Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 3, pp. 701-715, Warsaw 2012 50th Anniversary of JTAM IRRATIONAL ELLIPTIC FUNCTIONS AND THE ANALYTICAL SOLUTIONS OF SD OSCILLATOR Qingjie Cao Centre for Nonlinear Dynamics Research, School of Astronautics, Harbin Institute of Technology, Harbin, China, and Centre for Nonlinear Dynamics Research, Shijiazhuang Tiedao University, Shijiazhuang, China; e-mail: q.j.cao@hit.edu.cn Dan Wang, Yushu Chen Centre for Nonlinear Dynamics Research, School of Astronautics, Harbin Institute of Technology, Harbin, China Marian Wiercigroch Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, King’s College, Aberdeen, Scotland, UK The smooth and discontinuous (SD) oscillator is a strongly nonlinear system with an irra- tional restoring force proposed in P.R.E (2006), which leads to barriers for the conventional methods to investigate the dynamical behaviour directly. In this paper, two kinds of irra- tional elliptic functions and a kind of hyperbolic functions are defined in the real domain to formulate the analytical solutions of the system.The properties of the functions are obtained including differentiability, periodicity and parity.As the application of the defined irrational functions, the chaotic thresholds of the oscillator are also depicted by using the Melnikov method. Numerical analysis shows the efficiency of the proposed procedure. Key words: SD oscillator, irrational nonlinearity, irrational elliptic functions, threshold of chaos 1. Introduction The SD oscillator is a typical strong nonlinear systemwith an irrational restoring force (Cao et al., 2006), which is widely used in engineering, such as the cable-stayed bridge, isolation system and truss-structure and soonZhangandSun (2005),Yang andYang (2008), LuhandLin (1011), Hajirasouliha et al. (2011), Kirsch (1989). Based upon Taylor’s expansion, the conventional theoretical methods are confined by the harsh conditions of locality and smoothness. The global dynamical structure and local behaviour of the SD oscillator with irrational nonlinearity can hardly be depicted precisely. It is imperative to develop methodologies for the key problems in engineering. Methodologies can be found to get the approximate solutions of the irrational nonlinear systems in the literature. The homotopy perturbationmethod (He, 2006; Liao, 2004) was used to get an approximate expression for the periodic solutions of a irrational nonlinear system, and the harmonic balance method (Belendez et al., 2007, 2009) via the first Fourier coefficient is used to construct two approximate frequency-amplitude relations for a conservative nonlinear oscillatory system which has an irrational restoring force. Analytical approximations were investigated for the systemwhichhas an irrational restoring force by employing a combined method of Newton’s approach and the harmonic balance technique (Wu et al., 2003, 2004, 2006; Sun et al., 2007). A generalized Senator-Bapat (GSB) perturbation technique (Lai and Xiang, 2010) was given to solve the conservative oscillating system with an irrational nonlinearity. Approximate qualitative analyses were given for the SDoscillator in the past studies (Cao et al., 2006, 2008a,b; Tian et al., 2009, 2012). A triple linear approach (Cao et al., 2008a) was used 702 Q. Cao et al. to theoretically investigate the dynamics of the irrational nonlinear oscillator. The codimension- two bifurcation for the SD oscillator was proposed and studied in Tian et al. (2009). Periodic solution analysis was given (Cao et al., 2011) by using the averaging theorem in Nayfrh (1981). However, theproblemof analytical solutions for the oscillator given inCao et al. (2008a) remains open. Themotivation of this paper is to provide a solution to the open problemproposed inCao et al. (2008a) for irrational integrals.We propose a series of irrational elliptic functions (Greenhill, 1959;Whittaker andWatson, 1952) and hyperbolic functions defined in the real domain, which give analytical periodic solutions and homoclinic solutions of the irrational system. We discuss the chaotic threshold of the oscillator by the Melnikov methods with the hyperbolic functions defined in this paper. Meanwhile, the basic properties of the irrational elliptic functions and hyperbolic functions are gained as a part of the applied mathematic theory. This paper is organized as follows. In Section 2, brief introduction to the open problem proposed inCao et al. (2008a) is given. In the following section, Section 3, two kinds of irrational elliptic functions and a kind of hyperbolic functions are defined and the basic properties are obtained, which allows us to get the analytical solutions of the SD oscillator. Furthermore, in Section 4, the chaotic threshold and attractors of the oscillator are given by employing the Melnikov method as the application of the defined functions, and finally this paper is ended with conclusions and discussions. 2. Open problem of the SD oscillator Consider the non-dimensional system of the unperturbed SD oscillator, written as x′′+x ( 1− 1√ x2+α2 ) =0 (2.1) which was firstly proposed and investigated in Cao et al. (2008a). This system is strongly irra- tional nonlinear with smooth (α> 0) and discontinuous (α=0) behaviour. A pitchfork bifurcation occurs for the equilibria of the system at α = 1: a pair of centres (±1,0) anda saddle-like (0,0) co-existed for α=0, a pair of centres (± √ 1−α2,0) anda saddle (0,0) for 0<α< 1 and a unique center (0,0) for α­ 1. Even some effective works have been achieved, a triple linear systemwas proposed to get the approximate solutions Cao et al. (2008a), an equivalent form (Tian et al., 2012) was presented to get a step forward to the analytical solutions of the system and the GSB method (Lai and Xiang, 2010) was employed to solve the system, the theoretical solution of system (2.1) still remains open, the details seen in Cao et al. (2008a). 3. The definitions Generally, it is difficult to get the analytical solutions of the irrational system precisely. Here definitions are given to get the analytical solutions of the oscillator in the following sections. 3.1. Irrational elliptic functions of the first kind 3.1.1. Definitions The Hamilton function of (2.1) can be written as H =U(x)+ 1 2 y2 (3.1) Irrational elliptic functions... 703 where ẋ= y U(x)= 1 2 x2− √ x2+α2+ 1+α2 2 The phase portraits are plotted for different values of H(x,y) as shown in Fig. 1, the details seen in the corresponding captions. Fig. 1. Phase portraits: (a) for α=0.4 with the pair of centres and saddle, (b) for α=0with the pair of centres and saddle-like equilibrium (Cao et al., 2006) and (c) for α=1with the unique centre, respectively Introducing a notation k2/2=H, it follows H = 1 2 k2 = 1 2 y2+U(x) (3.2) Denoting x0 as the maximum intersect point of orbits with the x-axis, it follows that −2 √ 1−α ¬ x0 ¬ 2 √ 1−α for k ¬ 1−α and the orbits can be classified by k which is determined by x0, as shown in Table 1. Table 1.UnperturbedOrbits classified by parameter k Orbits Centres Small periodic orbits Homoclinic orbits Large periodic orbits k k=0 0 1−α 2H 2H =0 0< 2H < (1−α)2 2H =(1−α)2 2H > (1−α)2 Time τ from (x0,0) to (x,y) along the lower branch of the periodic orbit can be obtained by the following integral τ =− x ∫ x0 1 √ k2− ( √ x2+α2−1)2 dx (3.3) Letting x=± √ (kcosϕ+1)2−α2 (3.4) where ϕ∈ [0,π], k∈ [0,1−α), it follows that the time τ expressed in (3.3) is rewritten as τ = ϕ ∫ 0 1 √ 1− α2 (1+kcosϕ)2 dϕ (3.5) and denote I(k,α) = π ∫ 0 1 √ 1− α2 (1+kcosϕ)2 dϕ (3.6) 704 Q. Cao et al. The definition of ϕ= amτ is said to the angular form of τ, and the irrational elliptic functions of the first kind are defined as follows sd(τ,k,α) , sinϕ=sin(amτ) ad(τ,k,α) , cosϕ=cos(amτ) hd(τ,k,α) , √ [1+kcos(amτ)]2−α2 (3.7) 3.1.2. Properties The fundamental properties of irrational elliptic functions of the first kind defined above are listed bellow by denoting hdτ, sdτ, adτ instead of hd(τ,k,α), sd(τ,k,α), ad(τ,k,α) without confusion. (1) Identity sd2τ+ad2τ =1 hd2τ+α2 =(1+kadτ)2 (2) Parity From integral (3.5) and the above definitions, one can obtain that angle ϕ is an odd function of the time variable τ, sdτ is an odd function of τ and hdτ, adτ are even functions of τ, then sd(−τ)=−sdτ hd(−τ)=hdτ ad(−τ)= adτ (3) Differentiability From integral (3.5) and above definitions, the differentiation of the angle ϕ and the irra- tional elliptic functions of the first kind sdτ, hdτ, adτ are given as follows ∂ϕ ∂τ = hdτ 1+kadτ ∂sdτ ∂τ = adτhdτ 1+kadτ ∂adτ ∂τ =− sdτhdτ 1+k √ 1−sd2τ ∂hdτ ∂τ =−ksdτ (4) Periodicity The periodicity of the irrational functions can be obtained andwritten as the following by considering the periodicity of the integrand of (3.5) of period of 2π. When ϕ= amτ τ → τ+2I(k,α) am(τ +2I(k,α)) = amτ +2π am(−τ +2I(k,α)) =−amτ+2π then sd(τ +2I(k,α)) = sdτ sd(−τ+2I(k,α)) =−sdτ ad(τ +2I(k,α)) = adτ ad(−τ+2I(k,α)) = adτ which leads to that 2I(k,α) is the period of sdτ and adτ and follows that 2I(k,α) is the period of hdτ, that is hd(τ +2I(k,α)) =hdτ hd(−τ +2I(k,α)) =hdτ Irrational elliptic functions... 705 The graphs of the elliptic functions are plotted in Fig. 2, (a) for sd(t), (b) for ad(t) and (c) for hd(t), respectively, for parameters α=0.6 and k=0.1. Fig. 2. Graphs for the elliptic functions of the first kind: (a) for sd(t), (b) for ad(t) and (c) for hd(t) when parameters α=0.6 and k=0.1 Special values of the irrational elliptic functions of the first kind are listed in Table 2. Table 2. Special values of the irrational elliptic functions of the first kind ϕ=0, τ =0 ϕ=π, τ = I(k,α) sd(0)= 0, ad(0)= 1 sd(I)= 0, ad(I)=−1 hd(0) = √ (1+k)2−α2 hd(I) = √ (1−k)2−α2 The Taylor expansion of the integrand of Eq. (3.6) is written as 1 √ 1− α2 (1+kcosϕ)2 = 1√ 1−α2 − kα2 √ (1−α2)3 cosϕ+ 3k2α2 2 √ (1−α2)5 cos2ϕ (3.8) − 5k3 2 √ (1−α2)7 cos3ϕ+ 5k4α2(4+3α2) 8 √ (1−α2)9 cos4ϕ− 3k5α2(8+12α2+α4) 8 √ (1−α2)11 cos5ϕ+ · · · which leads to I(k,α) = 3k2α2π 4 √ (1−α2)5 + π√ 1−α2 − 15k 4α2(3α4+α2−4)π 64 √ 1−α2 + · · · (3.9) Equation (3.5) is rewritten as τ = z ∫ 0 1 √ 1− α2 (1+kcosϕ)2 dϕ (3.10) which leads to that τ can be written as follows τ = az+ bz3+ cz5+ · · · (3.11) where a= 1+k√ 1+2k+k2−α2 b= kα2 6 √ (1+2k+k2−α2)3 c= kα2(α2+8k2+7k−1) 120 √ (1+2k+k2−α2)5 Then, the anti-series form of τ in terms of the variable z in Eq. (3.11) can be written as z=Aτ+Bτ3+Eτ5+ · · · (3.12) 706 Q. Cao et al. where A= 1 a B=− b a4 E= 3b2−ac a7 The series forms of the irrational elliptic functions are given as follows sdτ = s1τ−s3τ3+s5τ5+ · · · (3.13) where s1 = √ 1+2k+k2−α2 1+k s3 = √ 1+2k+k2−α2 3!(1+k)4 (1+3k+3k2+k3−α2) s5 = √ 1+2k+k2−α2 5!(1+k)7 [15k4+6k5+k6+(α2−1)2 −3k2(5+4α2)+k3(20+7α2)+k(6+3α2−9α4)] adτ =1−a2τ2+a4τ4−a6τ6+ · · · (3.14) where a2 = (1+k)2−α2 2!(1+k)2 a4 = (1+k)2−α2 4!(1+k)5 [1+3k2+k3−α2+3k(1+α2)] a6 = 1 6!(1+k)8 [8k7+k8− (α2−1)3+8k(α2−1)2(1+3α2)−8k5(13α2 −7) −k6(31α2−28)+k4(70−109α2 +75α4)−8k3(7−2α2+13α4) +k2(28+31α2−14α4−45α6)] Furthermore, the series form of the analytical solutions of the periodic orbits is hdτ = p0−p2τ2+p4τ4−p6τ6−··· (3.15) where p0 = √ (1+k)2−α2 p2 = k √ (1+k)2−α2 2!(1+k) p4 = k √ (1+k)2−α2 4!(1+k)4 (1+3k+3k2+k2−α2) p6 = k √ (1+k)2−α2 6!(1+k)7 [15k4+6k5+k6+(1−α2)2+3k2(5+4α2) +k3(20+7α2)+k(6+3α2−9α4)] 3.1.3. The analytical expression of the periodic orbits When 0¬ k< 1−α, the analytical expression of the periodic orbits can be written as x=±hd(τ,k,α) (3.16) Irrational elliptic functions... 707 The first and the second derivatives of the solution x are obtained by the differential properties of irrational elliptic functions of the first kind, respectively dx dτ =∓ksdτ d 2x dτ2 =∓hdτ ( 1− 1√ hd2τ+α2 ) =−x ( 1− 1√ x2+α2 ) (3.17) which leads to x = ±hd(τ,k,α) is the periodic solution inside the homoclinic orbits of the oscillator. Moreover, the period of these orbits is given by Tk = 2I(k,α), and Tk increases monotonically in k with lim k→0 Tk = 2π√ 1−α2 lim k→1−α Tk →∞ For I(k,α), when k=0, then lim k→0 I(k,α) = π ∫ 0 1√ 1−α2 dx= π√ 1−α2 (3.18) and when α=0, then I(k,α) =π. 3.2. The hyperbolic functions 3.2.1. Definition TheHamiltonian k2/2= y2/2+U(x) represents the energies of the homoclinic orbits when k=1−α. Let τ = ϕ ∫ 0 1 √ 1− α2 [1+(1−α)cosϕ]2 dϕ (3.19) and the hyperbolic functions of the oscillator are defined as follows ϕ= tamτ lim k→1−α sd(τ,k,α) , tsd(τ,α) lim k→1−α ad(τ,k,α) = √ 1− tsd2(τ,α) , tad(τ,α) lim k→1−α hd(τ,k,α) = √ [(1−α)tad(τ,α)+1]2−α2 , thd(τ,α) (3.20) where 0 ¬ α < 1 and 0 ¬ k < 1−α, and the graphs of the hyperbolic functions thd(t) and tsd(t) are shown in Fig. 3. Fig. 3. Graphs of the hyperbolic function for α=0.4: (a) for thd(t) and (b) for tsd(t) 708 Q. Cao et al. 3.2.2. Properties The fundamental properties of the hyperbolic functions are obtained as following. (1) Identity tsd2τ+ tad2τ =1 thd2τ +α2 = [1+(1−α)tadτ]2 (2) Parity It can be seen that tsdτ is an odd function of τ and thdτ, tadτ are even functions of τ due to the odd property of ϕ, that is tsd(−τ)=−tsdτ thd(−τ)= thdτ tad(−τ)= tadτ (3) Differentiability Differentiation of the hyperbolic functions are gained by the definitions and identity pro- perties as follows ∂ϕ ∂τ = thdτ 1+(1−α)tadτ ∂tsdτ ∂τ = tadτthdτ 1+(1−α)tadτ ∂tadτ ∂τ =− tsdτthdτ 1+(1−α) √ 1− tsd2τ ∂thdτ ∂τ =−(1−α)tsdτ 3.2.3. Analytical expressions of the homoclinic orbits When k=1−α, the analytical expressions of the homoclinic orbits can be written as x=±thd(τ,k,α) (3.21) The first and second derivatives of the solution are obtained by the differential properties of the hyperbolic functions, respectively dx dτ =∓(1−α)tsdτ d2x dτ2 =∓thdτ ( 1− 1√ thd2τ +α2 ) =−x ( 1− 1√ x2+α2 ) (3.22) which leads to that x=±thd(τ,k,α) is the analytical solution of the homoclinic orbits of the SD oscillator. 3.3. Irrational elliptic functions of the second kind 3.3.1. Definition When α ­ 1, (0,0) is the unique centre and the phase portrait is shown in Fig. 1c. The Hamiltonian is given as follows H = 1 2 k2 = 1 2 y2+U(x)­ (1−α) 2 2 (3.23) where k >α−1 and α­ 1, then the time τ from (x0,0) to (x,y) along the lower branch of the periodic orbits can be obtained by the following τ =− x ∫ x0 1 √ k2− ( √ x2+α2−1)2 dx (3.24) Irrational elliptic functions... 709 where x0 is the maximum intersect point of the orbits with the x-axis. Let x=± √ (kcosϕ+1)2−α2 (3.25) where ϕ∈ [0,ϕ0] for x∈ [0,x0] and ϕ0 =arccos[(α−1)/k]. Integration (3.24) can be simplified as τ = ϕ ∫ 0 1 √ 1− α2 (1+kcosϕ)2 dϕ (3.26) where the time variable τ is a function of k, ϕ, α. Denote W(k,α) = ϕ0 ∫ 0 1 √ 1− α2 (1+kcosϕ)2 dϕ (3.27) Define ϕ=Amτ (3.28) which means ϕ as the angle of τ. Furthermore, irrational elliptic functions of the second kind are defined as follows Sd(τ,k,α) , sinϕ=sin(Amτ) Ad(τ,k,α) , cosϕ=cos(Amτ) Hd(τ,k,α) , √ [1+kcos(Amτ)]2−α2 (3.29) 3.3.2. Properties The fundamental properties of irrational elliptic functions of the second kind defined above are listed bellow by denoting Hdτ, Sdτ, Adτ instead of Hd(τ,k,α), Sd(τ,k,α), Ad(τ,k,α) without confusion. (1) Identity Sd2τ +Ad2τ =1 Hd2τ+α2 =(1+kAdτ)2 (2) Parity It can be seen that Sdτ is an odd function of τ and Hdτ, Adτ are even functions of τ due to odd property of ϕ, that is Sd(−τ)=−Sdτ Hd(−τ)=Hdτ Ad(−τ)=Adτ (c) Differentiability Derivatives of ϕ, Sdτ,Adτ,Hdτ with respect to the time variable τ are given as follows ∂ϕ ∂τ = Hdτ 1+kAdτ ∂Sdτ ∂τ = AdτHdτ 1+kAdτ ∂Adτ ∂τ =− SdτHdτ 1+k √ 1−Sd2τ ∂Hdτ ∂τ =−kSdτ 710 Q. Cao et al. (4) Periodicity For the symmetry of the phase portraits, the period T of the solution is four times of the time taken from x=x0 to x=0, which leads to the period of Hdτ is 4W(k,α), that is Hd(τ+4W(k,α)) =Hdτ Hd(−τ +4W(k,α)) =Hdτ The differential properties of the irrational elliptic functions of the second kind leads to the period of Sdτ is 4W(k,α) as follows ∂Hd(τ +4W(kα)) ∂τ =−kSd(τ+4W(k,α)) = ∂Hdτ ∂τ =−kSdτ (3.30) that is Sd(τ +4W(k,α)) =Sdτ Sd(−τ +4W(k,α)) =−Sdτ Furthermore, the period of Adτ can be obtained as 4W(k,α), that is Adτ = √ 1−Sd2τ = √ 1−Sd2(τ +4W(k,α)) =Ad(τ +4W(k,α)) (3.31) and Ad(τ +4W(k,α)) =Adτ Ad(−τ+4W(k,α)) =Adτ Special values of the irrational elliptic functions of the second kind are listed in Table 3. Table 3. Special values of the irrational elliptic functions of the second kind ϕ=0, τ =0 ϕ=π, τ =W(k,α) Sd(0)= 0,Ad(0)= 1 Hd(W)= 0 Hd(0)= √ (1+k)2−α2 Ad(W)= α−1 k , Sd(W)= √ 1− (α−1) 2 k2 3.3.3. Analytical expressions of the periodic orbits When the parameter k > α− 1, the expressions for the periodic orbits of the irrational system can be obtained as x=±Hd(τ,k,α) (3.32) The first and the second derivatives of the solution are obtained by the differential properties of irrational elliptic functions of the second kind, respectively dx dτ =∓kSdτ d2x dτ2 =∓Hdτ ( 1− 1√ Hd2τ+α2 ) =−x ( 1− 1√ x2+α2 ) (3.33) which leads to that x = ±Hd(τ,k,α) is the analytical solution of periodic orbits outside the centre (0,0) of the oscillator. Moreover, the period of these orbits is given by Tok =4W(k,α) Irrational elliptic functions... 711 4. Chaotic threshold The perturbed SD oscillator with viscous damping and an external harmonic forcing can be written as follows x′′+2ξx′+x ( 1− 1√ x2+α2 ) = f0cos(ωτ) (4.1) where x′ = y= dx/dτ, and which can be written as x′ = y y′ =−2ξy−x ( 1− 1√ x2+α2 ) +f0cos(ωτ) (4.2) 4.1. Melnikov method of the homoclinic orbits The Melnikov function (Guckenheimer and Holmes, 1983; Garcia-Margallo and Bejarano, 1998) of system (4.2) is given as M±(τ1)= ∞ ∫ −∞ f(x±(τ))∧g(x±(τ),τ + τ1) dτ = ∞ ∫ −∞ {−2ξy±(τ)2+y±(τ)f0cos[ω(τ + τ1)]} dτ (4.3) =−2ξ ∞ ∫ −∞ y±(τ) 2 dτ +f0 ∞ ∫ −∞ y±(τ)cos[ω(τ+ τ1)] dτ The following discussion concentrates on M+(τ1) for symmetry of the system.When 0¬α< 1, one can get the displacement and velocity of the homoclinic orbit as x= thd(τ) y=x′ =−(1−α)tsd(τ) (4.4) Then, theMelnikov function of the system can be obtained as follows M±(τ1)=−4ξA±2f0 sin(ωτ1)B (4.5) where A=(1−α)2 ∞ ∫ 0 tsd(τ)2 dτ B=(1−α) ∞ ∫ 0 tsd(τ)sin(ωτ) dτ =(1−α)πiRes[tsd(τ)sinωτ, iH(α)] where Res[f,z] means the residue of f at the point z. It can be seen that M±(τ1)= 0 has simple zero for τ1 if and only if the following inequality holds ∣ ∣ ∣ f0 ξ ∣ ∣ ∣­ 2 |A| |B| (4.6) when the inequality is satisfied, the Poincaré map of Eq. (4.2) might be chaotic in the sense of Smale horseshoe, the intersection between the stable and unstable manifolds. Furthermore, the thresholds of chaos occurrence are plotted in Fig. 4a: with solid line for α=0, k=1, thin for α = 0.4, k = 0.6 and dashed for α = 0.8, k = 0.2, respectively. Chaos might occur if the parameters over the corresponding curves. 712 Q. Cao et al. Fig. 4. Chaotic thresholds and attractors: (a) chaotic thresholds of the SD oscillator plotted with the solid curve for α=0, k=1, thin for α=0.4, k=0.6, dashed for α=0.8, k=0.2 and dotted for α=1, k=0.4, respectively; (b) attractors for α=0, ξ=0.015, f0 =1.1, ω=1.3; (c) for α=0.4, ξ=0.02, f0 =0.70, ω=1.06 and (d) for α=1, ξ=0.015, f0 =0.80, ω=1.3, respectively 4.2. Melnikov method of the subharmonic orbits One can get the subharmonicMelnikov function of the subharmonic orbitswhen α=1, that is M+1 (τ1)=M − 1 (τ1)= T ∫ 0 {−2ξ(yk(τ))2+yk(τ)f0cos[ω(τ + τ1)]} dτ (4.7) which can be simplified as follows M+1 (τ1)=−8ξP +4f0Q (4.8) where P = k2 W(k,α) ∫ 0 Sd(τ)2 dτ Q= k W(k,α) ∫ 0 Sd(τ)sin(ωτ) dτ It can be seen that M+1 (τ1) = 0 has simple zero for τ1 if and only if the following inequality holds ∣ ∣ ∣ f0 ξ ∣ ∣ ∣­ 2 |P| |Q| (4.9) when the inequality is satisfied, the subharmonic orbits would occur under the sense of Smale horseshoe transform. Furthermore, the curve of threshold is plotted for α=1, k=0.4 with the dotted curve in Fig. 4a. Irrational elliptic functions... 713 4.3. Attractors Numerical analysis is carried out by using the forth order Runge-Kuttamethod (Cartwright and Piro, 1992), the attractors of the SD oscillator in different conditions are given in Figs. 4b and 4c, respectively: (b) for α = 0, ξ = 0.015, f0 = 1.1, ω = 1.3; (c) for α = 0.4, ξ = 0.02, f0 = 0.70, ω = 1.06 and (d) for α = 1, ξ = 0.015, f0 = 0.80, ω = 1.3, which verified the conditions given by the thresholds found in this paper. 5. Conclusions Two kinds of irrational elliptic functions and a kind of irrational hyperbolic functions have been defined for the SD oscillator which is controlled by the parameter α in this paper. The fundamental properties of identity, parity, differentiability and the periodicity of the functions have been obtained and presented. The periodic solutions of the oscillator have been formulated by the irrational elliptic functions and the homoclinic solutions of the oscillator have been given by the hyperbolic functions. The chaotic threshold of the irrational systemhas been obtained by employing theMelnikov functionswith respect to the hyperbolic functions defined in this paper. The irrational elliptic functions defined herein may be applied to formulate a general irrational nonlinear system to get the precise solution; as a part of applied mathematics, further studies formore properties of the irrational functions defined in this paper are being investigated by the current authors. The properties of the defined irrational functions in complex domain remain open. Acknowledgement The first two authors acknowledge the financial supports, under the Grant 10872136, 11072065 and 10932006, National Natural Science Foundation of China. Appendix The method used to get the pole of the hyperbolic function tsd(τ) of Eq. (3.19) is shown as follows. With the half-angle formulae, let t=tan ϕ 2 dϕ dt = 2 1+t2 cosϕ= 1− t2 1+ t2 sinϕ= 2t 1+ t2 Then integral (3.19) is rewritten as τ =2 t ∫ 0 1+ t2+(1−α)(1− t2) (1+ t2) √ [1+ t2+(1−α)(1− t2)]2− (1+ t2)2α2 dt (A.1) Whittaker (1937) proved that the integral of motion is still real when t is replaced by√ −1t and the initial conditions β1, . . . ,βn by √ −1β1, . . . , √ −1βn, respectively, if the force is independent of time. The expression thus obtained represents the same motion with the same initial condition of the system. In this way, the variable is extended to the imaginary axis in the following. Letting τ = iu t= is 714 Q. Cao et al. then, integral (A.1) is written as u=2 s ∫ 0 1−s2+(1−α)(1+s2) (1−s2) √ [1−s2+(1−α)(1+s2)]2− (1−s2)2α2 ds (A.2) which leads to the following definitions tsd(τ,α) = tsd(iu,α) = 2t 1+ t2 = i 2s 1−s2 = i tsd(u,α) tad(u,α) . (A.3) Denoting H(α) = 2 1 ∫ 0 1−s2+(1−α)(1+s2) (1−s2) √ [1−s2+(1−α)(1+s2)]2− (1−s2)2α2 ds (A.4) leads to that u=H(α) when s=1 and tsd(τ,α) has a pole τ = iH(α) when s=1. References 1. Belendez A., Hernandez A., Belendez T., Neipp C., 2007, Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire, Journal of Sound and Vibration, 3, 21018-1029 2. BelendezA.,HernandezA.,BelendezT., NeippC., 2009,Approximate solutions of a nonli- near oscillator typified as amass attached to a stretched elasticwire by the homotopyperturbation method,Chaos, Solitons and Fractals, 39, 746-764 3. Cao Q.J., Wiercigroch M., Pavlovskala E.E., Grebogi C., Thompson J.M.T., 2006, Archetypal oscillator for smooth and discontinuous dynamics,Physical Review E, 74, 046218 4. Cao Q.J., Wiercigroch M., Pavlovskala E.E., Grebogi C., Thompson J.M.T., 2008a, Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics,Phi- los. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366, 635-653 5. Cao Q.J., Wiercigroch M., Pavlovskala E.E., Grebogi C., Thompson J.M.T., 2008b, The limiting case response of the archetypal oscillator for discontinuous dynamics, International Journal of Non-Linear Mechanics, 43, 462-473 6. Cao Q.J., Xiong Y.P., Wiercigroch M., 2011, Resonances behavior of SD oscillator at the discontinuous pPhases, Journal of Applied Analysis and Computation, 1, 183-191 7. Cartwright J.H.E., Piro O., 1992, The dynamics of Runge-Kutta methods, International Jo- urnal of Bifurcation and Chaos, 2, 427-449 8. Garcia-Margallo J., Bejarano J.D., 1998,Melnikov’smethod for non-linear oscillators with nonlinear excitations, Journal of Sound and Vibration, 212, 2, 311-319 9. Greenhill A.G., 1959,The Applications of Elliptic Functions, NewYork, Dover 10. Guckenheimer J., Holmes P., Nonlinear Oscillation. Dynamical System and Bifurcation of Vector Fields, Springer-Verlag, NewYork 11. Hajirasouliha I., Pilakoutas K., Moghaddam H., 2011, Topology optimization for the se- ismic design of truss-like structure,Computers and Structures, 89, 702-711 12. He J.H., 2006, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 1141-1199 13. KirschU., 1989,Optimal topologies for truss structures,ComputerMethods inAppliedMechanics and Engineering, 72, 15-28 Irrational elliptic functions... 715 14. Lai S.K., Xiang Y., 2010, Application of a generalized Senator-Bapat perturbation technique to nonlinear dynamical systems with an irrational restoring force, Computers and Mathematics with Applications, 60, 7, 2078-2086 15. Liao S., 2004,Beyond Perturbation:Introduction to the Homotopy Analysis Method, Boca Raton, FL: CRCPress 16. Luh G.C., Lin C.Y., 2011,Optimal design of truss-structures using particle swarm optimization, Computers and Structures, 89, 2221-2232 17. Nayfeh A.H., 1981, Introduction to Perturbation Techniques, Wiley, NewYork 18. Sun W.P.,Wu B.S., LimC.W., 2007, Approximate analytical solutions for oscillation of amass attached to a stretched elastic wire, Journal of Sound and Vibration, 300, 1042-1047 19. TianR.L., CaoQ.J.,YangS.P., 2009,The codimension-twobifurcation for the recentproposed SD oscillator,Nonlinear Dynamics, DOI 10. 1007/s, 11071-009-9517-9 20. Tian R.L., Yang X.W., Cao Q.J., 2012, Bifurcations and chaotic threshold for a nonlinear systemwith an irrational restoring force,Chinese Physics B, 21, 2, 020503 21. Whittaker E.T., 1937, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Fourth. edn, Cambridge University Press 22. Whittaker E.T., Watson G.N., 1952, A Course of Modern Analysis, Cambridge University Press 23. Wu B.S., Lim C.W., Ma Y.F., 2003, Analytical approximation to large-amplitude oscillation of a non-linear conservative system, International Journal of Non-Linear Mechanics, 38, 1037-1043 24. Wu B.S., Lim C.W., 2004, Large amplitude non-linear oscillations of a general conservative system, International Journal of Non-Linear Mechanics, 39, 859-870 25. WuB.S., SunW.P.,LimC.W., 2006,Ananalytical approximate technique for a class of strongly non-linear oscillators, International Journal of Non-Linear Mechanics, 41, 766-774 26. Yang B.S., Yang C., 2008, The analysis and simulation of passive vibration isolation system using zero dynamic stiffness, [In:]ModularMachine Tool andAutomaticManufacturing Technique, 2265, 01-0053-04 27. Zhang X.J., Sun B.N., 2005, Aerodynamic stability of cable-stayed-suspension hybrid bridges, Journal of Zhejiang University Science A, 6, 8, 869-874 Niewymierne funkcje eliptyczne i rozwiązania analityczne dla oscylatora typu SD Streszczenie Oscylator typu gładkiego i nieciągłego (smooth anddiscontinuous—SD) jest silnie nieliniowymukła- demmechanicznymz niewymierną siłą restytucyjną opisaną przezP.R.E.w 2006 r. Jej charakter stanowi barierę dla konwencjonalnychmetod badania dynamiki oscylatorów SD w sposób bezpośredni.W pracy zdefiniowano dwa rodzaje niewymiernych funkcji eliptycznych i jeden typ hiperbolicznych w dziedzinie liczb rzeczywistychdowyznaczenia rozwiązańanalitycznych rozważanegoukładu.Właściwości tych funk- cji obejmują różniczkowalność, okresowość i parzystość. Tak sformułowanych funkcji, jako przykład ich zastosowania, uzyto do określenia zakresówwystępowania drgań chaotycznych oscylatora przy wykorzy- staniu metodyMielnikowa. Symulacje numeryczne potwierdziły efektywność zaproponowanejmetody. Manuscript received March 12, 2012; accepted for print March 22, 2012