Eur. Phys. J. Plus (2013) 128: 80 DOI 10.1140/epjp/i2013-13080-6
THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations Ruilan Tian1 , Qiliang Wu1 , Xinwei Yang2,a , and Chundi Si1 1 2
Department of Mathematics and Physics, Shijiazhuang Tiedao University - Shijiazhuang, 050043, China School of Traffic, Shijiazhuang Institute of Railway Technology ShiJiaZhuang, 050041, China Received: 20 April 2013 c Societ` Published online: 31 July 2013 – a Italiana di Fisica / Springer-Verlag 2013 Abstract. The smooth-and-discontinuous oscillator under constant excitations (CSD) is an unsymmetrical system with an irrational restoring force, which leads to a barrier to detect the chaotic threshold using conventional nonlinear techniques. The goal of the present work is to overcome the trouble raised by constant excitations and irrational nonlinearity to investigate the necessary and insufficient condition for chaos. A smooth approximate system, whose behaviors are topologically equivalent to the ones of the original system, is proposed to investigate the chaotic boundary. The Hamiltonian system of the approximate system is analyzed and the homoclinic orbits in analytical form are obtained. The Melnikov method is employed to determine the distance between the stable and unstable manifolds under the perturbation of damping and external forcing. Bifurcation diagrams and numerical simulations are used to reveal the motion of chaos and the period for approximate system which is in good agreement with the original system. It is worth recalling that the approach for constructing the approximate function of smoothness is presented, which puts forward a step toward the solution of the irrational nonlinearity system.
1 Introduction Constant excitation and an irrational restoring force in dynamic systems have attracted much interest all over the world. There are a few works recently done on the study of the system subjected to constant excitation. Cveticanin studied the mechanical vibration of a one–degree-of-freedom mass-spring system under the influence of a constant positive excitation force [1] and vibrations of a free two-mass system with quadratic nonlinearity and a constant excitation force [2]. For studying the impurity effect on the photoinduced spin-state switching of transition metal complexes, the effect of fixed-spin doping on a phase switching of an Ising model under constant excitation is investigated in [3]. Based on the singularity theory, the effects of the constant excitation on the local bifurcation of the periodic solutions in the 1:2 internal resonant systems are analyzed in [4]. A new accurate analytical approach has been developed for solving the nonlinear TDOF oscillation systems with constant excitation [5]. Pure odd-order oscillators with constant excitation are also analyzed in [6]. Furthermore, for an irrational nonlinear system, Gourdon [7], Manevitch [8] and Gatti [9] investigated energy pumping in a strongly nonlinear system of two degree-of-freedom by expanding the irrational item into a Taylor series up to the cubic term. Lai [10] applied the generalized Senator-Bapat (GSB) perturbation technique [11], which is adopted for solving some special cases of the conservative oscillating system with an irrational nonlinearity. The energy balance method [12–19] is also used to solve this problem; Jamshidi and Ganji [12] applied the energy balance method and the variational iteration method [13,14] to a nonlinear oscillation of a mass attached to a stretched wire (irrational nonlinear system). Sun [15] obtained the approximate analytical solutions for the irrational nonlinear system on the basis of combining Newton’s method with the harmonic balance method. Zhang [16] applied He’s energy balance method to determine frequency-amplitude relations of nonlinear oscillators with discontinuous terms or high nonlinearities. Bel´edez [17,18] also introduced the energy balance method and the homotopy perturbation method [19] for treating the oscillator of an irrational nonlinear system. However, there is limited research on theoretical analysis of the criteria for the chaotic motion subjected to the constant excitation coupled with an irrational restoring force. In 2006, a smooth-and-discontinuous (SD) oscillator with an irrational restoring force was proposed and investigated by Cao et al. (see [20–22]), which can be used to a
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Eur. Phys. J. Plus (2013) 128: 80
L
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m X
F
k
k
l
l
Fig. 1. An elastic arch consisting of a curved and tapered beam pinned to rigid abutments.
study the transition from smooth to discontinuous dynamics depending on the value of the smoothness parameter α. Tian et al. studied the the codimension-two [23] bifurcation and Hopf [24] the bifurcation for the SD oscillator. By introducing a series of transformation, the Melnikov boundary of the strongly nonlinear system for both smooth and discontinuous regions is investigated in [25]. However, the effect of the constant excitation produced by gravity or other constant forces is not considered in these results. Although the case of equilibrium, the Hamiltonian phase portrait and type of strange attractors of the smooth-and-discontinuous oscillator under constant excitations (CSD), which is subjected to the constant excitation coupled with the irrational restoring force, has been discussed in [26], there is still an obstacle in theoretically analyzing the criteria for chaotic motion (see details in appendix A). The motivation of this paper is to detect the chaotic boundary of the CSD oscillator by means of the approximate system which efficiently reflects the nonlinear dynamics of the original system. This approach enables us to investigate the nonlinear dynamics theoretically for both perturbed and unperturbed dynamics. The proposed approximate system bears significant similarities to the original system of phase portraits and the bifurcation which successfully avoids the barrier of the associated irrational nonlinearity and constant excitation. This paper is organized as follows. In the next section, the CSD oscillator differing significantly from the SD oscillator due to the constant excitation is presented. In sect. 3, based on the trilinear function [21], a new smooth function is constructed to replace the irrational restoring force in the CSD oscillator, which includes two lines and two curves connecting at the three equilibria. Further, the analytical form of the homoclinic orbits is developed as a parametric function of the time variable t. The approach is valid for both the smooth and the limit discontinuous system. Section 4 discusses the Melnikovian function of the perturbed approximate system. The Melnikov method is applied to determine the distance between stable and unstable manifolds, and the boundary beyond which chaos will be probably generated, is obtained. Finally, bifurcation diagrams are considered and numerical simulations are also carried out in sect. 5, to confirm these analytical predictions.
2 The CSD oscillator The simple system comprises a linked pair of inclined springs of stiffness k, which are capable of resisting both tension and compression and pinned to rigid supports, as shown in fig. 1, L is the original length, X is the mass m displacement, l is the half distance between the rigid supports and F is the constant excitation produced by gravity or other constant force. Although the springs themselves provide linear restoring resistance, the resulting vertical response on the mass is a strongly nonlinear force because of the geometric configuration. The equation of motion can be written as L 2 ¨ = F. (1) mX + ω0 X 1 − √ X 2 + l2 System (1) can be rewritten in a dimensionless form by letting x = X/L, α = l/L ≥ 0, ω02 = 2k/m and ρ = F/mL, 1 2 = ρ. (2) x ¨ + ω0 x 1 − √ x2 + α2
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Equation (2) can be written in a two-dimensional system of first order by letting ω02 = 1 as follows: ⎧ ⎪ ⎨ x˙ = y . 1 ⎪ √ +ρ y ˙ = −x 1 − ⎩ x2 + α2
(3)
The Hamiltonian function of system (3) is H(x, y) =
1 2 1 2 2 x + y − x + α2 − ρx + α 2 2
(4)
and the dynamical behavior of the CSD oscillator differs significantly from those obtained in the SD oscillator. The reader can refer to ref. [26] for further details. Furthermore, the Hamiltonian function (4) is not readily analysed because of constant excitations and irrational nonlinearity (see details in appendix A). Hence, we will try to detect the chaotic threshold for the CSD oscillator actively by an analytical approach in the following sections, which might increase the research and application range of the SD oscillator.
3 Unperturbed approximate system Nowadays, as there are many interactions which cannot be exactly solved, approximate analytical solutions [27] or the approximation system [21], despite their errors and failures, are still appealing case studies and commonly used for solving nonlinear equations. Therefore, it makes sense for us to construct an approximation system for a further theoretical investigation of trajectories. In this section, based on the theory of strongly equivalence for bifurcation, the approximate function of smoothness is constructed to study the chaos of the original system, which is more accurate than the piecewise linear approximate function [21]. Consider the restoring force of the original system, 1 , f (x) = ρ − x 1 − √ x2 + α2 and the piecewise linear approximate function [21] (see figs. 2(a) and (b)). The blue curves are for the irrational restoring force f (x), the black lines are for the trilinear restoring force in [21], the red curves are the smooth approximate function. Obviously, from figs. 2(a) and (b), we know that the trilinear approximation [21] shown in red lines is nonsmooth at the two slope changing points and f (x) is a smooth function marked by blue curve, which leads to a more dramatic error between the original function and the linear approximation. Hence, the piecewise linear approximate function cannot provide excellent results with respect to the original function f (x) and might be a little better than some approximate function of smoothness. It is essential to introduce a new function to further study the smooth-and-discontinuous oscillator under constant excitations. Based on the equilibria type of the system (2) and the theory of strongly equivalence for bifurcation, let us construct the restoring force of the approximate function of smoothness g(x) satisfying g(xi ) = f (xi ) = 0 , i = 1, 2, 3, g (xi ) = f (xi ) where f (xi ) =
df (xi ) , dxi
g (xi ) =
dg(xi ) , dxi
i = 1, 2, 3.
To obtain a smooth function to alternate the nonlinear irrational restoring force, two triple functions, i.e. x2 ≤ x ≤ x1 and x1 ≤ x ≤ x3 , are introduced, the two curves satisfy the theory of strong equivalence mentioned above. The other two sections of the approach function for x ≤ x2 and x ≥ x3 are straight lines which are tangential to the function f (x) at the two centers x2 and x3 . Thus, a new smooth function is constructed to further fit the original system as follows: ⎧ x ≤ x2 d2 (x − x2 ), ⎪ ⎪ ⎪ ⎨ a (x − x )3 + b (x − x )2 + d (x − x ), x ≤ x ≤ x 1 1 1 1 1 1 2 1 , (5) g(x) = 3 2 ⎪ a2 (x − x1 ) + b2 (x − x1 ) + d1 (x − x1 ), x1 ≤ x ≤ x3 ⎪ ⎪ ⎩ d3 (x − x3 ), x3 ≤ x
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Eur. Phys. J. Plus (2013) 128: 80
(a)
f(x)
(b)
0
0
x2
x3
x1
x2
(c)
x1
x3
(d)
0
U(x)
0
x2
x1 x
x3
x2
x1 x
x3
Fig. 2. Restoring forces for α = 0.3, (a) ρ = 0.1; (b) ρ = 0.0, and potential wells for α = 0.3, (c) ρ = 0.1; (d) ρ = 0.0.
where di = f (xi ),
i = 1, 2, 3,
|Ai | |Bi | , bi = , i = 1, 2, |Ki | |Ki | (xi+1 − x1 )3 (xi+1 − x1 )2 Ki = , i = 1, 2, 3(xi+1 − x1 )2 2(xi+1 − x1 ) −d1 (xi+1 − x1 ) (xi+1 − x1 )2 Ai = , i = 1, 2, di+1 − d1 2(xi+1 − x1 ) (xi+1 − x1 )3 −d1 (xi+1 − x1 ) Bi = , i = 1, 2, 3(xi+1 − x1 )2 di+1 − d1 ai =
and | · | = det(·) (“·” refers to any 2-step matrix). This smooth function g(x), is shown by the red curve in figs. 2(a) and (b). The corresponding potential energies are shown in figs. 2(c) and (d). Obviously, we can see a satisfactory agreement between the smooth function and the original nonlinear problem in fig. 2. Further, the nonlinear restoring force g(x) is more approximate than the one of the trilinear system. So, in the next sections, we can use the new force g(x) to study the change in dynamics. 3.1 Hamiltonian function Let x˙ = y, the unperturbed approximate system can be written as x˙ = y . y˙ = g(x)
(6)
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(a)
(b) Γ4 Γ1
y
Γ3
Γ2
x2
x
x1
x
x3
Fig. 3. (a) Phase portraits for the Hamiltonian function (7) of the approximate system (6) and (b) the details of the homoclinic orbit of system (7).
The Hamiltonian function of eq. (6) can be obtained and written in the following form:
x 1 2 H(x, y) = y − g(x)dx 2 0 ⎧d 2 ⎪ (x − x2 )2 + M1 , ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a1 (x − x1 )4 + b1 (x − x1 )3 + 1 2 ⎨4 3 = y − ⎪ a2 2 b 2 ⎪ ⎪ (x − x1 )4 + (x − x1 )3 + ⎪ ⎪ 4 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d3 (x − x )2 + M , 3 2 2 where Mi =
x ≤ x2 d1 (x − x1 )2 , 2 d1 (x − x1 )2 , 2
x2 ≤ x ≤ x1 ,
(7)
x1 ≤ x ≤ x3 x3 ≤ x
ai bi d1 (xi+1 − x1 )4 + (xi+1 − x1 )3 + (xi+1 − x1 )2 , 4 3 2
i = 1, 2.
3.2 The solutions of homoclinic orbit Because the function g(x) is piecewise at the three points x1 , x2 and x3 , the homoclinic orbit is divided into four parts, (−,2) (−,2) (2,1) (2,1) (1,3) (1,3) for x ≤ x2 , Γ2 x− , x˙ ± for x2 < x ≤ x1 , Γ3 x+ , x˙ ± for x1 < x < x3 marked the orbit as Γ1 x− , x˙ − (3,+) (3,+) for x ≥ x3 . (See fig. 3.) and Γ4 x+ , x˙ + (2,1) (2,1) has been obtained as a parametric function of the The analytical form of the homoclinic orbits Γ2 x− , x˙ ± time variable t, ⎧ √ ⎪ 4Ce Ct ⎪ (2,1) ⎪ √ + x1 ⎪ x− = ⎪ ⎨ (B1 − e Ct )2 − 4A1 C (8) ,
√ √ √ 3 3 ⎪ ⎪ 2 e Ct 2 e2 Ct (B − e Ct ) 8C 4C 1 ⎪ (2,1) ⎪ ⎪ ⎩ x˙ ± = ± (B − e√Ct )2 − 4A C − ((B − e√Ct )2 − 4A C)2 1 1 1 1 where t ∈ (−∞, −t1 ] ∪ [t1 , +∞), 1 1 1 1 + A + B + 2C + B t1 = √ ln 2 C C , 1 1 1 (x2 − x1 )2 (x2 − x1 ) (x2 − x1 ) C A1 =
a1 , 2
B1 =
2b1 , 3
C = d1 .
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(1,3) (1,3) can be written as Similarly, the homoclinic orbits Γ3 x+ , x˙ ± ⎧ √ Ct ⎪ 4Ce ⎪ (1,3) ⎪ √ + x1 x+ = − ⎪ ⎪ Ct 2 ⎨ (B2 − e ) − 4A2 C ,
√ √ √ 3 3 ⎪ ⎪ 8C 2 e2 Ct (B2 − e Ct ) 4C 2 e Ct (1,3) ⎪ ⎪ ⎪ ⎩ x˙ ± = ± (B − e√Ct )2 − 4A C − ((B − e√Ct )2 − 4A C)2 2 2 2 2
(9)
where t ∈ (−∞, −t2 ] ∪ [t2 , +∞), 1 1 1 1 + A + B + 2C + B t2 = √ ln 2 C C , 2 2 2 2 (x3 − x1 ) (x3 − x1 ) (x3 − x1 ) C 2b2 , C = d1 . 3 (−,2) (−,2) have been found as The homoclinic orbits Γ1 x− , x˙ − A2 =
a2 , 2
B2 =
⎧ 2M1 ⎪ ⎨ x(−,2) = − cos −d2 t + x2 − −d2 , ⎪ ⎩ (−,2) x˙ − = 2M1 sin −d2 t
(10)
where t ∈ (−t1 , t1 ). (3,+) (3,+) can be written similarly as The homoclinic orbits Γ4 x+ , x˙ + ⎧ 2M2 ⎪ ⎨ x(3,+) = cos −d3 t + x3 + −d3 , ⎪ ⎩ (3,+) x˙ + = − 2M2 sin −d3 t
(11)
where t ∈ (−t2 , t2 ).
4 The dynamics of perturbed system In this section, we focus our attention on the existence of chaos for system (3) perturbed by a viscous damping and an external harmonic excitation. The Melnikov method will be used to investigate the Melnikov boundary beyond which chaos will be generated probably. Consider a viscous damping ξ and an external excitation of amplitude f with frequency ω, system (3) can be written in the form 1 − ρ = f cos ωt. (12) x ¨ + ξ x˙ + x 1 − √ x2 + α2 The nonlinear restoring force f (x) = ρ − x 1 − √x21+α2 can be replaced by the approximate function g(x) and system (12) yields x ¨ + ξ x˙ − g(x) = f cos ωt. (13) Thus, it is of interest to investigate the existence of chaos for system (13), so the Melnikov function is introduced to find the conditions that chaos maybe occur. Let x˙ = y, eq. (13) can be expressed as x˙ = y . (14) y˙ = −ξy + g(x) + f cos ωt
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We introduce the following notation: F (X) =
y g(x)
,
G(X, t) =
0 −ξy + f cos ωt
,
x X= . y
From the homoclinic orbit, we can obtain tr(DF ) = tr
0 1 g (x) 0
= 0,
and 2 (t) + y± (t)f cos ω(t + t0 ), F (X± (t))ΛG(X± (t), t + t0 ) = −ξy±
where the operator Λ is defined as aΛb = a1 b2 − a2 b1 , for any a = (a1 , a2 )T and b = (b1 , b2 )T . The Melnikov function is given by
M (t0 ) =
+∞
F (X± (t))ΛG(X± (t), t + t0 )dt
−∞ +∞
(−ξ x˙ 2± (t) + x˙ ± (t)f cos ω(t + t0 ))dt
=
−∞
+∞
= −∞
= −ξ
−ξ x˙ 2± (t)dt
+∞
+ −∞
x˙ ± (t)f cos ω(t + t0 )dt
+∞ −∞
− f sin ωt0
x˙ 2± (t)dt + f cos ωt0
+∞ −∞
x˙ ± (t) cos ωt dt
+∞ −∞
x˙ ± (t) sin ωt dt
= −ξI1 (ω) + f cos t0 I2 (ω) − f sin t0 I3 (ω) = −ξI1 (ω) + f I22 (ω) + I32 (ω) cos ω(t0 + ϕ),
(15)
where
I1 (ω) = I3 (ω) =
+∞
−∞
+∞ −∞
x˙ 2± (t)dt, x˙ ± (t) sin ωt dt,
I2 (ω) = ϕ=
+∞
−∞
x˙ ± (t) cos ωt dt,
I2 (ω) 1 arc cot . ω I3 (ω)
The equation M (t0 ) = 0 has simple zero for t0 if and only if the following inequality yields I1 (ω) f > ξ 2 . I2 (ω) + I32 (ω) The Melnikov boundary for system (13) obtained for α = 0.3, ρ = 0.2 is shown in fig. 4(a), and for α = 0.3, ρ = 0.0 is shown in fig. 4(b), for the different values of parameters ξ = 0.015, ξ = 0.030, ξ = 0.045 and ξ = 0.060. In the area beyond the boundaries, chaos will exist. Especially, the Melnikov boundary for system (13) at ρ = 0.0 can be obtained by an implicit analytical analysis (see details in appendix A).
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1.2
(a)
ξ=0.060 ξ=0.045 ξ=0.030 ξ=0.015
1.0
f
0.5
(b)
ξ=0.060 ξ=0.045 ξ=0.030 ξ=0.015
0.8
0.4
0.0 0.0
0.6
1.2
1.8
0.0 2.4 0.0
1.4
ω
2.8
ω
Fig. 4. Chaotic boundaries for system (13) detected by using the Melnikov method. The thin solid, dotted, dashed and the thick solid curves mark the boundaries for ξ = 0.015, ξ = 0.030, ξ = 0.045 and ξ = 0.060, (a) for α = 0.3, ρ = 0.2, (b) for α = 0.3, ρ = 0.0.
6
Lyapunov exponent
x
(a)
6
4
4
2
2
0
0
-2
-2
0.0 0.08
0.5
1.0
(c)
1.5 0.0 0.08
0.04
0.04
0.00
0.00
-0.04 0.0
0.5
f
1.0
-0.04 1.5 0.0
(b)
0.5
1.0
1.5
1.0
1.5
(d)
0.5
f
Fig. 5. Bifurcation diagrams for displacement x versus the external forcing amplitude f for ξ = 0.060, α = 0.3, ρ = 0.2, ω = 1.08 for (a) the original system (12) calculated numerically using the Runge-Kutta method of fourth order, and (b) the approximate system (13) constructed by the semianalytical method.
5 Numerical simulation In this section, the numerical simulation is used to obtain the bifurcation diagrams and the Lyapnnov exponents diagrams shown in fig. 5 for ρ = 0.0 and fig. 7 for ρ = 0.0, and the chaotic attractors are displayed in fig. 6 for ρ = 0.0 and fig. 8 for ρ = 0.0. Figure 5 demonstrates the comparison between the bifurcation diagrams calculated for the original system (12) and its smooth approximation (13) for ξ = 0.060, α = 0.3, ρ = 0.2 and ω = 1.08. As can be seen in fig. 5(a), no chaotic motion is observed below the boundary predicted by the Melnikov method which is marked by the red dashed line in fig. 5(b).
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(a)
1
1
y
0
0
-1
-1
-3
-2
-1
x
0
1
(b)
-3
2
-2
-1
x
0
1
2
Fig. 6. Chaotic attractors for the two systems at ξ = 0.060, α = 0.3, ρ = 0.2, ω = 1.08 and f = 0.5, (a) for the original system (12), and (b) for the approximate system (13).
4
Lyapunov exponent
x
(a)
4
2
2
0
0
-2
-2
0.0 0.06
0.5
1.0
(c)
1.5 0.0 0.06
0.03
0.03
0.00
0.00
-0.03 0.0
0.5
f
1.0
-0.03 1.5 0.0
(b)
0.5
1.0
1.5
1.0
1.5
(d)
0.5
f
Fig. 7. Bifurcation diagrams for displacement x versus the external forcing amplitude f for ξ = 0.060, α = 0.3, ρ = 0.0, ω = 1.08 for (a) the original system (12), and (b) the approximate system (13).
The structure of the typical chaotic attractor demonstrated in fig. 6 differs significantly from the SD oscillator. The attractors displayed in fig. 6 are calculated for the original system (12) (fig. 6(a)) and the smooth approximation (13) (fig. 6(b)) for the same parameters ξ = 0.060, α = 0.3, ρ = 0.2, ω = 1.08 and f = 0.5. Figure 7 depicts the bifurcation diagrams for displacement x versus the external forcing amplitude f for ξ = 0.060, α = 0.3, ρ = 0.0 and ω = 1.08. A good degree of correlation is obtained both in bifurcation diagrams and Lyapunov exponents shown in fig. 7 (more approximate than the trilinear approximation [21]), and the structure of chaotic attractors shown in fig. 8. In fig. 7(a), no chaotic motion is observed below the boundary predicted by the Melnikov method which is marked by the red dashed line in fig. 7(b). As can be seen in figs. 5, 6, 7 and 8, a good degree of correlation is obtained in both the attractors’ topological structure and the Lyapunov exponents.
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2
y
(a)
1
1
0
0
-1
-1
-2
-4
(b)
2
-3
-2
x
-1
0
1
2
-2
-4
-3
-2
-1
x
0
1
2
Fig. 8. Chaotic attractors for the two systems at ξ = 0.060, α = 0.3, ρ = 0.0, ω = 1.08 and f = 0.84, (a) for the original system (12), and (b) for the approximate system (13).
6 Conclusions A new and alternative approximate approach has been proposed to investigate the complicated nonlinear dynamics of the smooth-and-discontinuous oscillator under constant excitation. A smooth approximation of this oscillator, which is topologically equivalent to the original system, has been constructed to estimate approximately the dynamical behaviors of the CSD system. The approach presented here overcomes the barrier of constant excitations and irrational nonlinearity that requires solving the complicated integrals (A.4). For the perturbed system, the chaotic boundaries of the approximate system are obtained both for ρ = 0 and ρ = 0, which is in excellent agreement with the numerical simulations. The results of numerical simulations show the chaotic attractors of the approximate system have the same structure as the original system, while the chaotic attractors of the CSD oscillator are significantly different from the ones of the SD oscillator. The authors acknowledge the financial support of the National Natural Science Foundation of China (No 11002093, 11172183, 10932006 and 11202142) and the Science and technology plan project of Hebei Science and Technology Department (No 11215643).
Appendix A. System (12) can be written as X˙ = F (X)X + G(X),
(A.1)
where y x , y = x, ˙ F (X) = , y f (x) 0 1 + ρ, g(X) = . f (x) = −x 1 − √ −ξy + f cos ωt x2 + α2 X=
The Melnikov function is given by
M (t0 ) =
+∞ −∞ +∞
= −∞
[F (x± (t))ΛG(x± (t), t + t0 )] dt −ξ(y± (t))2 + y± (t)f cos ω(t + t0 ) dt,
(A.2)
where x± (t), y± (t) present the displacement and the velocity of the homoclinic orbit, respectively. The Hamiltonian function of system (3) is y2 x2 2 H(x, y) = + − x + α2 − ρx + α, 2 2
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because H(0, 0) = 0, we can get the function y(t) = ± 2 x2 + α2 − x2 + 2ρx − 2α ,
(A.3)
then (A.3) can be rewritten as dx . ±dt = √ 2 2 2 x + α − x2 + 2ρx − 2α
(A.4)
This is an implicit function for time t of x, but it can not be inverted. Even for the value of ρ = 0.0, we can investigate the function as follows:
∞ (F (x± (t)) ∧ G(x± (t), t + t0 ))dt M± (t0 ) =
−∞ ∞
= −∞
(−ξ x˙ 2± (t) + x˙ 2± (t)f cos ω(t + t0 ))dt
(A.5)
= −4ξA − 4f B sin ωt0 , where
0
A=
x2 + α2 − x2 − 2α dx,
x0
0
sin ωt dx.
B= x0
Let u =
√
x2 + α2 − α2 , then
0
A=
x0
0
= u0
= lim
u→0
x2 + α2 − x2 − 2α dx
2(1 − α)u − u2 (u + α)du √ , u2 + 2αu
(A.6)
N0 (u, α) N3 (u, α) uN1 (u, α)
and
x
t= x0
dx √ x2 + α2 − x2 − 2α
u
√
= u0
=−
u2
(u + α)du , + 2αu 2(1 − α)u − u2
N0 (u, α) ln(2αN2 (u, α)), uN1 (u, α)
where u(u + 2α) u(2 − 2α − x), N1 (u, α) = 2 α(1 − α) 2u − 4αu − u2 + 4α − 4α2 , N1 (u, α) + α(1 − α) arcsin(u + 2α − 1), N2 (u, α) = 4α − 4α2 − 2αu + u + x N3 (u, α) = 2N1 (u, α) + α(1 − α)((1 − 2α)) arcsin(u + 2α − 1), √ x0 = 2 1 − α, u0 = x20 + α2 − α2 .
N0 (u, α) =
(A.7)
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Eur. Phys. J. Plus (2013) 128: 80
So, the equation M (t0 ) = 0 has simple zero for if and only if the following inequality yields: f |A| > ξ |B| .
(A.8)
|A| In the area above the boundaries, chaos will happen in the area of | fξ | > |B| . Note the value of A and B are obtained with the help of the numerical integration here. However, it is still a barrier for us to integrate the eq. (A.4) generally for ρ = 0 in both theoretical derivation and numerical integration. Hence, the Hamiltonian (4) is not readily analysed. To allow further theoretical investigation of trajectories (particularly under harmonic excitation), it is useful to make a smooth approximation (introduced in this paper) to the irrational restoring force curve.
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