues to decrease then an in"nite-period homoclinic bifurcation occurs, because the modulation

period lengthens as the two limit cycles approach the saddle point and for f"!0.060 the two limit

cycles touch the saddle point and their periods become in"nite (Fig. 12a). Note the di!erence with the

usual homoclinic bifurcation, where a limit cycle touches the saddle point and then disappears. In

amplitude spectra (Fig. 12c and Fig. 12d) of the time history of the modulated motion (Fig. 12b), there is

a remarkable resemblance between the torus doubling route to chaos and the homoclinic bifurcation due

to the presence of the lower frequencies. In both cases the bifurcation point corresponds to the excitationof the in"nite period frequency.

(vi) for f"!0.061 a stable limit cycle surrounding the saddle point exists (Fig. 13a) and corresponds to

a modulated motion for the LieHnard oscillator (4) (Fig. 13b).

(vii) for f"!0.062 the stable and the unstable limit cycle coalesce and annihilate through a saddle-node

bifurcation or blue sky catastrophe due to the sudden disappearance of the attractor from the state space

of the system.

5. Conclusion and directions for future work

We have investigated a parametrically excited LieHnard system with the restoring force function

and the coe$cient of the velocity term expanded, respectively, up to third and second order in the

displacement.

Using an asymptotic perturbation method based on Fourier expansion and time rescaling, we have

derived a model system formed by two coupled equations for the amplitude and the phase of solutions. Their

"xed points correspond to limit cycles for the LieHnard system and we have determined stability of

steady-state response as well as response-parametric excitation and response-frequency curves.

For very low values of the parametric excitation we have proved the existence of a quasi-periodic motion,

characterized by the combination of the natural frequency with a low frequency connected to the parametric

excitation. Comparison with the solution obtained by the numerical integration con"rms the validity of our

analysis.We have used the PoincareH }Bendixson theorem, the Dulac's criterion and energy considerations to study

existence and characteristics of limit cycles of the model system. A limit cycle corresponds to a modulated

motion in the LieHnard system.

Moreover, an unusual in"nite-period homoclinic bifurcation is possible owing to the symmetry of the

model system (20), (21). Two stable limit cycles approach a saddle point and their periods diverge. At the

bifurcation point the two limit cycles merge and give rise to a greater stable limit cycle surrounding the saddle

point. Finally, the limit cycle and another unstable limit cycle annihilate through a saddle-node bifurcation.

We now illustrate some possible developments of our research line:

(i) application of the AP method beyond its leading order and derivation of the second-order approximate

solution;

(ii) application of the AP method to the model system (20), (21) in order to"nd secondary Hopf bifurcations;(iii) analysis of subharmonics and superharmonics resonances for the LieHnard system (4).

References

[1] D.W. Jordan, P. Smith, Nonlinear Ordinary Di!erential Equations, second ed., Oxford University Press, Oxford, England, 1987.

[2] L. Perko, Di!erential Equations and Dynamical Systems, Springer, New York, 1991.

[3] R. Grimshaw, Nonlinear Ordinary Di!erential Equations, Blackwell, Oxford, England, 1990.



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[4] A. Maccari, Approximate solution of a class of nonlinear oscillators in resonance with a periodic excitation, Nonlinear Dyn. 15

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parametrically excited liehnard systems

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