Vibration of generalized double well oscillators

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Vibration of Generalized Double Well Oscillators

Grzegorz Litak1,2∗, Marek Borowiec1,∗∗, and Arkadiusz Syta3,∗∗∗

1 Department of Applied Mechanics, Technical University of Lublin, Nadbystrzycka 36, PL-20-618 Lublin, Poland2 Institut fur Mechanik und Mechatronik, Technische Universitat Wien, Wiedner Hauptstraβe 8 - 10 A-1040 Wien, Austria2 Department of Applied Mathematics, Technical University of Lublin, Nadbystrzycka 36, PL-20-618 Lublin, Poland

Received 8/2/2008, revised , acceptedPublished online

Key words Duffing oscillator, Melnikov criterion, chaotic vibration

We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of adouble well dynamical system with a nonlinear fractional damping term and external excitation. The usual double wellDuffing potential having a negative square term and positivequartic term has been generalized to a double well potentialwith a negative square term and a positive one with an arbitrary real exponentq > 2. We have also used a fractionaldamping term with an arbitrary powerp applied to velocity which enables one to cover a wide range ofrealistic dampingfactors: from dry frictionp → 0 to turbulent resistance phenomenap = 2. Using perturbation methods we have found acritical forcing amplitudeµc above which the system may behave chaotically. Our results show that the vibrating systemis less stable in transition to chaos for smallerp satisfying an exponential scaling low. The critical amplitudeµc as anexponential function ofp. The analytical results have been illustrated by numericalsimulations using standard nonlineartools such as Poincare maps and the maximal Lyapunov exponent. As usual for chosen system parameters we haveidentified a chaotic motion above the critical Melnikov amplitudeµc.

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1 Introduction

A nonlinear oscillator with single or double well potentials of the Duffing type and linear damping is one of the simplestsystems leading to chaotic motion studied by [1, 2, 3, 4]. Theproblem of its nonlinear vibrations has attracted researchersfrom various fields of research across natural science and physics [4, 5, 6], mathematics [8] mechanical engineering [10, 11,14, 12, 13]; and finally electrical engineering [1, 2, 3]. This system, for a negative linear part of stiffness, shows homoclinicorbits, and the transition to chaotic vibration can be treated analytically via the Melnikov method [7]. Such a treatment hasbeen already performed successfully to selected problems with various potentials [8, 9, 14]. Vibrations of a single Duffingoscillator have got a large bibliography [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In the last decade coupledDuffing oscillators [19, 20, 21, 22, 23] with numerous modifications to potential and forcing parts have been studied. Onthe other hand the problem of nonlinear damping in chaotically vibrating system has not been discussed in detail. Someinsight into this problem can also be found in the context of self excitation effects

[15, 19, 22, 23, 25, 24, 26]. and dry friction effects [33, 34,35, 36] In the paper by Truebaet al. [16], the systematicdiscussion on square and cubic damping effects on global homoclinic bifurcations in the Duffing system has been given.Recently Truebaet al. [17] and Borowiecet al. [18] have analyzed a single degree of freedom nonlinear oscillator withthe Duffing potential and fractional damping. Different aspects fractionally damped systems have been studied recentlyby Mickenset al., Gottlieb, and Mickens [27, 28, 29]. On the other hand Maiaet al. and Padovan and Sawicki [30,32, 31] analyzed similar systems where fractional damping have been introduced in different way through a fractionalderivative. Awrejcewicz and Holicke [37] and more recentlyAwrejcewicz and Pyryev [38] applied Melnikov’s methodin the presence of dry friction for a stick-slip oscillator.More general introduction to the problem of non-smooth ordiscontinuous mechanical systems can be found in [39, 40].

In the present paper we revisit this problem looking for a global homoclinic bifurcation and transition to chaotic vi-brations in a system described by a more general double well potential where its usual positive quartic term has beengeneralized to term with an arbitrary real exponent grater than 2. Below we would also apply a nonlinear damping termwith a fractional exponent covering the gap between viscous, dry friction and turbulent damping phenomena.

∗ Corresponding author, e-mail:[email protected], Phone: +4881 538 1573, Fax: +4881 525 0808∗∗ [email protected]∗∗∗ [email protected]

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http://arXiv.org/abs/nlin/0610052v1

2 G. Litak, M. Borowiec, and A. Syta: Vibration of GeneralizedDouble Well Oscillators

The equation of motion has the following form:

x + αx |x|p−1+ δx + γsgn(x)|x|q−1 = µ cosωt, (1)

wherex is displacement andx velocity, respectively, while the external forceFx:

Fx = −δx − γsgn(x)|x|q−1, (2)

and corresponding potentialV (x) (Fig. 1a) is defined as:

V (x) =δx2

2+

γ|x|qq

, (3)

whereq > 2 is a real number. In spite of the definitionV (x) (Eq. 3) in terms of absolute value|x| it is still a function ofC2 class if onlyq > 2 (see Appendix A).

The non-linear damping term is defined by the exponentp:

dpt(x) = αx |x|p−1 . (4)

In Fig. 1b we have plotted the above function versus velocity(v = x) for few values ofp. Note that, the casep → 0 (seep = 0.1 in Fig. 1b for a relatively small velocity) mimics the dry friction phenomenon [33, 34].

-0.2

0.0

0.2

0.4

-1.0 -0.5 0.0 0.5 1.0

V(x

)

x

q=2.5

3.0

3.5

4.0

(a)

-2.0

-1.0

0.0

1.0

2.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

dam

ping

term

v

p=0.10.5

1.02.0

(b)

Fig. 1 External potentialV (x) = δx2

2+ γ|x|q

q(Eq. 3) forδ = −2 for a few values ofq (q = γ > 2 in Fig. 1a), Damping term for

variousp (Fig. 1b).

2 Melnikov Analysis

We start our analysis with the unperturbed Hamiltonian:H0

H0 =v2

2+ V (x). (5)

Note that for our choice of potentialsδ = −2 andγ = q (Fig. 1a)V (x) has the three nodal points (x = −1, 0, 1) wherethe middle one (x = 0) corresponds to the local peak at the saddle point. The existence of this point with a horizontaltangent enables occurrence of homoclinic bifurcations. This includes transitions from regular to chaotic solutions.To studythe effects of damping and excitation on the saddle point bifurcations, we apply small perturbations around the homoclinicorbits. Our strategy is to use a small parameterǫ to the Eq. 1 with perturbation terms. Uncoupling Eq. 1 into two differentialequations of the first order we obtain

x = v (6)

v = −ǫαv |v|p−1 − δx − γsgn(x)|x|q−1 + ǫµ cosωt,

whereǫα = α andǫµ = µ, respectively.


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-0.8

-0.4

0.0

0.4

0.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2

v

x

q=4.0

q=3.5q=3.0

q=2.5

>>

>>

Fig. 2 Left hand side homoclinic orbits for unperturbed Hamiltonian (Eq. 5). Note, in our case the potential has reflection symmetryover 0-y axis so the orbits appear in pairs for corresponding regionsx > 0 andx < 0.

At the saddle pointx = 0, for an unperturbed system (Fig. 1a), the system velocity reaches zerov = 0 (for infinite timet = ±∞) so the total energy has only its potential part which has been gauged out to zero too. Thus transforming Eqs. 3and 5 for a nodal energy (E = 0) and forδ < 0, γ > 0 we get the following expression for velocity:

v =dx

dt=

√

2

(

−δx2

2− γ|x|q

q

)

. (7)

Performing integration overx we get

t − t0 = ±∫

dx

x√

−δ − 2γ|x|q−2

q

, (8)

wheret0 represents here a time like integration constant.Integration in Eq. 8 has been performed analytically. Forq > 2, one can writex∗ as:

x∗ = x∗(t − t0) = ±(−δq

2γ

)1

q−2 1

cosh2

q−2

[

(q−2)2

√−δ(t − t0)

] . (9)

The corresponding velocityv∗ reads:

v∗ = v∗(t − t0) = ∓√−δ

(−δq

2γ

)1

q−2 tanh[

(q−2)2

√−δ(t − t0)

]

cosh2

q−2

[

(q−2)2

√−δ(t − t0)

] . (10)

Due to the reflection symmetry of potentialV (−x) = V (x) (Eq. 3) there are two symmetric solutions for unperturbedhomoclinic orbits with ’+’ and ’-’ signs in Eqs. 9-10. A family of right hand side homoclinic orbits(x∗, v∗) has been plottedin Fig. 2. In unperturbed case both stable and unstable manifolds (Poincare sections of the orbits are usually denoted byWs andWu) can be identified with the orbits discussed above while perturbations would influence them in a different way[12]. Existence of cross-sections of betweenWS andWU manifolds signals Smale’s horseshoe scenario of transition tochaos.

The distanced (Fig. 3) between them can be estimated by the Melnikov functionM(t0):

M(t0) =

∫ +∞

−∞h(x∗, v∗) ∧ g(x∗, v∗)dt (11)

where the corresponding differential formh means the gradient of unperturbed Hamiltonian (Eq. 3):

h =(

δx∗ + γsgn(x)|x|q−1)

dx + v∗dv, (12)



d

x

W

W

u

s

v

Fig. 3 Schematic plot of stable and unstable manifolds (Ws andWu) of perturbed system Eq. 6.d denotes the distance betweenmanifolds given by Melnikov functionM(t0) Eq. 11.

while g is a perturbation form (Eq. 5) to the same Hamiltonian:

g =(

µ cosωt − αv∗ |v∗|p−1)

dx. (13)

All differential forms are defined on homoclinic orbits(x, v) = (x∗, v∗) (Eqs. 9-10).Thus the Melnikov functionM(t0):

M(t0) =

∫ +∞

−∞v∗(t)

(

µ cos (ω(t + t0)) − αv∗(t) |v∗(t)|p−1)

dt

= − sinωt0

∫ +∞

−∞v∗(t)µ sin ωtdt −

∫ +∞

−∞αv∗2(t) |v∗(t)|p−1 dt (14)

= − sin(ωt0)µI1 − αI2,

whereI1 andI2 are integrals to be evaluated.sinωt0 appears because of the odd parity of the functionv∗(t) under theabove integral where

cos(ω(t + t0)) = cos(ωt) cos(ωt0) − sin(ωt) sin(ωt0). (15)

Thus a condition for a global homoclinic transition, corresponding to a hors-shoe type of stable and unstable manifoldscross-section (Fig. 2), can be written as:

∨

t0

M(t0) = 0 and∂M(t0)

∂t06= 0. (16)

For a perturbed system the above constraint together with the explicit form of Melnikov function Eq. 14 gives the criticalamplitudeµc:

µc

α=

∣

∣

∣

∣

I2

I1

∣

∣

∣

∣

, (17)

whereI1 andI2 are corresponding integrals given in Eq. 14. In case ofI1 we have the following integral

I1 =√−δ

(−δq

2γ

)1

q−2∫ −∞

−∞

tanh(

(q−2)2

√−δt

)

cosh2

q−2

(

(q−2)2

√−δt

) sin(ωt) dt (18)

to be evaluated numerically in general but for some cases canbe easily performed numerically (see Appendixes B and C)while, in analogy to [16, 17, 18],I2 can be expressed as

I2 = (−δ)p+12

(−δq

2γ

)

p+1q−2

∫ +∞

−∞

sinhp+1(

(q−2)2

√−δt

)

coshq(p+1)

q−2

(

(q−2)2

√−δt

) dt

= (−δ)p+12

(−δq

2γ

)

(p+1)q−2

B

(

p + 2

2,

p + 1

(q − 2)

)

, (19)



0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0

µ /α

ω

c

q=2.5

p=0.10.5

1.0

2.0

(a)0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

µ /α

ω

c

q=3.0

p=0.10.5

1.0

2.0

(b)

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

µ /α

ω

c

q=3.5

p=0.10.5

1.0

2.0

(c)0.0

1.0

2.0

3.0

4.0

0.0 1.0 2.0 3.0 4.0µ

/α

ω

c

q=4.0

p=0.10.5

1.02.0

(d)

-2.0

0.0

φ2.0

4.0

6.0

0.0 1.0 2.0 3.0

(µ /

α )

p

Ln

c

q=3.5q=2.5

ω=3.0ω=2.0ω=1.0

v

(e)0.0

0.4

0.8

1.2

1.6

2.0 3.0 4.0 5.0 6.0

κ

q

’1’

’2’

(f)

Fig. 4 Critical amplitudeµc/α versus frequencyω for few values ofp (p = 0.1 0.5, 1.0, 2.0) and differentq (q = 2.5 in Fig. 4a,q = 3.0 in Fig. 4b,q = 3.5 in Fig. 4c,p = 4.0 in Fig. 4d. ln(µc/α) versus the exponentp for three values ofω andq = 2.5 and 3.5(Fig. 4e). Dependence of the slopeκ (κ = tan φ in Fig. 4e) on the exponentq – Fig. 4f (’1’ corresponds to present calculations forω = 1 while ’2’ is a fitting trail κ = 1/(q/1.3 + 1)).

where B(r, s) is the Euler Beta function dependent of arbitrary complex arguments with real parts (Re r > 0 andRe s > 0)defined as

B(r, s) =Γ(r)Γ(s)

Γ(r + s), (20)

while Γ(r) denotes the Euler Gamma function:

Γ(z) =

∫ ∞

0

e−ssz−1ds for Re z > 0. (21)

In Fig. 4a–d we plotted the results of Melnikov analysis for acritical amplitudeµc/α for few values ofp (p = 0.1 0.5,1.0, 2.0) and differentq (q = 2.5 in Fig. 4a,q = 3.0 in Fig. 4b,q = 3.5 in Fig. 4c,p = 4.0 in Fig. 4d) Forµ > µc the



-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.0 0.1 0.2 0.3 0.4λ

µ

1

ω=1.1

(a)

Fig. 5 Maximal Lyapunov exponentλ1 versusµ (Fig. 5a), bifurcation diagram together with size of attractor xmax andxmin versusµ(Fig. 5b) forω = 1.1.

system can transit to chaotic vibrations. Note, in spite of some quantitative changes all four figures (Fig. 4a-d) have similarshape and the sequence of corresponding curves with particular exponentsp = 0.1 0.5, 1.0, 2.0 is preserved for anyω andq. This gives us a conviction thatp may play some independent role. In fact plottingln(µc/α) in Fig. 4e versusp and wehave got straight lines with characteristic slope independent onω but changing withq. Dependence of the slopeκ:

κ = tanφ (22)

defined in Fig. 4e, versusq has been plotted in Fig. 4f forω = 1. Note, the curve ’1’ corresponds to present calculationsfor while ’2’ is a fitting curve:

κ =1

q1.3 − 1

(23)

The above scaling is not a surprise taking into account the structure ofM(t0) (Eq. 12). In this expression the exponentp isentering to the second integral independent ofω. On can also look into the analytic formulae forµc in cases ofq = 4 and3 in the Appendix B (Eq. B.5-B.8) where thep appears as an exponent.

3 Results of Numerical Simulations

To illustrate the dynamical behaviour of the system it is necessary to simulate the proper equations. Here we have usedthe Runge-Kutta method of the forth order and Wolf algorithm[41] to identify the chaotic motion. In our numerical codewe started calculations from the same initial conditions(x0, v0) = (0.45, 0.1) for any new examined value ofµ. Thesystem parametersδ = −2 andγ = q have be chosen the same as for analytic calculations. We haveperformed numericalcalculations for different choices of system parameters:α, ω, p andq, hut here, for or technical reasons, we limited ourdiscussion toα = 0.1, ω = 1.1, p = 0.5 andq = 2.5.

In Fig. 5a we have plotted the maximal Lyapunov exponent versus external forcing amplitudeµ. Here one can clearlysee points ofλ1 sing changes. Forµ ∈ [0.23, 0.27] and[0.33, 0.38] we have gotλ1 > 0 indicating chaotic vibrations. InFig. 5b we have plotted the corresponding bifurcation diagram. Thus a black region means chaotic motion. This result, as



0.0

1.0

2.0

3.0

4.0

5

0.0 0.5 1.0 1.5 2.0

µ /α

ω

c

µ/α=0.5

µ/α=2.4

ω=1.1

(a)

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.50 0.55 0.60 0.65 0.70 0.75 0.80

v

x

ω=1.1

(b)

-0.8

-0.4

0.0

0.4

0.8

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

v

x

ω=1.1

(c)

Fig. 6 Critical amplitudeµc/α versusω the dashed line corresponds toω = 1.1 (Fig. 6a). Phase portrait and Poincare maps forω = 1.1, α = 0.1 and two differentµ (µ = 0.05 in Fig. 6b whileµ = 0.24 in Fig. 6c). The results have been obtained forp = 0.5 andq = 2.5.

well as others, calculated for different sets of system parameters, is consistent with the Melnikov results. For comparisonwe have plotted the Melnikov curve again (Fig. 6a with two trial pointsµ = 0.05 andµ = 0.24 (for α = 0.1). There is nodoubt that Fig. 6b shows the regular synchronized motion represented by a single loop on a phase portrait and a singularpoint on Poincare stroboscopic map. On the other hand Fig. 6cshows clearly a strange attractor of chaotic vibration withcomplex structure of the Poincare map.

4 Summary and Conclusions

We have examined criteria for transition to chaotic vibrations in the double well system with a damping termdpt(v) =v|v|p−1 described by a fractional exponentp and nonlinear potential with negative square term (relatednegative stiffness)and a positive term with higher exponent|x|q whereq > 2. In spite of non-smoothness of corresponding vector-fields(h andg – Eqs. 11 and 12, respectively) it has been proven in the Appendix A that extra terms to the Melnikov integral



[40] projected out. Thus the critical value of excitation amplitudeµ above which the system vibrate chaotically has beenestimated, in by means of the Melnikov theory [7]. For some selected values of the exponentq (q =4,3, 8/3, 2/5) it waspossible to derive a final formula forµc while for other cases one of Mielnikov’ integrals has be calculated numerically.

The analytical results have been confirmed by simulations. In this approach we used standard methods of analysis asPoincare maps, bifurcation diagrams and Lyapunov exponent.

The Melnikov method, is sensitive to a global homoclinic bifurcation and gives a necessary condition for excitationamplitudeµ = µc1 system in its transition to chaos [8, 9]. On the other hand thelargest Lyapunov exponent [41], measuringthe local exponential divergences of particular phase portrait trajectories gives a sufficient conditionµ = µc2 for thistransition which has obviously a higher value of the excitation amplitudeµ = µc2 > µc1.

Above the Melnikov transition predictions (µ > µc1) we have obtained transient chaotic vibrations [9, 10, 11, 12, 18]as we expected drifting to a regular steady state away the fractal attraction regions separation boundary. This is typicalbehaviour of the system which undergo global homoclinic bifurcation.

Acknowledgments

This paper has been partially supported by the Polish Ministry of Science and Information. GL would like to thank MaxPlanck Institute for the Physics of Complex Systems in Dresden for hospitality.

Appendix A

Starting with the perturbation equation (Eq. 6) we write it in a two element vector form

q = h + ǫg, (A.1)

where

q = [x, v]

h = [v,−δx − γsgn(x)|x|q−1] (A.2)

g = [0,−αv |v|p−1+ µ cosωt].

On the other hand the homoclinic orbit

q∗(t − t0) = [x∗(t − t0), v∗(t − t0)], (A.3)

wheret0 is usually defined by simple zero of Melnikov integral [7]. Inthe limit of extreme timet → ±∞ the system state[x, v] reaches a saddle point[x, v] = [0, 0] (see Figs. 1a, 2). Consequently, in the aim to examine the Melnikov criterionfor chaos appearance, the vector fieldsh andg are defined on the homoclinic orbit (Fig. 2) as:

h(q∗) = [v∗(t − t0),−δx∗(t − t0) − γsgn(x∗(t − t0)|x∗(t − t0)|q−1] (A.4)

and

g(q∗, t) = [0,−αv∗(t − t0) |v∗(t − t0)|p−1+ µ cosωt]. (A.5)

The perturbed stable and unstable manifoldsW s andWu read [40]

qu,s(t, t0) = q∗(t − t0) + ǫqu,s1 (t, t0) + O(ǫ2) (A.6)

respectively.Note the perturbation correction to the homoclinic orbitq

u,s1 (t, t0) in the above expression (Eq. A.6) should be found

by solving the following linear differential equation about the examined timet (or a system stateq = q∗(t − t0)):

qu,s1 (t, t0) =

(

∂h1

∂v− ∂h2

∂x

)

|q=q∗(t−t0)

qu,s1 (t, t0) + g(q∗(t − t0), t) (A.7)

Note that the above vector fieldsh(x, v) andg(x, v, t) are notC2 functions. Namelyh is of C2 only if q ≥ 2 and ofC1

if 1 ≤ q < 2. Similarly g is of C2 only if p = 1 or p ≥ 2 and ofC if 0 < p < 1 andC1 1 < p < 2, respectively. In case



W

v=f(x)

W

u

s

v

x

Fig. A.1 Stable and unstable manifolds and one of lines of discontinuity v = f(x). This line crosses manifold at[xd, vd] for thespecific timet = td xd = x(td) andvd = v(td).

of 1 ≤ q < 2 the linex = 0 separates the whole phase space(x, v) into two parts whereh(x, v) is enough smooth (C2).The same can be applied to the linev = 0 andg(x, v, t) as a possible set for0 < p < 1 and1 < p < 2. This line crossesmanifold at[xd, vd] for the specific timet = td such thatxd = x(td) andvd = v(td) (Fig. A.1).

According to Kunze and Kupper [40] the (C2) space separation includes additional terms to the Melnikov integral.Thus the Mielnikov function M(t0):

M(t0) = M0(t0) +∑

td

(

h⊥,+(q∗(t−d )) · qu,+1 (t0 + t−d , t0) − h⊥,−(q∗(t−d )) · qu,−

1 (t0 + t−d , t0) (A.8)

+ h⊥,−(q∗(t+d )) · qs,−1 (t0 + t+d , t0) − h⊥,−(q∗(t+d )) · qs,+

1 (t0 + t+d , t0))

,

whereqs,±1 , qu,±

1 are stable and unstable manifold perturbation solutions (Eq. A.7) for t in the vicinity of td but t > td for’+’ sign andt < td for ’−’ sign, respectively.

M0(t0) is defined as for smooth vector fields:

M0(t0) =

∫ ∞

−∞h⊥(q∗(t − t0) · g(q∗(t + t0), t) dt (A.9)

andh⊥ = [−h2, h1].Once first discontinuity are identified forx = 0, td → ±∞, one have to examineφ±

1 (t, t0) = qu,s,±1 (t, t0) and

φ±2 (t, t0) = qu,s,±

1 (t, t0):

{

φ1 = (1 + δ + γ(q − 1)xq−2)φ1

φ2 = (1 + δ + γ(q − 1)xq−2)φ2 − αv|v|p−1 + µ cosωtfor x > 0 (A.10)

and{

φ1 = (1 + δ + γ(q − 1)(−x)q−2)φ1

φ2 = (1 + δ + γ(q − 1)(−x)q−2)φ2 − αv|v|p−1 + µ cosωtfor x < 0 (A.11)

Note substitutingx = 0 for q > 2 to Eqs. A.10 and A.11 we get the same equations forφ1/2 and consequently the sameexpressions. This means automatically no extra terms to theMelnikov integral (Eq. A.8) caused by thex = 0 discontinuity.Interestinglyq ≤ 2 would lead to a different result but for such case the is no homoclinic orbit in the unperturbed systemdescribed byH0 (Eqs. 5,3).

Let us non focus onv = 0 discontinuity. In this case

{

φ1 = (1 + δ + γ(q − 1)|x∗|q−2)φ1

φ2 = (1 + δ + γ(q − 1)|x∗|q−2)φ2 − α(v∗)p + µ cosωtfor v > 0 (A.12)

{

φ1 = (1 + δ + γ(q − 1)|x∗|q−2)φ1

φ2 = (1 + δ + γ(q − 1)|x∗|q−2)φ2 + α(v∗)p + µ cosωtfor v < 0 (A.13)



Note, excluding natural odd numbers for thep exponent, the above equations (Eqs. A.12 and A.13) are usually differentfor any otherp ≥ 0. However both solutions[φ−

1 , φ−2 ] and[φ+

1 , φ+2 ] have to be projected into

h⊥|v=0 = [−h2, h1]|v=0 = [δx∗(td − t0) + γsgn(x∗(t − t0)|x∗(td − t0)|q−1, 0] (A.14)

and the differences in solutions inφ−2 andφ+

2 are effectively projected out. Interestingly this is also valid for p = 0 (a dryfriction case).

Finally for q > 2 andp ≥ 0 the Mielnikov function M(t0) can be treated as a

M(t0) = M0(t0). (A.15)

Appendix B

In this appendix we show how to get homoclinic orbits and analytically for some specific cases of exponentq: q = 4, 3,2.67 and 2.5.

In case ofq = 4 we follow works by Truebaet al. [16] and Borowiecet al. [18] (and Eqs. 9-10)

x∗ = x∗(t − t0) = ±√

−2δ

γ

1

cosh(√

−δ(t − t0))

v∗ = v∗(t − t0) = ±√

2

γδ

tanh(√

−δ(t − t0))

cosh(√

−δ(t − t0)) (B.1)

where ’+’ and ’−’ signs are related to left– and right–hand side orbits, respectively,t0 is a time like integration constant.On the other hand forq = 3 we have

x∗ = x∗(t − t0) = ∓ 3δ

2γ

1

cosh2(√

−δ(t−t0)2

)

v∗ = v∗(t − t0) = ∓3δ√−δ

2γ

tanh(√

−δ(t−t0)2

)

cosh2(√

−δ(t−t0)2

) , (B.2)

Consequently forq = 2 23 = 8/3 ≈ 2.67 we have

x∗ = x∗(t − t0) = ±(−4δ

3γ

)3/21

cosh3(√

−δ(t−t0)3

)

v∗ = v∗(t − t0) = ∓(−4δ

3γ

)3/2 √−δ

tanh(√

−δ(t−t0)3

)

cosh3(√

−δ(t−t0)3

) , (B.3)

And for q = 2.5

x∗ = x∗(t − t0) = ∓(

5δ

4γ

)21

cosh4(√

−δ(t−t0)4

)

v∗ = v∗(t − t0) = ±(

4δ

3γ

)2 √−δ

tanh(√

−δ(t−t0)4

)

cosh4(√

−δ(t−t0)4

) , (B.4)

The results for a Melnikov integral can be easily found in theabove cases. Evaluating the corresponding integral (Eq.11) after some algebra the last condition (Eq. 16) yields to acritical value of excitation amplitudeµc. Thus forq = 4[4, 16, 17, 18] we have:

µc = α2p/2(−δ)p+1/2

πωγp/2B

(

p + 2

2,p + 1

2

)

cosh

(

πω

2√−δ

)

, (B.5)



while in case ofq = 3 [42, ?, 43]:

µc = α3p(−δ)3p/2+2

2p+1πω2γpB

(

p + 2

2, p + 1

)

sinh

(

πω√−δ

,

)

(B.6)

for q = 8/3:

µc = α2

65 (p+1)(−δ)11p/10−2/5

335 (p+1)π(9ω2 − δ)ωγ3p/5+9/10

B

(

p + 2

2,3(p + 1)

2

)

cosh

(

3πω

2√−δ

)

, (B.7)

and finally forq = 5/2:

µc = α52p(−δ)5p/2+3/2

24p+3π(4ω2 − δ)ω2γ2pB

(

p + 2

2, 2(p + 1)

)

sinh

(

2πω√−δ

)

. (B.8)

Appendix C

The integralI1 can be evaluated analytically in some specific cases of exponentsq corresponding to homoclinic orbitsEqs. B.1-B.4 numbered by the corresponding power indexm applied to hyperboliccos function in the denominators. Letus consider integralsI1 for givenm = 1,2,3 and 4 related to variousq exponentsq = 4, 3, 8/3 and 5/2, respectively. Tobetter clarity we will use new notationI1 → I1(m) for givenm:

I1(m) =

∫ +∞

−∞v∗(t)µ sin ωtdt = Cm

∫ +∞

−∞

tanh(τ)

coshm(τ)sin(ωmτ)dτ (C.1)

=ωmCm

m

∫ +∞

−∞

cos(ωmτ)

coshm(τ)dτ =

ωmCm

mJm(ωm),

while constantsCm andωm are defined as follows:

Cm =√−δ

(−(m + 1)δ

mγ

)m/2

, ωm =mω√−δ

(C.2)

Evaluating the integralJm(ωm) (C.1), for positive integerm, twice by parts we have got the following recurrence identity

Jm+2(ωm+2) =ω2

m+2 + m2

m(m + 1)Jm(ωm+2) for m = 1, 2, 3, ... (C.3)

Thus onlyJ1 andJ2 need to be calculated. Below we evaluate them on the complex plane by summing correspondingresidue.

z=x+iy

x

y y

x

complex plane

R 8

k=321

−1−2

0

Fig. C.1 Deformed contour integration schema and imaginary poles.

∮

f(z)dz = 2πi

N∑

k=1

Res[f(z), zk], (C.4)



where

Res[f(z), zk] =1

(m − 1)!lim

z→zk

dm−1

dzm−1[(z − zk)mf(z)] , (C.5)

for m = 1 or 2, in our case.The examined functionf(z) is defined as:

f(z) =2m

(exp(z) + exp(−z))mexp (iωmz). (C.6)

Note that on the real axis (Fig. C.1)Re z = τ it can be written as

Imf(τ) =cos(ωmτ)

coshm τ. (C.7)

The multiplicity of each pole of the complex functionf(z) (Eq. C.6) is given by

zk =(π

2+ πk

)

i for k = 1, 2, 3, ... (C.8)

NoteJm (Eq. C.1) can be easily determined form = 1 or 2. Namely, after summation of all poles in the upper half-plane(Fig. C.1), we get form = 1

J1 =

∫ +∞

−∞dτ

cos(ω1τ)

cosh τ=

π

cosh(

πω1

2

) (C.9)

while for m = 2 we obtain

J2 =

∫ +∞

−∞dτ

cos(ω2τ)

cosh2 τ=

πω2

sinh(

πω2

2

) . (C.10)

On the other hand, in case ofm = 3 andm = 4 (and also for any largerm), we can use the recurrence relation (Eq.C.3):

J3 =π(ω2

3 + 1)

2 cosh(

πω3

2

) , J4 =πω4(ω

24 + 4)

6 sinh(

πω4

2

) . (C.11)

Consequently using Eq. C.1

I1(1) =

(−2δ

γ

)1/2πω

cosh(

πω2√−δ

) , I1(2) =

(−3δ

2γ

)

2πω2

√−δ sinh

(

πω√−δ

) , (C.12)

I1(3) =

(−4δ

3γ

)3/2π(9ω2 − δ)

2√−δω cosh

(

3πω2√−δ

) , I1(4) =

(−5δ

4γ

)28π(4ω2 − δ)

3(−δ) sinh(

2πω√−δ

) .



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Vibration of generalized double well oscillators

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