Mathematical Problems in Engineering Volume 2010, Article ID 760479,24pages doi:10.1155/2010/760479

*Research Article*

**Limiting Phase Trajectories and Resonance Energy** **Transfer in a System of Two Coupled Oscillators**

**L. I. Manevitch,**

^{1}**A. S. Kovaleva,**

^{2}**and E. L. Manevitch**

^{3}*1**N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, 4, Kosygina street,*
*Moscow 119991, Russia*

*2**Space Research Institute, Russian Academy of Sciences, Moscow 117997, Russia*

*3**Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign,*
*Champaign, IL 61820, USA*

Correspondence should be addressed to L. I. Manevitch,lmanev@chph.ras.ru Received 30 July 2009; Accepted 6 November 2009

Academic Editor: Jos´e Balthazar

Copyrightq2010 L. I. Manevitch et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectoriesLPTs. The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duﬃng oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.

**1. Introduction**

The problem of passive irreversible transfer of mechanical energy referred to as energy pumping in oscillatory systems has been studied intensively over last decades; see, for example, 1, 2 for recent advances and references. In this case, the key role of transient process is evident, in contrast to great majority of conventional problems of nonlinear dynamics, in which the main attention has been given to nonlinear normal modesNNMs, characterized by the conservation of energy. Recent studies3,4; have shown that the NNM approach is eﬀective in the case of weak energy exchange, while the concept of the limiting phase trajectories LPTcan be used to describe intense energy exchange between weakly coupled oscillators or oscillatory chains.

The limiting phase trajectory LPT has been introduced 3 as a trajectory corresponding to oscillations with the most intensive energy exchange between weakly coupled oscillators or an oscillator and a source of energy; the transition from energy exchange to energy localization at one of the oscillators is associated with the disappearance of the LPT. Recently, the LPT ideas have been applied to the analysis of resonance-forced vibrations in a 1DOF and 2DOF nondissipative system3–6. An explicit expression of the LPT in a single oscillator5,6is prohibitively diﬃcult for practical utility. The purpose of the present paper is to show that explicit asymptotic solutions can be obtained for a class of problems that are associated with the maximum energy exchange, and introduce relevant techniques.

The paper is organized as follows. The first part is concerned with the construction of the LPT for the Duﬃng oscillator subject to 1 : 1 resonance harmonic excitation. In Section 2 we briefly reproduce the results of 5, 6. We derive the averaged equations determining the slowly varying envelope and the phase of the nonstationary motion and then construct the LPT for diﬀerent types of motion. Section 3 introduces the nonsmooth temporal transformations as an eﬀective tool of the nonlinear analysis.Section 4 examines the transformations of the LPT into stationary motion in the weakly dissipated system.

In the remainder of the paper Section 5, we analyse the dynamics of a 2DOF
system. The system consists of a linear oscillator of mass*M*the source of energycoupled
with a mass *m* an energy sink by a nonlinear spring with a weak linear component.

Excitation is due to an initial impulse acting upon the mass*M. It is shown that motion of*
the overall structure can be divided into two stages. The first stage is associated with the
maximum energy exchange between the oscillators; here, motion is close to beating in a
nondissipative system. At the second stage, trajectories of both masses in the damped system
are approaching to rest. The task is to construct an explicit asymptotic solution describing
both stages of motion for each oscillator. To this end, we reduce the equations of a 2DOF
system to an equation of a single oscillator and then find beating oscillations characterizing
the most intense energy exchange in a nondissipative system. A special attention is given to a
diﬀerence between the dynamics of the Duﬃng oscillator and an equivalent oscillator in the
presence of dissipation. While the Duﬃng oscillator is subjected to harmonic excitation, the
2DOF system is excited by an initial impulse applied to the linear oscillator. The response of
the linear oscillator, exponentially vanishing at*t* → ∞, stands for an external excitation for
the nonlinear energy sink. We examine the transformation of beating to damped oscillations.

We show that a solution of the system, linearized near the rest state, is suﬃcient to describe the second part of the trajectory.

**2. Resonance Oscillations of the Duffing Oscillator**

**2.1. Main Equations**We investigate the transient response of the Duﬃng oscillator in the presence of resonance 1:1. The dimensionless equation of motion is

*d*^{2}*u*

*dt*^{2} 2εγ*du*

*dt* *u*8αεu^{3} 2εFsin1*εst,* 2.1

where*ε >*0 is a small parameter. We recall that maximum energy pumping from the source
of excitation into the oscillator takes place if the oscillator is initially at rest; this corresponds
to the initial conditions

*t* 0^{}*,* *u* 0, *du*

*dt* 0. 2.2

An orbit satisfying conditions2.2*is said to be the limiting phase trajectory*3.

In order to describe the nonlinear dynamics, we invoke a complex-valued transforma-
tion7–9. Introduce the variables*ψ*and*ψ*^{∗}, such that

*u* 1

2i

*ψ*−*ψ*^{∗}

*,* *v* 1

2

*ψψ*^{∗}
*,*
*ψ* *viu,* *ψ*^{∗} *v*−*iu,*

2.3

where *i* √

−1; the asterisk denotes complex conjugate. It will be shown that only one
complex function is suﬃcient for a complete description of the dynamics. Inserting *ψ,* *ψ*^{∗}
from2.3into2.1, a little algebra shows that2.1is equivalent to the followingstill exact
equation of motion

*dψ*

*dt* −*iψεiα*

*ψ*−*ψ*^{∗}_{3}
*εγ*

*ψψ*^{∗}

−2εFsin1*εst* 0. 2.4

Applying the multiple scales method10,11, we construct an approximate solution of2.4as an expansion

*ψt, ε ψ*_{0}τ0*, τ*_{1} *εψ*_{1}τ0*, τ*_{1} · · ·*,*
*d*

*dt*

*∂*

*∂τ*_{0} *ε* *∂*

*∂τ*_{1}*,* *d*^{2}
*dt*^{2}

*∂*^{2}

*∂τ*_{0}^{2} 2ε *∂*^{2}

*∂τ*_{0}*∂τ*_{1} · · ·*,* 2.5
where *τ*_{0} *t* and *τ*_{1} *εt* are the fast and slow time-scales, respectively. A similar
representation is valid for the function*ψ*^{∗}. Then we substitute expressions2.5in2.4and
equate the coeﬃcients of like powers of*ε. In the leading order approximation, we obtain*

*∂ψ*0

*∂τ*_{0} −*iψ*0 0,
*ψ*_{0}τ0*, τ*_{1} *ϕ*_{0}τ1e^{iτ}^{0}

2.6

A slow function *ϕ*0τ1 will be found at the next level of approximation. Equating the
coeﬃcients of order*ε*leads to

*∂ψ*1

*∂τ*_{0} *dϕ*0

*dτ*_{1}*e*^{iτ}^{0}− *iψ*1*iα*

*ψ*0−*ψ*_{0}^{∗}_{3}
*γ*

*ψ*0*ψ*_{0}^{∗}
*iF*

*e*^{iτ}^{0}^{sτ}^{1}^{}−*e*^{−iτ}^{0}^{sτ}^{1}^{}

0, 2.7

In order to avoid the secular growth of*ψ*1τ0*, τ*1in*τ*0, that is, avoid a response not
uniformly valid with increasing time, we eliminate resonance terms from2.7. This yields
the following equation for*ϕ*_{0}:

*dϕ*0

*dτ*_{1} *γϕ*0−3iα*ϕ*0^{2}*ϕ*0 −iFe^{isτ}^{1}*,* *ϕ*00 0. 2.8
Next we introduce the polar representation

*ϕ*_{0} *ae*^{iδ}*,* 2.9

*where a and* *δ* represent a real amplitude and a real phase of the process*ϕ*0τ1. Inserting
2.9into2.8and setting separately the real and imaginary parts of the resulting equations
equal to zero,2.8is transformed into the system

*da*

*dτ*_{1} *γa* −*F*sinΔ,
*adΔ*

*dτ*_{1} −*sa*3αa^{3}−*F*cosΔ,

2.10

*where a>*0,Δ *δ*−sτ1. It now follows from2.3,2.9that

*ut, ε aτ*1sint Δ*τ*_{1} *Oε.* 2.11

This means that the amplitude*aτ*1and the phaseΔτ1completely determine the process
*ut,ε in the leading-order approximation. Note that a* 0 if the oscillator is not excited.

**2.2. Critical Parameters and LPTs of the Undamped Oscillator**

In this section, we recall main definitions and results concerning the dynamics of the nondissipative system. In the absence of damping, system2.10is rewritten as

*da*

*dτ*_{1} −*F*sinΔ,
*adΔ*

*dτ*_{1} −*sa*3αa^{3}−*F*cosΔ.

2.12

It is easy to prove that system2.12conserves the integral of motion

*H* 3

4*αa*^{4}− *sa*^{2}

2 −*Fa*cosΔ *H*_{0}*,* 2.13

where *H*0 depends on initial conditions. In the phase plane, the LPT corresponds to the
contour*H* 0, as only in this case a system trajectory goes through the point*a* 0. Taking
*H*_{0} 0, we obtain the following expression:

*H* *a*

3
4*αa*^{3}−*s*

2*a*−*F*cosΔ

0. 2.14

Formula 2.14*implies that the LPT has two branches, the first branch is a* 0; the
second branch satisfies the cubic equation

3
4*αa*^{3}−*s*

2*a*−*F*cosΔ 0. 2.15

Equality2.15determines the second initial condition*a0*^{} 0, cosΔ0^{} 0. We
suppose that da/dτ1 *>*0 at*τ*_{1} 0^{}; under this assumption,Δ0^{} −π/2. Hence the initial
conditions for the LPT take the form

*τ*_{1} 0^{}*, a0*^{} 0,Δ0^{} −*π*

2*.* 2.16

Throughout this paper, we write 0 instead of 0^{}, except as otherwise noted.

Next we determine critical parameters of system 2.12. The steady states of 2.12 satisfies the equations

*da*

*dτ*1 0, *dΔ*

*dτ*1 0. 2.17

The second equation is equivalent to the equality

−sa3αa^{3}−*F*sgncosΔ 0, 2.18

where cosΔ ±1. We analyze the properties of 2.18 considering the properties of its
discriminant*D*_{1}

*D*1 1
9α^{2}

*F*^{2}
4 − *s*^{3}

81α

*.* 2.19

If *D*1 *<* 0, 2.18 has 3 diﬀerent real roots; if *D*1 *>* 0, 2.18 has a single real and two
complex conjugate roots; if*D*1 0, two real roots will merge; see, for example, 12. The
latter condition gives the first critical value of the parameter*α*

*α*^{∗}_{1} 4s^{3}

81F^{2}*.* 2.20

A straightforward investigation proves that, if*α > α*^{∗}_{1}strong nonlinearity, there exists
only a single stable centre*C*_{}:0,*a*_{} Figure 1; if*α < α*^{∗}_{1}weak nonlinearity, there exist two

0.5
1
1.5
*a*

0 *π/2* *π* Δ

−π/2

−π

a

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

*a*

0 50

*τ*_{1}

100 150 200

b

**Figure 1: Phase portrait**a*and plot of aτ*1 b*for s* *0.4, F* 0.13,*α* 0.187 *α*^{∗}_{1}.

stable centres*C*_{−}:−π,*a*_{−}, C:0,*a*_{}, and an intermediate unstable hyperbolic point O:−π,
*a*0, see Figures2and3.

We now suppose that*α < α*^{∗}_{1}, that is the system may exhibit both types of oscillations,
either nearΔ −π, or nearΔ 0 Figures2–4. In both cases, the LPT begins at a 0,Δ

−π/2 but the run of the LPT depends on the relationship between the parameters. In order
to find a critical value*α*^{∗}_{2} *< α*^{∗}_{1} ensuring the transition from small to large oscillations, we
analyze2.15. Consider the discriminant of2.15

*D*2 4

*α*^{2} *F*^{2}− 2s^{3}
81α

2.21

In the critical case*α* *α*^{∗}_{2}, an unstable hyperbolic point coincides with the maximum of the
left branch of the LPT atΔ −πFigure 2. This means that*D*2 0 at*α*^{∗}_{2}, or

*α*^{∗}_{2} 2s^{3}
81F^{2}

*α*^{∗}_{1}

2 *,* 2.22

which defines a boundary between small quasilinearα < α^{∗}_{2}and large nonlinearα > α^{∗}_{2}
*oscillations. In particular, for s* *0.4, F* 0.13 we obtain*α*^{∗}_{2} 0.0935Figure 2.

Figures3 and 4 are plotted for*α < α*^{∗}_{2} and*α > α*^{∗}_{2}, respectively. InFigure 3a, one
can see the LPT encircling the center*C*_{−}of relatively small oscillations;Figure 4ashows the
LPT encircling the centre*C*_{} of large oscillations; this case is associated with the maximum
energy absorption. Figures 3b and 4b *demonstrate the behavior of the function aτ*1
corresponding to the respective branch of the LPT. Note that both branches of the LPT begin
*at the same point a* 0,Δ −π/2.

If*α* *α*^{∗}_{1} 2α^{∗}_{2}, the above-mentioned coincidence of the stable and unstable points at
Δ −πresults in the transformation of the phase portraitsFigure 1and disappearance of
*the stable centre C*_{−}.Figure 1ademonstrates a single stable fixed point atΔ 0.

1
2
*a*

0 *π/2* *π*

−*π/2* Δ

−*π*

**Figure 2: Passage from small to large oscillations:***α* 0.0935 *α*^{∗}_{2}.

1
2
*a*

0 *π/2* *π* Δ

−*π/2*

−*π*

a

0 0.2 0.4 0.6 0.8 1

*a*

0 50 100 150 200

*τ*1

b

**Figure 3: Phase portrait**a*and plot of aτ*1 b*for quasilinear oscillations: s* *0.4, F* 0.13,*α* 0.093*< α*^{∗}_{2}.

1
2
*a*

0 *π/2* *π* Δ

−*π/2*

−*π*

a

0 0.5 1 1.5 2

*a*

0 50 100 150 200

*τ*1

b

**Figure 4: Phase portrait**a*and plot of aτ*1 b*for strongly nonlinear oscillations: s* *0.4, F* 0.13,*α*^{∗}_{2}*< α*
0.094*< α*^{∗}_{1}.

0 0.2 0.4 Ψ 0.6

0.8 1 1.2

×10^{−3}

0.25 0.5 0.75 1

*a*
a

0 0.5 Ψ 1

1.5 2

0 0.5 1 1.5 2

*a*
b
**Figure 5: Potential**Ψ*a*in the smallaand largebscales.

**2.3. Reduction to the 2nd-Order Equation**

For the further analysis, it is convenient to reduce the equation of the LPT to the second-order form. Using2.15to excludeΔ, we obtain the following equation:

*d*^{2}*a*

*dτ*_{1}^{2} *f*a 0,

*a* 0, *da*

*dτ*_{1} *F* at*τ*1 0,

2.23

where

*fa * *a*
4

3
2*αa*^{2}−*s*

9
2*αa*^{2}−*s*

*d*Ψ*a*
*da* *,*
Ψ*a * *a*^{2}

8 3

2*αa*^{2}−*s*
_{2}

*.*

2.24

Note that2.23can be treated as the equation of a conservative oscillator with potentialΨa, yielding the integral of energy

*E* 1

2*v*^{2} Ψ*a * 1

2*F*^{2}*,* *v* *da*

*dτ*_{1}*.* 2.25

Figure 5depicts the potentialΨain the small and large scales for system3.1with
the parameters*α* *1/3, s* 1/4. The phase portrait is given inFigure 6.

The amplitude of oscillations*A*0is defined by the following equalityseeFigure 6b:

*E* Ψ*A*_{0} 1

2*F*^{2}*.* 2.26

−0.1

−0.05
0
*ν*

0.05 0.1 0.15

0 0.2 0.4 0.6 0.8 1

*a*
a

−2

−1.5

−0.5 0.5

−1
0
*ν*

1 1.5 2

0 0.5 1 1.5 *A*02

*a*
b

**Figure 6: Phase portraits of**2.23in the smallaand largebscales.

From2.25we have*v* *da/dτ*1 ±

*F*^{2}−2Ψa. Thus, the half-period of oscillations
is defined as

*T*_{1}A0
_{A}_{0}

0

*da*

*F*^{2}−2Ψa*.* 2.27

It follows from Figures1and6 that a high-energy system is weakly sensitive to the shape of the potential; the orbit is close to the straight line until it reaches the wall of the potential well. This implies that motion of system 2.23 is similar to the dynamics of a particle moving with constant velocity between two motion-limiters. A connection between the smooth and vibro-impact modes of motion in a smooth nonlinear oscillator has been revealed in13,14; a detailed exposition of this approach can be found in2, Chapter 6.

The vibro-impact hypothesis suggests that the time*T*_{1}^{∗}to reach*A*_{0}is calculated as

*T*_{1}^{∗} *A*0

*F* *.* 2.28

Note that *T*_{1}^{∗} *< T*_{1}, as the vibro-impact approximation ignores the deceleration of
motion in the vicinity of*A*0, when the velocity falls below the maximum level*F. Formally,T*_{1}^{∗}
can be found from2.27by lettingΨa 0.

**3. Analysis of the Transient Dynamics**

In what follows, we consider the dynamics of a weakly damped oscillator with strong
nonlinearityα > α^{∗}; it is often the case of particular interest.

As seen inFigure 7, the damped system exhibits strongly nonlinear behavior on the
time interval0, T_{1}^{∗}; an instant*T*_{1}^{∗} *corresponds to the first maximum of the function aτ*1.
After that motion becomes similar to smooth oscillations about the stationary point. This
allows separating the transient dynamics into two stages. While on the interval 0≤ *τ*_{1} ≤*T*_{1}^{∗}

0 1 2 3 Δ

−1

−2

−3

1
2
3
*a*

a

0.5
1
1.5
*a*

0 50 100 150

*τ*_{1}
b

**Figure 7: Phase portrait**a*and plot of aτ*1 bfor strongly nonlinear oscillations in a dissipative system.

motion is close to the LPT of the undamped system, at the second stage,*τ*1 ≥ *T*_{1}^{∗}, motion is
similar to quasilinear oscillations.

In the remainder of this section we investigate a segment of the trajectory on the
interval 0, *T*_{1}^{∗}. The task is to calculate an instant *T*_{1}^{∗} *and the values aT*_{1}^{∗} and ΔT_{1}^{∗}
determining the starting point for the second interval of motion. For simplicity, we assume
that dissipation is suﬃciently small and may be ignored on the interval0,*T*_{1}^{∗}. This allows
one to approximate the first part of the trajectory by a corresponding segment of the LPT of
the nondissipative system.

As mentioned above, the dynamics of a strongly nonlinear oscillator is similar to free motion of a particle moving with constant velocity between two motion-limiters. This allows us to employ the method of nonsmooth transformations1,2in the study of strongly nonlinear oscillations.

At the first step, we introduce nonsmooth functions*τ*φ*and eφ dτ/dφ*defined as
follows:

*τ* *τ*
*φ* 2

*π*
arcsin

sin*πφ*

2
*,*
*e*

*φ*

1, 0*< φ*≤1, *e*
*φ*

−1, 1*< φ*≤2,

3.1

where*φ* Ωτ1, the frequencyΩwill be found below. Plots of functions 3.1are given in
Figure 8. In a general setting, the solution of2.12is constructed in the form

*aτ*1 *X*_{1}τ *e*
*φ*

*Y*_{1}τ, Δ*τ*_{1} *X*_{2}τ *e*
*φ*

*Y*_{2}τ,
*d*

*dτ*_{1} Ω

*e* *∂*

*∂τ* *∂*

*∂φ*

*.* 3.2

We recall that*∂e/∂φ* *δφ*−*n, whereδφ*−*nis Dirac’s delta-function, n* 1, 2, . . . .
We exclude*δ-singularity by requiring*

*Y*1,2 0 at*τ* 1,2, . . . . 3.3

−4 −3 −2 −1 0 1 2 3 4 *φ*
1

*τ*

a

−4 −3 −2 −1 0 1 2 3 4 *φ*

1

−1
*e*

b

**Figure 8: Functions***τ*φ*and eφ.*

This implies that

*da*
*dτ*_{1} Ω

*e∂X*_{1}

*∂τ* *∂Y*_{1}

*∂τ*

*,*
*dΔ*

*dτ*_{1} Ω

*e∂X*2

*∂τ* *∂Y*2

*∂τ*

3.4

provided*Y*_{1,2} 0 at*τ* 1, 2,. . . .To derive the equations for*X** _{i}*,

*Y*

_{i}*, i*1, 2, we insert3.4into 2.12

*and separate the terms with and without e. This yields the set of equations*

Ω*∂Y*_{1}

*∂τ* −Fsin*X*_{2}cos*Y*_{2}*,*
Ω*∂X*1

*∂τ* −Fsin*Y*_{2}cos*X*_{2}*,*
Ω

*X*_{1}*∂Y*2

*∂τ* *Y*_{1}*∂X*2

*∂τ*

*sX*_{1}−3αX_{1}^{3}−9αX_{1}*Y*_{1}^{2} −Fcos*Y*_{2}cos*X*_{2}*,*
Ω

*X*1*∂X*2

*∂τ* *Y*1*∂Y*2

*∂τ*

*F*sin*Y*2sin*X*2*.*

3.5

It is easy to prove that3.5are satisfied by*Y*_{1} 0,*X*_{2} 0. Under these conditions, the
variables*X*_{1}and*Y*_{2}satisfy the equations similar to2.12

Ω*∂X*1

*∂τ* *F*sin*Y*_{2} 0,
ΩX1*∂Y*2

*∂τ* *sX*1−3αX_{1}^{3}*F*cos*Y*2 0,

3.6

with the initial conditions*X*1 0,*Y*2 −π/2 at*τ* 0. System3.6is integrable, yielding the
integral of motion similar to2.14

3

4*αX*_{1}^{3}− *s*

2*X*_{1}−*F*cos*Y*_{2} 0. 3.7

Then, it follows from3.3and3.6that

*∂X*1

*∂τ* 0 at*Y*_{2} 0, τ 1. 3.8

It is worth noting that equalities3.8*have a clear physical meaning: they represent the condition*
*of maximum of* *X*1 *atY*2 0.

For the further analysis, it is convenient to transfer3.6into the second-order form.

Using3.7to exclude*Y*_{2}, the resulting equation and the initial conditions are written as
Ω2*∂*^{2}*X*1

*∂τ*^{2} *fX*1 0,
*X*_{1} 0, Ω*dX*_{1}

*dτ* *F,*

3.9

*where f*X1is defined by 2.24. A precise solution of3.9, expressed in terms of elliptic
functions, is prohibitively diﬃcult for practical utility 6, 13. In order to highlight the
substantial dynamical features, the solution to3.6,3.9is expressed in terms of successive
approximations

*X*_{1} *x*_{0}*x*_{1}· · ·*,* *Y*_{2} *y*_{0}*y*_{1}· · ·*,* Ω Ω01*ε*_{1}· · ·, 3.10
where it is assumed that|x1τ| |x0τ|,|y1τ| |y0τ|, ε1 1 on an interval of interest.

The validity of this assumption will be tested below by numerical simulations. Since the
vibro-impact approximation is insensitive to the presence of the potential, the function*x*_{0}
is chosen as the solution of the equation

*∂*^{2}*x*0

*∂τ*^{2} 0 3.11

*with the initial conditions x*_{0} 0,Ω0*∂x*_{0}/∂τ *F atτ* 0. It follows from3.11that

*x*0τ *A*0*τ,* *A*0Ω0 *F.* 3.12

From3.1and the maximum condition we have

Ω0*T*_{1}^{∗} 1, *T*_{1}^{∗} Ω^{−1}_{0} *A*_{0}
*F* *,*
*x*0τ *Fτ*1*,* 0≤*τ*1≤*T*_{1}^{∗}*, x*0

*τ*
*T*_{1}^{∗}

*x*01 *A*0*.*

3.13

We now recall that system3.9is conservative; it possesses the integral of energy

*E*_{1} 1
2*V*_{1}^{2} 1

Ω^{2}Ψ*X*_{1} 1

2Ω^{2}*F*^{2}*,* *V*_{1} *dX*1

*dτ* *,* 3.14

whereΨX1is defined by2.24. By analogy with2.26, we obtain

Ψ*X*1 1

2*F*^{2} at*τ* 1. 3.15

Inserting3.10,3.12into3.15and ignoring small terms, we then have the equation
to determine*A*_{0}

Ψ*A*_{0} 1

2*F*^{2}*.* 3.16

Given*A*_{0}, we obtain from*T*_{1}^{∗} *A*_{0}*/F*see2.28.

*The approximation x*1is governed by the following equation:

Ω^{2}_{0}*∂*^{2}*x*1

*∂τ*^{2} −fx0,
*x*1τ −Ω^{−2}_{0}

_{τ}

0

τ−*ξfA*0*ξdξ,*

3.17

Given*A*_{0} andΩ0, formula3.17yields

*x*_{1}τ −*A*_{0}*τ*^{3}
4Ω^{2}_{0}

9α^{2}

56 A0*τ*^{4}− 3sα

5 A0*τ*^{2}*s*^{2}
6

*.* 3.18

We now find the function*Y*_{2}τ. Arguing as above, we construct*Y*_{2} *y*_{0}*y*_{1}, where
*y*_{0}can be found from the first equation3.6, in which we let*X*_{1} *x*_{0}. This yields

*y*_{0} −arcsin
*F*

*A*_{0}Ω0

−*π*

2*,* 0*< τ <*1. 3.19
The term*y*1τis defined by the second equation3.6. As before, we take*X*1 *x*0and
exclude cosY2by3.7to get

*∂y*1

*∂τ*
1
Ω0

−*s*
2 9

4*αA*^{2}_{0}*τ*^{2}

*,*
*y*_{1}τ, t0 1

Ω0

−*sτ*
2 3

4*αA*^{2}_{0}*τ*^{3}

*.*

3.20

0 0.5 1 1.5 2 2.5 3 3.5 4
*τ*1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

*a*
*x*_{0}
*x*1

a

0.5 1 1.5 2 2.5

*τ*1

0
0.2
0.4
0.6
0.8
1
*a*

1.2 1.4 1.6 1.8

b

**Figure 9: LPT of system**2.12 a*and solution aτ*1of system2.10 bsolid line—numerical solution;

dashed line—the leading-order approximation*x*0; dot-and-dash line—the first order approximation*x*0
*x*1.

As an example, we calculate the LPT for system2.12with the parameters

*γ* 0, *s* 0.2, *α* 0.333, *F* 1 3.21

and compare the results with the numerical solution for system2.10, in which*γ* 0.04.

Calculations by formulas3.13,3.16give*A*0 1.67,*T*_{1}^{∗} Ω^{−1}_{0} 167. Thus we have
*the maximum M* *X*_{1}T_{1}^{∗}≈1.67 at*T*_{1}^{∗} ≈1.67 in the leading-order approximation and*M*_{1} ≈
1.67 at*T*_{1}^{∗} ≈*2 for the numerical solution aτ*1 Figure 9a; for system2.10with*γ* 0.05
*the numerical solution gives the maximum M* ≈1.56 at*T*_{1}^{∗} ≈ 2.1. This confirms that small
dissipation can be ignored over the interval 0≤*τ* ≤*T*^{∗}.

*It is easy to check by a straightforward calculation that the correction x*1is negligible.

In a similar way, one can evaluate the small term*y*1.

**4. Quasilinear Oscillations**

In this section, we examine quasilinear oscillations on the second interval of motion,*τ >*1. It
is easy to see that an orbit of the dissipative system tends to its steady state as*τ* → ∞. The
*steady state O:*a0,Δ0for system2.10is determined by the equality

*a*^{2}

*s*−3αa^{2}_{2}
*γ*^{2}

*F*^{2}*,* 4.1

or, for suﬃciently small*γ,*

*γa*_{0} −FsinΔ0*,* *sa*_{0}−3αa^{3}_{0} −FcosΔ0*,*
Δ0≈ −*γa*_{0}

*F* *O*
*γ*^{3}

*,* *a*_{0}

*s*−3αa^{2}_{0}

−F*O*
*γ*^{2}

*.* 4.2

Let*ξ* *a*−*a*0*, β* Δ−Δ0 denote deviations from the steady state. In addition, we
must impose the matching conditions

*a*_{0}*ξ* *x*^{∗}_{0}*,* *dξ*

*dτ*_{1} 0 at*τ*_{1} *T*_{1}^{∗}*,* 4.3

where*x*^{∗}_{0} *x*_{0}T_{1}^{∗},*T*_{1}^{∗}is determined by2.28.

We suppose that the contribution of nonlinear force in oscillations near*O*is relatively
small. Under this assumption, one can consider the system linearized near*O*

*dξ*

*dτ*1 *Fβ* −*γξ,*
*dβ*

*dτ*_{1} − *k*1

*a*_{0}*ξ* −*γβ,*

4.4

where*k*1 9αa^{2}_{0}−*s. Ifk*1*>*0, the solution of system4.4takes the form
*ξz c*_{0}*e*^{−γτ}^{1}^{−T}^{1}^{∗}^{}cos *κ*

*τ*_{1}−*T*_{1}^{∗}

*,* *βz rc*_{0}*e*^{−γτ}^{1}^{−T}^{1}^{∗}^{}sin*κ*

*τ*_{1}−*T*_{1}^{∗}

*,* *τ*_{1}−*T*_{1}^{∗}*>*0,
4.5
where we denote*c*0 *x*^{∗}_{0}− *a*0,*κ*^{2} *Fk*1*/a*0*>*0,*r* *κ/F. In particular, taking the parameters*
3.21we find*x*_{0}^{∗} 1.46,*a*_{0} 1.065,Δ0 0.1,*k*_{1} 3.2, and, therefore,*c*_{0} 0.395,*κ* √

3.

Figure 10demonstrates a good agreement between a numerical solution of2.10with
parameters3.21 solid lineand an approximate solution found by matching the segment
3.12 dot-and-dashwith the solution4.5of the linearized systems dash at the point
*T*_{1}^{∗}. Despite a certain discrepancy in the initial interval of motion, the numerical and analytic
solutions approach closely as*τ*1increases. This implies that a simplified model3.12,3.16
matched with solution4.5suﬃces to describe a complicated near-resonance dynamics.

Arguing as above, one can obtain the solution in case*k*_{1} *<*0. Denoting*k*_{2} *F|k*1|/a0

and assuming*γ* *k, we find a solution similar to*4.5with coshkτ1−*T*_{1}^{∗}and sinhkτ1−
*T*_{1}^{∗}in place of cosκτ1−*T*_{1}^{∗}and sinκτ1−*T*_{1}^{∗}, respectively.

We now correlate numerical and analytic results. As seen in Figure 11, the first
*maximum of the slowly varying envelope of the process ut,ε*equals*M*1 ≈1.5; it is reached
at the instant*t*^{∗}≈20, or*T*_{1}^{∗} ≈2; the second maximum*M*_{2}≈1.4 is at*t*^{∗}≈60,*T*_{1}^{∗} ≈6, the third
maximum*M*_{3} ≈1.3 is at*t*^{∗} ≈100,*T*_{1}^{∗} ≈10, and so forth. When these results are compared
with that ofFigure 10, it is apparent that the numerically constructed envelope is in a good
agreement with the asymptotic approximations of the function*aτ*1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

*a*

0 2 4 6 8 10 12 14 16 18 20

*τ*1

**Figure 10: Transient dynamics of system**2.10: numericssolid; segment3.12 dot-and-dash; solution
4.5 dash.

1.5
1
0.5
0.5
1
1.5
*u*

100 200 300 400*t*

**Figure 11: Numerical integration of**2.1:*ε* *0.1, s* 0.2,*α* *0.333, F* 1,*γ* 0.05.

**5. Dynamics of a 2DOF System**

**5.1. Reduction of a 2DOF System to a Single Oscillator**

In this section we present a reduction of the dynamical equations of a 2DOF system to an
equation of a single oscillator. The system consists of a linear oscillator of mass*M*the source
of energycoupled with a mass*m*an energy sinkby a nonlinear spring with a weak linear
component. For brevity, we consider the nonlinear spring with cubic nonlinearity. In this case,
the equations of motion and the initial conditions have the following form:

*Md*^{2}*x*1

*dt*^{2} *γdx*1

*dt* *k*1*x*1*k*3x1−*x*2^{3}*Dx*1−*x*2 *η*
*dx*1

*dt* −*dx*2

*dt*

0,

*md*^{2}*x*_{2}

*dt*^{2} −*k*_{3}x1−*x*_{2}^{3}−*Dx*1−*x*_{2}−*η*
*dx*_{1}

*dt* −*dx*_{2}
*dt*

0,

5.1

with the initial conditions

*t* 0 : *x*1 *x*2 0 : *dx*1

*dt* *V*0*>*0, *dx*2

*dt* 0. 5.2

Here *x*_{1} and *x*_{2} *are the displacements of the masses M and m, respectively;* *k*_{1} *>* 0
is the stiﬀness of linear spring; *k*_{3} *>* 0 is the coeﬃcient of nonlinear coupling between the
*linear oscillator and the sink; D is the coeﬃcient of linear coupling*D <0 corresponds to a
system with multiple states of equilibrium; the coeﬃcients*γ* and*η*characterize dissipation
in the linear oscillator and the coupling, respectively. For simplicity, we let*γ* 0. Note that
energy transfer cannot be activated in a nonexited system. In the absence of external forcing,
it requires nonzero initial conditions for at least a single unit.

In what follows we assume that*m/M* *ε*^{2}1. Then, we introduce the dimensionless
time variable*τ*0 *ω*0*t, whereω*0

*k*1*/M. In these notations, system*5.1becomes

*d*^{2}*x*1

*dτ*_{0}^{2} *x*1*ε*^{2}*cx*1−*x*2^{3}*ε*^{3}*dx*1−*x*2 *ε*^{3}*η*
*dx*1

*dτ*0 −*dx*2

*dτ*0

0,
*d*^{2}*x*2

*dτ*_{0}^{2} −*cx*1−*x*2^{3}−*εdx*1−*x*2−*εη*
*dx*1

*dτ*_{0} − *dx*2

*dτ*_{0}

0,
*τ*_{0} 0 : *x*_{1} *x*_{2} 0; *v*_{1} *εv*_{0}*,* *v*_{2} 0,

5.3

where we denote

*ε*^{2}*c* *k*_{3}
*k*1

*,* *ε*^{3}*d* *D*
*k*1

*,* *ε*^{3}*η* *η*

*k*1*M,* *εv*_{0} *V*_{0}
*ω*0

*,* *vi* *dx*_{i}*dτ*0

*.* 5.4

*In addition, we consider the relative displacement u* *x*_{2}–*x*_{1},*du/dτ*_{0} *v satisfying*
the equation

*d*^{2}*u*
*dτ*_{0}^{2} *d*^{2}*x*1

*dτ*_{0}^{2} *εηdu*

*dτ*0 *cu*^{3}*εdu* 0, 5.5

with the initial conditions*τ*0 *0: u* *0, v* −ε v0. Using5.3,5.5, the variable*x*1 can be
excluded. We recall that oscillations in the damped system vanish at rest*O*1:x1 *x*2 0,*v*1

*v*_{2} 0as*τ*_{0} → ∞. This implies that the eﬀect of dissipation, whatever small it might be,
must be considered in the approximate solution; otherwise, the convergence to*O*_{1} is ignored.

Hence the solution of the first equation5.3should be written as

*x*_{1}τ0 *εv*_{0}*h** _{ε}*τ0sin

*τ*

_{0}−

*ε*

^{2}

*c*

_{τ}_{0}

0

*h** _{ε}*τ0−

*s*sinτ0−

*su*

^{3}sds

*ε*

^{3}

*. . . ,*

*h*

*ε*τ0

*e*

^{−ε}

^{3}

^{η/2τ}^{0}

*,*

5.6

and, by5.6,
*d*^{2}*x*1

*dτ*_{0}^{2} −*εv*0*h**ε*τ0sin*τ*0−*ε*^{2}*cu*^{3}τ0 *ε*^{2}*c*
_{τ}_{0}

0

*h**ε*τ0−*s*sinτ0−*su*^{3}sds*ε*^{3}· · ·*.* 5.7

Next, we insert 5.7 into 5.5 and ignore small terms insubstantial for the asymptotic analysis. As a result, we obtain the following equation:

*d*^{2}*u*

*dτ*_{0}^{2} *εηdu*

*dτ*_{0} *cu*^{3}*εdu* *εv*_{0}*h** _{ε}*τ0sin

*τ*

_{0}−

*εc*

*τ0,*

_{ε}*c*

*τ0*

_{ε}*εcI*

_{ε}*,*

*I*

_{ε}_{τ}_{0}

0

*h** _{ε}*τ0−

*s*sinτ0−

*su*

^{3}sds,

5.8

*with the initial conditions u0 0, v0 *−ε v0. We will show that, under conditions of 1:1
resonance,*I**ε*τ0 ∼ *ε*^{−1}*, c**ε*τ0∼ 1; this means that the integral term should be taken into
consideration in the asymptotic analysis.

Formula5.8represents a nonhomogeneous integro-diﬀerential equation with respect
*to u, that is above transformations reduce the original 2DOF system to a single oscillator of a*
*more complicated structure. The initial condition v0 *−ε v0implies that initially the system
is close to rest, and the trajectory of system approaches the LPT of5.8. Thus the task is to
construct the LPT for the integro-diﬀerential equation5.8.

**5.2. Equations of the Resonance Dynamics**

To study the system subject to 1:1 resonance, we rewrite5.8in the form
*du*

*dτ*_{0} −*v* 0,
*dv*

*dτ*_{0} 12εσu*εμ*

*cu*^{3}−*u*
*ε*

*ηv*−*v*0*h**ε*τ0sin*τ*0

*εc**ε* τ0 0,

*u0 *0, *v0 *−*εv*_{0}*.*

5.9

In5.9, we denote*μ* 1/ε,*σ* *d/2. The resonance conditions imply that the parenthetical*
expression with factor*εμ*is relatively small compared to all other terms of order 1.

As inSection 2, we use complex-valued transformations2.5and the multiple time- scale method. Inserting2.5into5.9, we then have

*dψ*

*dτ*0 −*i1εσψεμi*
*c*

8

*ψ*−*ψ*^{∗}31
2*ψ*−*ψ*^{∗}

*εη*
2

*ψψ*^{∗}

−*iσψ*^{∗}−*v*_{0}*h** _{ε}*τ0sin

*τ*

_{0}

−*εC** _{ε}*τ0 0,

*C*

*τ0*

_{ε}*εic*

8
_{τ}_{0}

0

*h** _{ε}*τ0−

*s*sinτ0−

*s*

*ψ*−*ψ*^{∗}_{3}
*ds*

5.10

Then we construct an approximate solution of5.10in terms of expansions2.5with the
slow and fast tine scales*τ*_{0} *t,τ*_{1} *εt, respectively.*

For the resonance eﬀect to be considered in a proper way, the leading-order equation and its solution should be

*∂ψ*0

*∂τ*0 −*i1εσψ*0 0,
*ψ*0τ0*, τ*1 *ϕ*0τ1e^{i1εστ}^{0}*.*

5.11

The function*ϕ*_{0}τ1can be found from the equation

*∂ψ*_{1}

*∂τ*_{0} *dϕ*_{0}

*dτ*_{1}*e*^{i1εστ}^{0}−*i1εσψ*1*iμ*
*c*

8

*ψ*_{0}−*ψ*_{0}^{∗}31
2*ψ*_{0}−*ψ*_{0}^{∗}

*η* *ψψ*^{∗}

2 −*iσψ*^{∗}−*v*_{0}*h** _{ε}*τ0sin

*τ*

_{0}

*C*

_{0ε}τ0 0,

5.12

where*C*0ετ0 *εic/8*_{τ}_{0}

0*h**ε*τ0−*s*sinτ0−*sψ*0− *ψ*_{0}^{∗}^{3}*ds.*

To avoid secularity, we separate the resonance terms including*e*^{i1εστ}^{0} and then
equate the sum to zero. First, we evaluate *C*_{0ε}τ0. To do so, we present the cubic term as
ψ0−*ψ*_{0}^{∗}^{3} −3|ϕ0|^{2}*ϕ*0*e*^{i1εστ}^{0}*nonresonance terms, and then writeC*0ετ0in the form

*C*_{0ε}τ0 −ε3c

16*ϕ*_{0}τ1^{2}*ϕ*_{0}τ1
_{τ}_{0}

0

*e*^{−ε}^{3}^{η/2τ}^{0}^{−s}

*e*^{iτ}^{0}^{−s}−*e*^{−iτ}^{0}^{−s}

*e*^{i1εσs}*ds*· · ·

−ε3c

16*ϕ*_{0}τ1^{2}*ϕ*_{0}τ1S1ετ0−*S*_{2ε}τ0 · · · *,*

5.13

where the nonresonance terms are omitted. Here we denote

*S*_{1ε}τ0
_{τ}_{0}

0

*e*^{−ε}^{3}^{η/2τ}^{0}^{−s}*e*^{iτ}^{0}^{−s}*e*^{i1εσs}*ds* *e*^{i1εστ}^{0}
*ε*

*iσε*^{2}*η*

1−*e*^{−iσε}^{3}^{η/2τ}^{0}
*,*

*S*2ετ0
_{τ}_{0}

0

*e*^{−ε}^{3}^{η/2τ}^{0}^{−s}*e*^{−iτ}^{0}^{−s}*e*^{i1εσs}*ds* *e*^{i−ε}^{3}^{η/2τ}^{0}
_{τ}_{0}

0

*e*^{εi2σε}^{2}^{η/2s}*ds*∼*O1.*

5.14

Ignoring*S*_{2}compared with*S*_{1}, we calculate

*C*0ετ0 −i
3c

16σ

*ϕ*0τ1^{2}*ϕ*0τ1

1−*h*1ετ1e^{−iστ}^{1}

*e*^{i1εστ}^{0}nonresonance terms,
5.15
*h*1ετ1 *e*^{−ε}^{2}^{η/2τ}^{1}*.* 5.16

If we sum the resonance constituents in all other terms of5.12and then equate the total sum to zero, we obtain the equation

*∂ϕ*_{0}

*∂τ*_{1} *iμ*

−3c

8 *ϕ*_{0}^{2}*ϕ*_{0}1
2*ϕ*_{0}

*η*

2*ϕ*_{0} *i*

2*v*_{0}*h*_{1ε}τ1e^{−iστ}^{1}

− *i* 3c

16σ *ϕ*_{0}τ1^{2}*ϕ*_{0}τ1

1−*h*_{1ε}τ1e^{−iστ}^{1}
0.

5.17

Then we insert the polar representation*ϕ*_{0} *ae** ^{iδ}*into5.17and set separately the real
and imaginary parts of the resulting equations equal to zero. In these transformations, the
last term in5.17can be omitted if 2μσ

*μd>1. Under this assumption, we obtain*

*da*

*dτ*1 −*γa*−*h*_{1ε}τ1FsinΔ,
*adΔ*

*dτ*1

*μa*

−s*αa*^{2}

−*h*_{1ε}τ1FcosΔ,

5.18

whereΔ *δσ τ*_{1}, and*σ* *d/2,γ* *η/2,α* 3c/8, s 1/21−*εd, F* *v*_{0}*/2.*

By analogy with2.16, we accept the initial conditions
*a0 *0, Δ0 −*π*

2*.* 5.19

In the absence of dissipationη 0, system5.18takes the form
*da*

*dτ*1 −FsinΔ,
*adΔ*

*dτ*1

*μa*

−s*αa*^{2}

−*F*cosΔ,

5.20

which is very similar to2.10. Critical parameters, stable centers, and the LPT for 5.20
are derived in the same way as in Section 3. However, as mentioned above, steady state
positions of dissipated systems2.10and system5.18are diﬀerent. This reflects the fact
that, while2.10is subjected to persistent harmonic excitation, the eﬀect of an initial impulse
exponentially decreases with time. Therefore, the first segment of the trajectory5.18can be
approximated by a corresponding solution of5.20but the second segments convergences
to zero as*τ*_{1} ∞.

**5.3. Dynamical Analysis of the Oscillator***5.3.1. Beating in a Nondissipative System*

We now compare analytic and numerical results. We recall that numerical and experimental
studies 3,4have shown the eﬀective energy exchange for sets of parameters. Following
3,4, we choose*ε* 0.316ε^{2} 0.1, c *0.8, d* 0.632εd 0.2,*v*0 2.215εv0 0.7. Note
that in this case*μd* 2, and integral terms in5.17can be ignored.

0 5 10 15 20 25 30 35 40 45 50
*τ*0

0
0.2
0.4
0.6
0.8
*a* 1

1.2 1.4 1.6 1.8

a

0 10 20 30 40 50 60 70 80 90

−2

−1.5

−1

−0.5 0 0.5 1 1.5

b

**Figure 12: LPT of system**5.20 a*and beating solution uτ*0of5.8.

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

0 10 20 30 40 50 60 70 80 90 100

*x*1

*x*2

**Figure 13: Plots of***x*1τ0 solidand*x*2τ0 dash.

The LPTFigure 12ais calculated as a solution of5.20*with initial conditions a0*
0,Δ0 −π/2. One can see that the LPT approximates the envelope of the process*u*2τ0
Figure 13with an error about 10%. Thus we may conclude that the observed intense energy
exchange is associated with motion over the LPT.

We now calculate the critical parameter*α*^{∗}_{1}. Here we have*μ*^{2} *10, F* 1.1075,*εd* 0.2,
*s* 0.4, and, by2.20

*α*^{∗}_{1} 3

8*,* *c*^{∗}_{1} 4μ^{2}*s*^{3}

27F^{2}*,* *c*_{1}^{∗} 32μ^{2}*s*^{3}

81F^{2} 0.205. 5.21

Since the accepted value *c* 0.8 *> c*^{∗}_{1} , the resonance exchange takes place in the
domain of large oscillations demonstrated inFigure 1.