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Harmonic Balance, Melnikov Method and Nonlinear Oscillators Under Resonant Perturbation Michele Bonnin1 ∗ † 1 Politecnico di Torino, Torino, Italy SUMMARY The Subharmonic Melnikov’s method is a classical tool for the analysis of subharmonic orbits in weakly perturbed nonlinear oscillators, but its application requires the availability of an analytical expression for the periodic trajectories of the unperturbed system. On the other hand, spectral techniques, like the Harmonic Balance, have been widely applied to the analysis and design of nonlinear oscillators. In this manuscript we show that bifurcations of subharmonic orbits in perturbed systems can be easily detected computing the Melnikov’s integral over the Harmonic Balance approximation of the unperturbed orbits. The proposed method significantly extend the applicability of the Melnikov’s method since the orbits of any nonlinear oscillator can be easily detected by the Harmonic Balance technique, and the integrability of the unperturbed equations is not required anymore. As examples, several case studies are presented, the results obtained are confirmed by extensive numerical experiments. 1. I Nonlinear oscillators subject to periodic perturbations and coupled nonlinear oscillators are enduring problems in the classical theory of synchronization. While the full range of dynamical behavior, including chaos, is exhibited by these types of dynamical systems, periodic orbits are perhaps of primary importance from the point of view of applications. If the single unperturbed oscillator is not structurally stable, a weak perturbation can drastically change the phase portrait. In the case of aggregates of structurally stable oscillators, an appropriate perturbation can introduce mutual entrainment or synchronization in the network, and this phenomenon is believed to play a major role in self organization in nature. The importance of synchronization in the self organization lies in the fact that what looks like a single process on a macroscopic level often turns out to be a collective oscillation resulting from the mutual synchronization among the tremendous number of the constituent oscillators [1]. It is thus of primary importance to understand how synchronization is achieved under the effect of perturbations or due to the presence of couplings. In these cases, control ∗ Correspondence to: Michele Bonnin, Politecnico di Torino, Corso Duca degli Abruzzi, 24, I-10129 Torino, Italy, Tel. +390115644199, Fax. +390115644099 † e-mail: michele.bonnin@polito.it 2 M B parameters can be the amplitude and frequency of the external forcing or the coupling strength. The problem is simplified if the amplitude of the external input or the strength of the connections are assumed to be small. This allow to employ perturbative techniques as the classical Melnikov’s method [2], based on the idea to make use of the computable solutions of the unperturbed system to determine the solutions of the perturbed one. The critical point is the required availability of an analytic expression for the unperturbed solutions. Unfortunately, the periodic trajectories of almost all non trivial oscillators can only be determined through numerical integrations. Consequently, Melnikov’s method has been mainly applied to integrable systems [3–6]. On the other hand, spectral techniques, like the Harmonic Balance, have been widely used to study periodic solutions of nonlinear systems [9–12]. The idea is to expand the periodic solution and the vector field in Fourier series and to equate the coefficients of the same order harmonics. A system of nonlinear ordinary differential equations is then transformed in a systems of nonlinear algebraic equations whose unknowns are the amplitude and the period of the harmonics. In this manuscript nonlinear oscillators under the effect of weak perturbation are investigated. As a main contribution, we show that the Melnikov’s integrals can be accurately evaluated over the analytical expression of the periodic orbit obtained through the Harmonic Balance technique. The proposed approach, permits to investigate both limit cycles bifurcating from continuous family of periodic orbits in non hyperbolic oscillators subject to external forcing, and frequency entrainment in hyperbolic oscillators driven by resonant excitations. The paper is organized as follows. In Section 2 the Harmonic Balance approach is outlined, under the assumption that the dynamical system is described by a system of differential equations. In Section 3 the Melnikov’s method for subharmonic orbits is illustrated in details. Non hyperbolic oscillators driven by an autonomous perturbation are analyzed in subsection 3.1, and the stability analysis of the bifurcating limit cycle is studied in subsection 3.2. The case of a non hyperbolic oscillator under the effect of a non autonomous periodic perturbation is analyzed in subsection 3.3. Subsection 3.4 is devoted to the analysis of dynamical systems whose free oscillation is a limit cycle. In Section 4 some key examples are presented. In section 5 conclusions are drawn. 2. H B T We consider a system of nonlinear ordinary differential equations (ODE’s), of the form ¡ ¢ ẋ(t) = f x(t), µ , (1) where x ∈ Rn , f : Rn 7→ Rn is a nonlinear vector field, and µ ∈ Rm is a vector of real parameters. We assume that there exists a set of parameters values for which system (1) admits a T periodic orbit γ(t) = γ(t + T ) described by a regular curve γ(t) ⊂ Rn . Therefore γ(t) can be developed in Fourier series, giving γ(t) = +∞ X Γ k ei k ω t k=−∞ where Γk is the vector of the complex valued Fourier coefficients and ω = 2π/T . (2) N O U R P 3 For computational purposes, the Fourier series has to be truncated to a suitable number of harmonics, high enough to accurately represent the solution γ(t), thereby obtaining γ̂(t) = N X Γ k ei k ω t . (3) k=−N The coefficients Γk and the angular frequency ω identify the T periodic function γ̂(t), which is not, in general, a solution of (1). However, the higher the number N of harmonics in (3), the smaller the error made approximating the exact solution γ(t) with the truncated Fourier series γ̂(t). In other words γ̂(t) approaches γ(t) as N tends to infinity. It is well known that any function of a T periodic argument is itself T periodic, hence also the vector field can be expanded in Fourier series ¡ ¢ f γ(t), µ = N X F k ei k ω t , (4) k=−N where the vector of the Fourier coefficients can be determined as Z π X N 1 imωt Fk = f Γm e , µ e−i k ω t d(ω t). 2π −π m=−N (5) Such integrals can in all cases be evaluated numerically, but in many cases they can be expressed in a closed analytical form [12]. By substituting equation (4) and the derivative of (3) in (1), we obtain the following equation N N X X i k ω Γ k ei k ω t = F k ei k ω t . (6) k=−N k=−N Due to the orthogonality of the base functions, equation (6) implies that the coefficients of the same order harmonics are equal, that is i k ω Γk − Fk = 0. (7) System (7) will be referred to as the Harmonic Balance system. Two main situation might be encountered. If the nonlinear system (1) is autonomous (i.e f (x(t)) does not depend explicitly on time) the period of the solution is unknown. Therefore system (7) has 2N + 1 equations and 2N + 2 unknowns, the 2N + 1 spectral coefficients Γk and ω. Take into account that Γk ei k ωt = Γk ei( k ωt+φk ) where Γk = Γk ei φk . Since we are interested in steady state behaviors, one of the initial phases φk is irrelevant and can be set to any desired value. As a consequence one of the associated coefficients, for example Im{Γ1 }, can be imposed to be null and the equation Im{Γ1 } = 0 is added to system (7). The second important case is the one in which the vector field is non-autonomous and contains a periodic forcing term ( f (x(t), t) = f (x(t), t + T ǫ )). In this situation the periodic solutions of the forced nonlinear system (1) are expected to have either the same period of the forcing term (harmonic solutions, T = T ǫ ) or be resonant with the perturbations (mT = n T ε with m, n ∈ Z). A m : 1 resonance with the external forcing is called subharmonic of order m and the periodic solutions are called subharmonics. A 1 : n resonance is called ultraharmonic and a m : n resonance is called ultrasubharmonic. The 4 M B angular frequency is now determined to be ω = mn ωε = mn 2π and the unknowns in system Tε (7) are simply the 2N + 1 spectral coefficients. In both cases the system of differential equations (1) is reduced to a system of nonlinear algebraic equation (7) involving the aforementioned unknowns and which can be efficiently solved exploiting standard numerical techniques. Once the harmonic balance system is solved, equation (3) gives the analytical, even though approximated, expression of the periodic trajectory. Any point (γ0 ∈ γ(t)) belonging to the orbit can be written as γ0 = N X Γk ei k ω t0 (8) k=−N and the associated flow leaving from γ0 is ϕ(t, γ0 ) = N X k=−N Γk e i k ω (t0 +t) = N X Γ0k ei k ω t . (9) k=−N 3. M’ M S O Melnikov’s method represents one of the few cases in which global information on specific systems can be obtained analytically. The original Melnikov’s method [2], is applied to homoclinic orbits passing through a hyperbolic saddle point. It defines an integral function which measure the first variation of the separation between the perturbed stable and unstable manifolds of the hyperbolic saddle point [4, 6–8, 13]. The subharmonic Melnikov’s method, on the contrary, introduces an integral function which measure the distance between two consecutive intersections of a perturbed orbit and a suitable cross section. If there exist a cross section and a perturbed orbit such that the distance is zero, the two intersections coincide and the perturbed orbit is periodic. The prospecting of a periodic orbit is therefore reduced to the research of a zero of the integral function [5,14,15]. The integral function, usually called Melnikov’s integral, yields a formal way for evaluating the distance, provided that an explicit expression of the unperturbed periodic trajectory is known. Recently some significant improvements have been developed for the classical subharmonic Melnikov’s method [16, 17]. These enhancements allow to investigate either limit cycles which bifurcate from continuous family of periodic orbits or weakly perturbed limit cycle oscillators. However, in both cases the integrability of the unperturbed system is still required. In the remaining of this section, we recall classical results and recent developments concerning the subharmonic Melnikov’s method. Before proceed to main results, we introduce some fundamental concepts on the integration of both homogeneous and inhomogeneous variational equations of planar autonomous differential equations along a given trajectory. We consider a smooth plane vector field f : R2 7→ R2 , f = ( f1 , f2 )⊤ with flow ϕ(t, x) (i.e. ϕ̇(t, x) = f (ϕ(t, x))), and the orthogonal vector field f ⊥ = (− f2 , f1 )⊤ . The solutions can be expressed in terms of geometric quantities which involve the divergence N O U R P ∂ f2 ∂x ∇ · f , the curl ∇ ∧ f = − ∂ f1 ∂y 5 and the curvature def K= f (x) ∧ f˙(x) . k f (x)k3 (10) Theorem 1 (Diliberto’s Theorem [16, 18]) Suppose ϕ(t, x) is the flow of the differential equation ¡ ¢ ẋ(t) = f x(t) (11) and consider a point x0 ∈ R2 . If f (x0 ) , 0, then the fundamental matrix solution Φ(t) satisfying det(Φ(0)) = 1, of the variational equation ẏ(t) = D f (ϕ(t, x0 )) y(t) (12) with D f is the jacobian matrix of f , is such that Φ(t) f (x0 ) = f (ϕ(t, x0 )) where (13) Φ(t) f ⊥ (x0 ) = α(t) f (ϕ(t, x0 )) + β(t) f ⊥ (ϕ(t, x0 )) (14) Z t½ ¡ ¢ ¡ ¢¾ 2K k f ϕ(s, x0 ) k − ∇ ∧ f ϕ(s, x0 ) β(s) ds α(t) = (15) def 0 def β(t) = Rt k f (x0 )k2 ∇· f (ϕ(s,x0 ))ds 0 e . k f (ϕ(t, x0 )) k2 (16) For the proof of the theorem the reader is referred to [16, 18] Diliberto’s theorem contains all the information about the solutions of the linear variational equation along the trajectories of the plane vector field f . The next lemma gives an explicit formula for the solution of the inhomogeneous linear variational equation along a trajectory of f . Lemma 2 (Variational Lemma [16]) Let f : R2 7→ R2 and g : R2 7→ R2 denote smooth vector fields. Consider x0 ∈ R2 and let ϕ(t, x) denotes the flow of f . If f (x0 ) , 0 then the solution of the initial value problem ż(t) = D f (ϕ(t, x0 )) z(t) + g (ϕ(t, x0 )) (17) z(0) = 0, is £ ¤ z(t) = [N(t, x0 ) + α(t) M(t, x0 )] f (ϕ(t, x0 )) + β(t) M(t, x0 ) f ⊥ (ϕ(t, x0 )) . where def M(t, x0 ) = Z 0 t t f (ϕ(s, x0 )) ∧ g (ϕ(s, x0 )) ds, k f (ϕ(s, x0 )) k2 β(s) (18) (19) ) ( ¢ ¡ ¢ ¡ ¢ ¡ ¢ 1 α(s) ¡ N(t, x0 ) = f ϕ(s, x0 ) ∧ g ϕ(s, x0 ) ds, h f ϕ(s, x0 ) , g ϕ(s, x0 ) i − 2 β(s) 0 k f (ϕ(s, x0 ))k (20) while α and β are defined as in the statement of Diliberto’s theorem. def Z 6 M B Once again the interested reader can found a proof of the lemma in [16]. Having these results at disposal it is possible to investigate the existence of periodic orbits in the perturbed system, which bifurcate from a continuous family of periodic orbits of the unperturbed one. We make one basic assumption about the unperturbed system, in particular we assume that it possesses a period annulus, i.e. a one-parameter family of periodic orbits γα (t), α ∈ (α1 , α2 ) with period T (γα (t)) > 0. It is always possible to find a cross section Σ transverse to the flow of f in the period annulus (see figure 1). Then there is some ε0 > 0 such that the flow of the perturbed system is also transverse to Σ. γα(t) Σ Figure 1: The period annulus of the unperturbed system and the cross section Σ. For the sake of simplicity we fix the reference frame so that y = 0 on the cross section Σ. Let us consider a point x0 ∈ Σ belonging to the period annulus. For the perturbed system we define the trajectory ϕ(t, x0 , ε) leaving from x0 , the parameterized Poincaré map P(x0 , ε) : Σ × R 7→ Σ, as P(x0 , ε) = ϕ(T (x0 , ε), x0 , ε) on the Poincaré section Σ and the period function T (x0 , ε) : Σ × R 7→ R+ , which assign to x0 the time required for the return on Σ. Setting ε to zero, the corresponding quantities for the unperturbed system are obtained. The idea of the bifurcation theory is to find the periodic trajectories of the perturbed system as fixed points of the parameterized Poincaré map P, that is, to find initial conditions x0 lying on the Poincaré section Σ such that ϕ(T (x0 , ε), x0 , ε) = x0 . With this goal in mind, the displacement function δ(x0 , ε) = ϕ(T (x0 , ε), x0 , ε) − x0 and the normalized displacement function ¢⊤ def ¡ ∆(x0 , ε) = f ⊥ (x0 ) · δ(x0 , ε) (21) are defined. It is evident that ∆(x0 , 0) = 0 for all x0 ∈ Σ, and that ∆(x0 , ε) = 0 if and only if the trajectory ϕ(t, x0 , ε) passing through x0 is periodic. With the choice made about the reference frame, it follows that P(x0 , ε) = (P(x0 , ε), 0)⊤ and δ(x0 , ε) = (d(x0 , ε), 0)⊤ . Before to analyze the influence of small perturbations on the periodic orbits of the unperturbed system, one final lemma is presented, which provides useful information about the functions α(t) and β(t) defined in Diliberto’s theorem (equations (15) and (16)). Lemma 3 Let f : R2 7→ R2 be a plane vector field with flow ϕ(t, x), and x0 ∈ Σ. If x0 is contained in a period annulus, the functions α(t) and β(t) defined in Diliberto’s theorem are such that α (T (x0 )) = −T ′ (x0 ) β (T (x0 )) = 1, (22) where T (x) is the period function. N O U R P 7 Proof By hypothesis ϕ(t, x) is the flow of f = ( f1 , f2 )⊤ , let ψ(t, x) be the flow of the orthogonal field f ⊥ = (− f2 , f1 )⊤ , thus ϕ̇(t, x) = f (ϕ(t, x)) (23) ψ̇(t, x) = f ⊥ (ψ(t, x)) . (24) Consider a point x(s) = ψ(s, x0 ) on the trajectory of the orthogonal vector field, and still belonging to the period annulus. Abbreviate T (s) = T (ψ(s, x0 )) and consider that, since ψ(s, x0 ) lies in the period annulus, the following equation holds ϕ (T (s), ψ(s, x0 )) − ψ(s, x0 ) = 0. A differentiation with respect to s yields ϕ̇ (T (s), ψ(s, x0 )) T ′ (s) + Dϕ (T (s), ψ(s, x0 )) ψ̇(s, x0 ) − ψ̇(s, x0 ) = 0 (25) and taking into account equations (23) and (24) we obtain £ ¤ f ϕ (T (s), ψ(s, x0 )) T ′ (s) + Dϕ (T (s), ψ(s, x0 )) f ⊥ (ψ(s, x0 )) − f ⊥ (ψ(s, x0 )) = 0. An evaluation at s = 0, remembering that ψ(0, x0 ) = x0 (which implies T (0) = T (ψ(0, x0 )) = T (x0 )) gives f (ϕ (T (x0 ), x0 )) T ′ (x0 ) + Dϕ (T (x0 ), x0 )) f ⊥ (x0 ) − f ⊥ (x0 ) = 0. (26) Applying Diliberto’s theorem the required formulas can be obtained. We begin by observing that both Dϕ(t, x0 ) f ⊥ (x0 ) and α(t) f (ϕ(t, x0 )) + β(t) f ⊥ (ϕ(t, x0 )) are solutions of the initial value problem (12) with initial condition y(0) = f ⊥ (x0 ), thus Dϕ(t, x0 ) f ⊥ (x0 ) = α(t) f (ϕ(t, x0 )) + β(t) f ⊥ (ϕ(t, x0 )) . (27) Substituting equation (27) evaluated at t = T (x0 ) into (26) we derive the following equation f (ϕ(T (x0 ), x0 )) T ′ (x0 ) + α(T (x0 )) f (ϕ(T (x0 ), x0 )) + β(T (x0 )) f ⊥ (x0 ) − f ⊥ (x0 ) = 0. (28) By hypothesis x0 belongs to the period annulus, therefore ϕ(T (x0 ), x0 ) = x0 , and considering that f and f ⊥ are orthogonal, equation (28) holds if and only if α(T (x0 )) = −T ′ (x0 ) and β(T (x0 )) = 1. which prove the thesis of the lemma. ¤ 3.1. Autonomous Perturbations We now consider the case of a nonlinear dynamical systems with an autonomous perturbation ẋ = f (x) + ε g(x), (29) and denote by ϕ(t, x, ε) the associated flow ϕ̇(t, x, ε) = f (ϕ(t, x, ε)) + ε g (ϕ(t, x, ε)) . (30) The following theorem [16,17] provides a mean to identify periodic trajectories in a period annulus of the unperturbed system from which limit cycles emerge. 8 M B Theorem 4 (Andronov-Poincaré) Suppose for ε = 0 system (29) has a period annulus. If the integral function Z T (x0 ,0) R t ¡ ¢ ¡ ¢ e− 0 ∇· f (ϕ(s,x0 )) ds f ϕ(t, x0 ) ∧ g ϕ(t, x0 ) dt (31) M(x0 ) = 0 has a simple zero at x0 , that is M(x0 ) = 0 and ∂M(x0 )/∂x , 0, then, for small value of ε, there is a limit cycle ϕ (t, x0 , ε) of the perturbed system (29) passing through Σ at x0 , emerging from the periodic orbit ϕ(t, x0 ) = ϕ(t, x0 , 0) of the unperturbed system. Proof By differentiating both the normalized displacement function and the displacement function with respect to ε we have ¡ ¢⊤ ∂δ(x0 , 0) ∂∆(x0 , ε) = f ⊥ (x0 ) · , ∂ε ∂ε ∂δ(x0 , ε) ∂T (x0 , ε) ∂ = ϕ̇(T (x0 , ε), x0 , ε) + ϕ(T (x0 , ε), x0 , ε). ∂ε ∂ε ∂ε (32) The latter, evaluated at ε = 0, gives ∂δ(x0 , 0) ∂T (x0 , 0) ∂ = ϕ̇(T (x0 , 0), x0 , 0) + ϕ(T (x0 , 0), x0 , 0). ∂ε ∂ε ∂ε (33) We now consider that ϕ(t, x0 , 0) is a periodic trajectory of the unperturbed system, hence the following equations hold ϕ̇(t, x0 , 0) = f (ϕ(t, x0 , 0)) (34) ϕ (T (x0 , 0), x0 , 0) = x0 . (35) By introducing equations (34) and (35) in equation (33) we obtain ∂δ(x0 , 0) ∂T (x0 , 0) ∂ = f (x0 ) + ϕ(T (x0 , 0), x0 , 0), ∂ε ∂ε ∂ε (36) and consequently, equation (32) becomes ¢⊤ ∂ ∂∆(x0 , 0) ¡ ⊥ = f (x0 ) · ϕ(T (x0 , 0), x0 , 0). ∂ε ∂ε (37) To find the proper expression of ∂ϕ/∂ ε we consider that ϕ(t, x0 , ε) is a trajectory of the perturbed system leaving from x0 ¡ ¢ ϕ̇(t, x0 , ε) = f (ϕ(t, x0 , ε)) + ε g ϕ(t, x0 , ε) ϕ(0, x0 , ε) = x0 . A differentiation with respect to ε and an evaluation at ε = 0 yield the variational initial value problem Ã ! d ∂ϕ(t, x0 , 0) ∂ϕ(t, x0 , 0) + g (ϕ(t, x0 , 0)) = D f (ϕ(t, x0 , 0)) dt ∂ε ∂ε (38) ∂ϕ(0, x0 , 0) = 0, ∂ε N O U R P 9 which is of the same kind of (17), hence the solution is given by the Variational Lemma (equation (18)). Inserting such solution in equation (37) we have ∂∆(x0 , 0) = β(T (x0 )) M(T, x0 ) k f (ϕ(T, x0 ))k2 . ∂ε Keeping into account that ϕ(t, x0 , 0) = ϕ(t, x0 ) and the second one of (22) it can be inferred that Z T (x0 ,0) R t ¡ ¢ ¡ ¢ ∂∆(x0 , 0) = e− 0 ∇· f (ϕ(s,x0 ))ds f ϕ(t, x0 ) ∧ g ϕ(t, x0 ) dt. (39) ∂ε 0 Since x0 belongs to the period annulus ∆(x0 , 0) = 0. If there exist an ε̄ > 0 and a continuous function h : (−ε̄, ε̄) 7→ Σ such that ∆(h(ε), ε) = 0, then for each ε ∈ (−ε̄, ε̄) there is a periodic trajectory ϕ(t, h(ε), ε) of the perturbed vector field passing through the point h(ε). We take the Taylor expansion of ∆(x0 , ε) in the neighborhood of ε = 0 ∆(x0 , ε) = ∆(x0 , 0) + ε ∂∆(x0 , 0) + O(ε2 ). ∂ε and observe that, as a consequence of (39), the hypothesis of the theorem can be recast as ∂∆(x0 , 0) =0 ∂ε and ∂2 ∆(x0 , 0) , 0, ∂ε ∂x therefore the implicit function theorem ensures the existence of the required function h(ε). ¤ Integral (31) is often referred to as the Melnikov’s integral and is the basic measure of the distance between x0 and the first iteration of the Poincaré map for the perturbed system. For every initial condition x0 the associated period T (x0 ) is uniquely determined. Since the integral is evaluated over the period T (x0 ), the dependency on time in (31) is not written explicitly. Remark If the unperturbed vector field is hamiltonian, ∇ · f = 0 and the Melnikov’s integral simply reduces to Z T (x0 ,0) ¡ ¢ ¡ ¢ f ϕ(t, x0 ) ∧ g ϕ(t, x0 ) dt. M(x0 ) = 0 Remark The proposed approach is not unique. Here a cross section Σ is fixed, and the Melnikov’s integral is evaluated as the initial condition x0 ∈ Σ is varied. This approach is preferred here in view of the joint application of the Melnikov’s method and the HarPN Γ0k and monic Balance technique, since any initial condition can be written as x0 = k=−N PN the associated flow is given by ϕ(t, x0 ) = k=−N Γ0k ei k ω t . Another approach is to keep fixed x0 and evaluate the Melnikov’s integral on a cross section which moves along the unperturbed trajectory, however the two approaches are completely equivalent. 3.2. Stability of Subharmonic Orbits The natural context to study stability and bifurcations of periodic orbits is the Poincaré map. Thus it is not surprising that Melnikov’s method, with its close relation to Poincaré map permits a simple approach to such problem. 10 M B If the perturbative vector fields is time dependent and periodic, to investigate the stability properties of the emerging limit cycles a symplectic transformation to action angle coordinates has to be performed, but such transformation requires the unperturbed vector field to be hamiltonian. Conversely, if the forcing vector field is autonomous, the stability analysis is much simpler. The following theorem provides information about the stability of the limit cycle emerging from a period annulus: Theorem 5 Suppose system (29) has, for ε = 0, a period annulus containing a point x0 ∈ Σ from which a limit cycle of the perturbed system emerges, and consider the function ( # ) Z T (x0 ,0) " ¡ ¢ ¡ ¢ ∂T (x0 , 0) d ζ(x0 , 0) = ε ∇ · f (x0 ) + ∇ · f ϕ(t, x0 ) dt . ∇ · g ϕ(t, x0 ) + ∂ε dε 0 (40) If ζ(x0 , 0) < 0, then the limit cycle is asymptotically stable, while if ζ(x0 , 0) > 0 then the limit cycle is asymptotically unstable. Proof Since δ(x, 0) = 0 for all x in the period annulus, the scalar displacement d(x, 0) is also null. The Taylor expansion of the scalar displacement d(x, ε) in the neighborhood of (x, 0) has the form ∂d(x, 0) d(x, ε) = ε + O(ε2 ), (41) ∂ε and a differentiation with respect to x gives ∂2 d(x, 0) ∂d(x, ε) =ε + O(ε2 ). ∂x ∂ε ∂x (42) By hypothesis the limit cycle of the perturbed system (29) passes through x0 , hence d(x0 , ε) = 0. Equation (42) implies that, for sufficiently small values of ε, d(x, ε) crosses the value δ(x0 , ε) = 0 as x passes through x0 , with positive or negative slope depending on the signs of ε and of the mixed partial derivative. We consider the case of negative slope, first ∂2 d(x, 0) ∂d(x, ε) ε <0 ⇒ < 0. ∂ε ∂x ∂x We take an initial condition x ∈ Σ such that x−x0 < 0, which implies d(x, ε) = P(x, ε)−x > 0, and P(x, ε) − x0 < 0 (otherwise the trajectories would intersect, see figure 2), it follows P(x, ε) − x0 − (x − x0 ) > 0 ⇒ x − x0 < P(x, ε) − x0 < 0. If we consider now P(x, ε) as the new initial condition and we iterate the process, we see that the sequence of points of the intersections with Σ converges towards x0 . We consider now the second possible situation, that is x − x0 > 0, which implies δ(x, ε) = P(x, ε) − x < 0 and P(x, ε) − x0 > 0. It gives P(x, ε) − x0 − (x − x0 ) < 0 ⇒ 0 < P(x, ε) − x0 < x − x0 , and again the sequence of intersections with the Poicaré section converges towards x0 . Therefore, if ζ(x0 , 0) < 0, the limit cycle ϕ(t, x0 , ε) passing through x0 is stable. By the same argument the limit cycle is unstable if ζ(x0 , 0) > 0. N O U R P 11 40 30 20 10 P(x) x 0 x0 −10 Σ −20 −30 −40 −80 −60 −40 −20 0 20 40 60 80 Figure 2: A stable limit cycle. The sequence of intersections of the trajectory with the Poincaré section converges to x0 . In order to complete the proof of the theorem it is necessary to devise the proper expression of ζ(x0 , 0). It is convenient to rewrite system (29) in the form ẋ = h(x, ε) (43) where h(x, ε) = f (x) + ε g(x), hence the flow ϕ(t, x, ε) is a solution of ϕ̇(t, x, ε) = h (ϕ(t, x, ε), ε) . (44) To simplify the notation ϕ(T (x, ε), x, ε) = x̄ hereafter. Considering the directional derivatives of the scalar Poincaré map along the cross section Σ, we obtain ∂ T (x, ε) ∂P(x, ε) τ̂( x̄) = ϕ̇( x̄) + Dϕ( x̄) τ̂(x), ∂x ∂x (45) where τ̂(x) is the versor tangent to the cross section Σ in x. If the cross section is not a simple straight line τ̂( x̄) and τ̂(x) may differ. It is always possible to project τ̂ in the directions parallel to h and h⊥ τ̂(x) = a h(x, ε) + b h⊥ (x, ε) τ̂( x̄) = c h( x̄, ε) + d h⊥ ( x̄, ε) (46) where a= hτ̂(x), h(x, ε)i , kh(x, ε)k2 hτ̂( x̄), h( x̄, ε)i c= , kh( x̄, ε)k2 b= hτ̂(x), h⊥ (x, ε)i h(x, ε) ∧ τ̂(x) = , kh⊥ (x, ε)k2 kh(x, ε)k2 hτ̂( x̄), h⊥ ( x̄, ε)i h( x̄, ε) ∧ τ̂( x̄) d= = . kh⊥ ( x̄, ε)k2 kh( x̄, ε)k2 (47) Substituting equations (44) and (46) in (45) we obtain ¢ ¡ ¢ ∂ T (x, ε) ∂P(x, ε) ¡ c h( x̄, ε) + d h⊥ ( x̄, ε) = h( x̄, ε) + Dϕ( x̄) a h(x, ε) + b h⊥ (x, ε) . (48) ∂x ∂x The application of Diliberto’s theorem to the last term of (48) yields, after some algebraic manipulations ! Ã ! Ã ∂P(x, ε) ∂P(x, ε) ∂ T (x, ε) − − a − b α(T (x, ε)) h( x̄, ε)+ d − b β(T (x, ε)) h⊥ ( x̄, ε) = 0. c ∂x ∂x ∂x 12 M B Taking into account equation (16) and the orthogonality of the vector fields we have ! ÃZ T (x,ε) ∂P(x, ε) h(x, ε) ∧ τ̂(x) ∇ · h(ϕ(t, x, ε), ε) dt . (49) = exp ∂x h( x̄, ε) ∧ τ̂( x̄) 0 To evaluate the derivative of (49) with respect to ε at ε = 0, we consider that x̄ = ϕ(T (x, 0), x, 0) = x. Therefore the first factor is equal to one and its derivative ! Ã ! Ã ∂h( x̄, ε) ∂h(x, ε) ∧ τ̂(x) (h( x̄, ε) ∧ τ̂( x̄)) − (h(x, ε) ∧ τ̂(x)) ∧ τ̂( x̄) ∂ε ∂ε ∂ h(x, ε) ∧ τ̂(x) = ∂ε h( x̄, ε) ∧ τ̂( x̄) (h( x̄, ε) ∧ τ̂( x̄))2 (50) is null. For the second factor we take into account that Z T (x,ε) ∂ ∂T (x, ε) ∇ · h (ϕ(t, x, ε), ε) dt = ∇ · h (ϕ(T (x, ε), x, ε), ε) ∂ε 0 ∂ε Z T (x,ε) d ∇ · h (ϕ(t, x, ε), ε) dt, (51) + dε 0 and since h(x, ε) = f (x) + ε g(x), the following equation holds d d d ∇ · h (ϕ(t, x, ε), ε) = ∇ · f (ϕ(t, x, ε)) + ∇ · g(ϕ(t, x, ε)) + ε ∇ · g(ϕ(t, x, ε)). (52) dε dε dε Taking into account equations (51), (52), and the considerations made about (50), a differentiation of equation (49) with respect to ε and an evaluation at (x, ε) = (x0 , 0) yields Z T (x0 ,0) Z T (x0 ,0) ¡ ¢ ¡ ¢ d ∂T (x0 , 0) ∂2 P(x0 , 0) ∇ · g ϕ(t, x0 ) dt + = ∇ · f (x0 ) + ∇ · f ϕ(t, x0 ) dt. ∂x ∂ε ∂ε dε 0 0 where ϕ(t, x0 ) = ϕ(t, x0 , 0). Keeping in mind that ∂2 P(x0 , 0) ∂2 d(x0 , 0) = ∂ε ∂x ∂ε ∂x the thesis of the theorem follows. ¤ Remark In general, stability of periodic orbits is determined computing Floquet’s multipliers. Recently some authors [19] have proved that Floquet’s multipliers can be accurately computed by exploiting numerical algorithms described in [20] evaluated over the Harmonic Balance solutions. However that approach is not reliable for weakly perturbed non hyperbolic oscillators. In fact, in this case at least two Floquet’s multipliers lie on the unit circle, and since their moduli depend continuously on perturbative terms, under the effect of a weak perturbation their value remains very close to the unity. Therefore, it is difficult to discriminate between the variation due to the perturbations and the numerical inaccuracies. Remark If the unperturbed system is hamiltonian ∇ · f = 0, the function ζ(x0 , 0) reduces to Z T (x0 ,0) ¡ ¢ ∇ · g ϕ(t, x0 ) dt. (53) ζ(x0 , 0) = ε 0 It is worth noting that the stability of the bifurcating limit cycles depends not only on the perturbation g(x) and its strength ε, but also on the unperturbed system, since the forcing vector field is evaluated over the free oscillation. N O U R P 13 3.3. Non Autonomous Perturbations This section is devoted to study the system ẋ = f (x) + ε g(x, t) (54) where both f (x) and g(x, t) are smooth functions, g is a T periodic function and for ε = 0 system (54) has a period annulus. Since the perturbed system is expected to exhibit periodic trajectories resonant with the perturbation, it is natural to search for the persistence of periodic orbits of the unperturbed system passing through x0 , whose period T (x0 ) is commensurable with the period T of the forcing term, that is m, n ∈ Z. m T = n T (x0 ) It is possible to find periodic trajectories of the perturbed system finding conditions on the functions f and g such that, for small values of ε , 0, fixed points of the parametrized Poincaré map remain. We consider the time derivative of the displacement function Dδ(x, ε) ẋ = ϕ̇(mT, x, ε) − ẋ = f (ϕ(mT, x, ε)) − f (x) Since ϕ(mT, x, 0) = x, an evaluation at ε = 0 gives, Dδ(x, 0) f (x) = 0, (55) therefore Dδ(x, 0) is singular and the implicit function theorem cannot be applied directly. However, different kinds of reduction can be made depending on the degeneracy of the period annulus. The most degenerate condition is the case of an isochronous annulus, for which the period function is constant, i.e. T ′ (x) = 0, for all x belonging to the annulus. Theorem 6 Suppose system (54) has, for ε = 0 an isochronous period annulus. If there are two positive integers m and n such that the unperturbed system has a mT/n periodic trajectory passing through x0 and the functions M(x0 ) = Z 0 mT n e− Rt 0 ∇· f (ϕ(s,x0 )) ds ¡ ¢ ¡ ¢ f ϕ(t, x0 ) ∧ g ϕ(t, x0 ) dt (56) and mT n ( ) ¡ ¢ ¡ ¢ ¢ ¡ ¢ 1 α(t) ¡ N(x0 ) = h f ϕ(t, x0 ) , g ϕ(t, x0 ), t i − f ϕ(t, x0 ) ∧ g ϕ(t, x0 ), t dt, ¡ ¢ β(t) k f ϕ(t, x0 ) k2 0 (57) have both a simple zero at x0 ∈ Σ, then the periodic orbit of the unperturbed system persists for small values of ε. Z Proof For the sake of simplicity we focus on a subharmonic orbit of the unperturbed system passing through x ∈ Σ which is in m : 1 resonance with the external forcing. We make use of the Taylor expansion of the displacement function in a neighborhood of ε = 0 δ(x, ε) = ε ∂ ∂δ(x, 0) + O(ε2 ) = ε ϕ(mT, x, 0) + O(ε2 ). ∂ε ∂ε (58) 14 M B It was shown in section 3.1 that ∂ϕ/∂ε is a solution of the inhomogeneous linear variational equation (17), hence, according to (18), it can be written as £ ¤ ∂ ϕ(mT, x, 0) = [N(x) + α(mT ) M(x)] f (ϕ(mT, x, 0)) + β(mT ) M(x) f ⊥ (ϕ(mT, x, 0)) ∂ε (59) As a consequence of (22), α(mT ) = −m T ′ (x) = 0 since the period annulus is isochronous, and β(mT ) = 1. Thus ∂ ϕ(mT, x, 0) = N(x) f (x) + M(x) f ⊥ (x) ∂ε Introducing such expression in (58), the following equation is obtained ¡ ¢ δ(x, ε) = ε N(x) f (x) + M(x) f ⊥ (x) + O(ε) . Hence the implicit function theorem can be applied to determine when there is an implicit solution of the equation δ(x, ε) = 0 at some point (x0 , 0). ¤ Before to consider the case of a regular period annulus, we introduce some preliminary considerations. It is possible to split the displacement function into its tangent and radial projections δ(x, ε) = σ(x, ε) f (x) + ρ(x, ε) f ⊥ (x) (60) where hδ(x, ε), f ⊥ (x)i hδ(x, ε), f (x)i ρ(x, ε) = . k f (x)k2 k f (x)k2 In what follows, we need the directional derivatives of σ(x, ε) and ρ(x, ε), thus we recall that £ ¤ Dδ(x, 0) = D ϕ(T, x, 0) − x = Dϕ(T, x, 0) − I, (61) σ(x, ε) = and applying Diliberto’s theorem we obtain Dδ(x, 0) f ⊥ (x) = α(T ) f (x) + β(T ) f ⊥ (x) − f ⊥ (x) Dδ(x, 0) f (x) = 0. (62) (63) Now we can compute the derivative of σ(x, ε) in the direction f ⊥ (x) and f (x) at (x, ε) = (x, 0), we obtain D σ(x, 0) f ⊥ (x) = D σ(x, 0) f (x) = hD δ(x, 0) f ⊥ (x), f (x)i hδ(x, 0), D f (x) f ⊥ (x)i + = α(T ) (64) k f (x)k2 k f (x)k2 hD δ(x, 0) f (x), f (x)i hδ(x, 0), D f (x) f (x)i + = 0, k f (x)k2 k f (x)k2 (65) where (62) and (63) were used, and since δ(x, 0) = 0. In a similar way it is possible to prove that D ρ(x, 0) f ⊥ (x) = β(T ) − 1 D ρ(x, 0) f (x) = 0. (66) (67) We can now introdue the theorem dealing with a regular period annulus, that is when T (x) , 0. Even in this situation the implicit function theorem cannot be applied directly, since equation (55) still holds, then a Lyapunov-Schmidt reduction is applied. ′ 15 N O U R P Theorem 7 Suppose system (54) has, for ε = 0 a regular period annulus. If there are two positive integers m and n such that the unperturbed system has a mT/n periodic trajectory passing through x0 and the function M(x0 ) = Z mT n e 0 − Rt 0 ¡ ¢ ∇· f ϕ(s,x0 ) ds ¡ ¢ ¡ ¢ f ϕ(t, x0 ) ∧ g ϕ(t, x0 ), t dt (68) has a simple zero at x0 ∈ Σ, i.e. M(x0 ) = 0 and ∂M(x0 )/∂x , 0, then the periodic orbit of the unperturbed system persists for small values of ε. Proof Using equation (64) and the first of (22), we have Dσ(x, 0) f ⊥ (x) = −mT ′ (x) which is different from zero because the annulus is regular. Applying the implicit function theorem to the function σ(x, 0), it is possible to define a smooth manifold S such that σ vanishes on S. In addition, for any x ∈ ϕ(t, x, 0) equation (65) implies that S is transverse to the section Σ and ϕ(t, x, 0) ⊂ S. Now we focus on the restriction of the radial projection ρ to S. Equations (66) and (67) imply that Dρ(x, 0) = 0. On the other hand, as a consequence of (58) and (59) we have ∂δ(x, 0) = [N(x) + α(mT )] f (x) + β(mT )M(x) f ⊥ (x). (69) ∂ε while differentiation of (60) with respect to ε and an evaluation at ε = 0 yield ∂δ(x, 0) ∂σ(x, 0) ∂ρ(x, 0) ⊥ = f (x) + f (x). ∂ε ∂ε ∂ε (70) Comparing the last two equations and taking into account the second of (22) we infer ∂ρ(x, 0) = M(x). ∂ε The hypotheses M(x0 ) = 0 and ∂M(x0 )/∂x , 0 imply ∂ρ(x0 , 0)/∂ε = 0 and ∂2 ρ(x0 , 0)/∂ε∂x , 0. Therefore there is an ε̄ > 0 and a smooth function h : (−ε̄, ε̄) 7→ R2 such that ρ(h(ε), ε) ≡ 0, as a consequence ρ(h(ε), ε) = σ(h(ε), ε) = 0 and δ(h(ε), ε) = 0, which proves the statement of the theorem. ¤ One more result, due to Chow et al. [21], is worth noting. This result is related to system possessing a homoclinic orbit to a hyperbolic saddle point as the separatrix of regions filled with periodic orbits. It implies that a countable sequence of subharmonic saddle-node bifurcations of periodic orbits converges to a homoclinic bifurcation (see [14, 15] for details). Theorem 8 Suppose for ε = 0 system (54) has a homoclinic orbit q0 (t) to a hyperbolic saddle point p0 . Assume the interior of q0 (t) ∪ p0 is filled with a continuous family of periodic orbits qα (t) whose period tends monotonically to infinity as the periodic orbits approach the homoclinic orbit. Let Z mT R ¡ ¢ t ¡ ¢ ¡ ¢ α m M (x) = e− 0 ∇· f q (s) ds f qα (t) ∧ g qα (t), t dt (71) 0 16 M B be the Melnikov’s integral associated to the mT periodic orbit. Then lim Mm (x) = M(x) m→+∞ where M(x) = Z +∞ e −∞ − Rt 0 ¡ ¢ ∇· f q0 (s) ds ¡ ¢ ¡ ¢ f q0 (t) ∧ g q0 (t), t dt is the Melnikov’s integral whose simple zeroes are associated to homoclinic bifurcations. 3.4. Limit Cycle Oscillators This section is devoted to the analysis of oscillators whose free oscillation is a limit cycle, i.e. the unperturbed system is structurally stable. When we consider the forced oscillator on the manifold R2 × S 1 , the three dimensional system has for ε = 0, an hyperbolic torus corresponding to the limit cycle. The flow on the torus will be periodic or quasi periodic depending on wether or not some resonant condition is satisfied. In either case, the orbit corresponding to the limit cycle will no longer be structurally stable, the stability of the limit cycle being transferred to the torus. The existence of frequency entrained oscillations in a limit cycle oscillator driven by a resonant forcing term is described by the following theorem Theorem 9 (Limit cycle subharmonic bifurcation theorem [16]) Let us consider the dynamical system ¡ ¢ ¡ ¢ ẋ(t) = f x(t) + ε g x(t), t and suppose that it admits a limit cycle ϕ(t, x0 ) whose period is resonant with the period ¡ ¢ ¡ ¢ of the external forcing g x(t), t = g x(t + T ), t + T . If x0 is a simple zero of the bifurcation function £ ¤ B(x0 ) = 1 − β(T ) N(x0 ) + α(T )M(x0 ), (72) namely, B(x0 ) = 0 and ∂B(x0 )/∂x , 0 then the perturbed system has a limit cycle ϕ(t, x0 , ε) passing through x0 . Proof With the projections of σ and ρ defined in equations (64)-(67) we have two possibilities. If β(T ) , 1 we can apply the implicit function theorem to the radial projection. Conversely, if α(T ) , 0 the implicit function theorem can be applied to the tangent projection, in both cases we can define a smooth function h : (−ε, ε) 7→ R2 such that either ρ(h(ε), ε) = 0 or σ(h(ε), ε) = 0. Now we need to identify the bifurcation function. Let us consider the Taylor expansion of σ and ρ in the neighborhood of ε = 0 dσ(h(0), 0) + O(ε2 ) dε dρ(h(0), 0) + O(ε2 ), ρ(h(ε), ε) = ρ(h(0), 0) + ε dε σ(h(ε), ε) = σ(h(0), 0) + ε (73) (74) where ∂σ(h(0), 0) ∂h(0) dσ(h(0), 0) = + Dσ(h(0), 0) dε ∂ε ∂ε ∂ρ(h(0), 0) ∂h(0) dρ(h(0), 0) = + Dρ(h(0), 0) , dε ∂ε ∂ε (75) (76) N O U R P 17 and comparing (69) and (70) we have ∂σ(h(0), 0) = N(h(0)) + α(T )M(h(0)) ∂ε ∂ρ(h(0), 0) = β(T )M(h(0)). ∂ε (77) Since h(0) ∈ Σ, we can express the vector field ∂h(ε)/∂ε as a linear combination of f and f⊥ ∂h(0) = a f (h(0)) + b f ⊥ (h(0)) ∂ε and taking into account (64)-(67) ∂h(0) = b α(T ) ∂ε ¡ ¢ ∂h(0) Dρ(h(0), 0) = b β(T ) − 1 . ∂ε Dσ(h(0), 0) (78) (79) Thus dσ(h(0), 0) = b α(T ) + N(h(0)) + α(T )M(h(0)) dε £ ¤ dρ(h(0), 0) = b β(T ) − 1 + β(T )M(h(0)). dε (80) (81) For the case in which β(T ) , 1 we consider equation (81), which implies b= β(T ) M((h(0)) 1 − β(T ) which substituted in (80) yields £ ¤ 1 − β(T ) N(h(0)) + α(T )M(h(0)) dσ(h(0), 0) = dε 1 − β(T ) The case in which α(T ) , 0 is similar, (80) implies b=− N(h(0)) + α(T )M(h(0)) α(T ) which substituted in (81) gives £ ¤ 1 − β(T ) N(h(0)) + α(T )M(h(0)) dρ(h(0), 0) = dε α(T ) as required.¤ 4. A In the previous section we shoved that the Melnikov technique provides a mean to determine conditions such that periodic trajectories survive under the effect of an external forcing. It was shown how the problem can be reformulated in terms of zeroes of an integral 18 M B function evaluated over the periodic solution of the unperturbed system. Thus, an analytic expression of the free oscillation must be available. Unfortunately, as stated in the introduction, trajectories of almost all non trivial oscillators can be only determined through numerical integration. Even for integrable systems, trajectories are often expressed in terms of special functions, making the integral equations often unsolvable. In some cases, the problem can be solved recurring to the method of the residues, while in other cases it is necessary to consider some approximation of the function to be integrated [5, 14, 15]. In this section we show that the Melnikov’s integrals can be accurately evaluated over the Harmonic Balance approximation of the unperturbed system. The proposed approach presents two main advantages, the Harmonic Balance technique yields the analytical approximations of the periodic trajectory of any nonlinear oscillator. Hence, the integrability of the unperturbed system is not required anymore, and the applicability of the Melnikov’s method is considerably extended. Moreover, the unperturbed trajectory is always approximated by a truncated Fourier series, and as a consequence the Melnikov’s integrals often result quite simple to solve making use of the orthogonality of the base functions. 4.1. Hamiltonian system with an autonomous perturbation As a first application of the proposed technique, we consider the following dynamical system ẋ = y (82) ẏ = x − x3 + ε(α y − β x2 y). For ε = 0, the system is hamiltonian and has centers at (x, y) = (±1, 0) and a hyperbolic saddle at (0, 0). There are two homoclinic trajectories passing through the hyperbolic saddle which separate regions filled with periodic orbits, the phase portrait is presented in figure 3. The periodic orbits surrounding each center and lying inside the homoclinic trajectory will be called inner orbits while the larger closed paths surrounding the centers and the homoclinic trajectories will be called outer orbits. Since the unperturbed system is integrable, all the trajectories can be determined analytically, however the periodic orbits are expressed in terms of Jacobi elliptic functions and the resulting Melnikov’s integral can be solved only considering their Fourier series [14, 15]. Theorem 4 establishes the conditions under which a limit cycle of the perturbed system emerges from periodic solutions of the unperturbed system. We are interested in determine the values of the bifurcation parameters satisfying such conditions. As a first step, we have to determine the proper expression of the unperturbed trajectories of period T = 2 π/ω. Following section 2 we search for solution of the system ẋ = y ẏ = x − x3 , in the form Ã x(t) y(t) ! ! N Ã X Xk ei k ω t . = Yk k=−N (83) (84) N O U R P 19 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Figure 3: The phase portrait of the unperturbed Duffing oscillator obtained through the Harmonic Balance technique. The nonlinear term is expressed as follow x3 = N X F k ei k ω t , (85) k=−N where 1 Fk = 2π Z π N 3 X −i k ω t i h ω t e Xh e d(ωt). −π h=−N Such integral can be expressed in a closed analytical form, in terms of the coefficients Xk only. By substituting (84) and (85) in (83), computing the proper derivatives and equating the coefficients of the same order harmonics we obtain i k ω Xk − Yk = 0 ³ ´ (86) k2 ω2 + 1 Xk − Fk = 0. System (86) can be efficiently solved exploiting standard numerical algorithms, giving the coefficients of the Harmonic Balance approximation of the free oscillation. According to theorem 4, a limit cycles emerges from a periodic orbit of period T = PN Γk ei k ωt0 , if the Melnikov’s integral has a simple zero, 2 π/ω, passing through γ0 = k=−N namely if M(γ0 ) = Z 0 T (γ0 ) ¡ ¢ ¡ ¢ f ϕ(t, γ0 ) ∧ g ϕ(t, γ0 ) dt = Z 0 T (γ0 ) ³ ´ y(t) α y(t) − β x2 (t) y(t) dt = 0. (87) Thus, for every t0 ∈ (0, T ], the Harmonic Balance coefficients describing the flow ϕ(t, γ0 ) can be computed as Γ0k = Γk ei k ω t0 . By introducing in (87) such coefficients we have M(γ0 ) = α I(γ0 ) − β J(γ0 ) = 0 20 M B where I(γ0 ) = Z T (γ0 ) 0 2 N X 0 i k ω t dt Yk e (88) k=−N 2 N 2 Z T (γ0 ) X N X J(γ0 ) = Xk0 ei k ω t Yk0 ei k ω t dt. 0 k=−N (89) k=−N The two integrals can be analytically computed for any number of harmonics, and they only involve summations of the Harmonic Balance coefficients. Thus a limit cycle arises when α J(γ0 ) = . (90) β I(γ0 ) In figure 4 the values of the ratio α/β at which a limit cycle arise, are depicted as a function of the frequency. For α/β < 0.75 the perturbed system has not periodic solution. At α/β = 0.75, an outer limit cycle arise from the unperturbed orbit of frequency ω = 0.6 through a saddle node bifurcation, and is the unique periodic solution for 0.75 < α/β < 0.8. For α/β = 0.8 two limit cycles, one surrounding each center, have birth trough saddle-node bifurcations, thus for α/β > 0.8 there is a limit cycle surrounding each center and a large limit cycle enclosing all three equilibria. 1.5 1.25 α β 1.0 0.75 0.5 0 0.5 ω 1 1.5 Figure 4: Bifurcation curves for the hamiltonian system with an autonomous perturbation as a function of the frequency. (Solid line) Birth of a limit cycle from an outer periodic orbit. (Dashed line) Birth of a limit cycle from an inner periodic orbit. In the present example, as ω → 0, i.e. T → +∞, both the inner and the outer periodic orbits of the unperturbed system approaches the homoclinic trajectory. Consequently, the values of the ratio α/β at which limit cycles arise, converge at the value (α/β = 0.8 here) at which a homoclinic tangency occurs, as stated by theorem 8. The convergence of the bifurcation curves for ω → 0 is evident in figure 4. According to theorem 5, the stability of the bifurcating limit cycles can be determined evaluating the sign of equation (53) on the unperturbed periodic solution ϕ(t, x0 ), thus we consider N 2 Z T (γ0 ) ³ Z T (γ0 ) ½ N ¯ X ´ ¯¯2 X 0 i k ω t ¾ 2πε ¯ 0 2 α − β dt = ¯Xk ¯ . ε α − β x (t) dt = ε α − β Xk e ω 0 0 k=−N k=−N (91) N O U R P 21 By substituting in equation (91) the Harmonic Balance coefficients we obtain that the inner limit cycles are asymptotically unstable, while the outer limit cycle is asymptotically stable. The values at which limit cycles have birth and their stability are confirmed by numerical integration of the state equations. 4.2. The Duffing oscillator As an example of a nonlinear oscillator with a periodic perturbation we consider the Duffing equation with linear negative stiffness and weak damping and sinusoidal forcing ẋ = y (92) ẏ = x − x3 + ε (γ cos(ω t) − δ y), where the forcing amplitude γ and the damping δ are assumed as bifurcation parameters. In the unperturbed limit the Duffing oscillator is an integrable system for which exact results are available. The reader is referred to [14, 15] to compare the accuracy and simplicity of the proposed approach with respect to classical results. The unperturbed system is the same of the previous example, but since the forcing term is periodic, we are now interested in subharmonic orbits of the forced system which are in m : 1 resonance with the external forcing, therefore we search for harmonic balance solutions of the unperturbed system in the form ! Ã ! X N Ã x(t) Xk i mk ω t e = (93) Yk y(t) k=−N and we express the nonlinear term as 3 x = N X k F k ei m ω t (94) k=−N where 1 Fk = 2π Z 0 T N 3 X h i ω t Xh e m e−i k ω t d(ω t). (95) h=−N Repeating the steps outlined in the previous example we obtain the following harmonic balance system i mk ωXk − Yk = 0 ·³ ¸ ´2 (96) k ω + 1 X − F = 0. m k k Exploiting the Harmonic Balance solution of system (96), it is easy to verify that for each family of periodic orbits dT/dx , 0, and then each family is regular. In particular for every ω the system admits one and only one trajectory. We are interested in determine for which values of the bifurcation parameters γ and δ, a m order subharmonic orbit of the unperturbed system survive under the effect of the external forcing and the presence of damping. We compute the subharmonic Melnikov’s integral for the resonant orbit Z mT Z mT ¡ ¢ ¡ ¢ ¡ ¢ m M (γ0 ) = f ϕ(t, γ0 ) ∧ g ϕ(t, γ0 ), t dt = y(t) γ cos(ωt) − δ y(t) dt, 0 0 22 M B where y(t) is determined introducing in (93) the Harmonic Balance coefficients obtained solving system (96). Thus 2 Z mT X Z mT X N ³ N ´³ ´ γ m 0 i mk ω t i ωt −i ωt 0 i mk ωt M (γ0 ) = Yk e e +e dt − δ Yk e dt 2 0 k=−N 0 k=−N and solving the integral we obtain N ¯ X ¯ 2π 2 ¯¯Y 0 ¯¯ . γ Re{Y 0 } − δ Mm (γ0 ) = m k ω k=−N (97) According to theorem 7, a m-order subharmonic orbit with period mT/ω, appears in the perturbed system when N ¯ X γ 1 ¯¯Y 0 ¯¯¯2 . = (98) δ Re{Ym0 } k=−N k The bifurcation curves corresponding to the birth of a subharmonic of order m in the perturbed system are easily determined by substituting in equation (98) the Harmonic Balance coefficients of the corresponding unperturbed mT -periodic orbit. Since the ratio γ/δ is a constant, the bifurcation curves are straight lines. Some of such curves are outlined in figure 5, where S m refers to an inner orbit while Ŝ m refers to an outer orbit. It is readily seen that, as predicted by theorem 8, the bifurcation curves accumulate on the curve S 0 associate to the homoclinic tangency. Ŝ 1 Ŝ 3 Ŝ 5 S0 S4 S3 S2 S1 0.1 γ 0.05 0 0 0.05 δ 0.1 Figure 5: Bifurcation curves for the subharmonics of the Duffing oscillator. S m : m-order subharmonic orbit inside the homoclinic trajectories; Ŝ m : outer orbit. Note the rapid convergence of S m and Ŝ m to the homoclinic tangency S 0 . 4.3. Coupled Van der Pol Oscillators As a last application of the proposed technique, we consider two coupled Van der Pol oscillators running in resonance u̇ = v 2 v̇ = −u + δ(1 − u )v (99) ẋ = τy³ ´ ẏ = τ −x + δ(1 − x2 ) y + εu. 23 N O U R P Here the system (x(t), y(t))⊤ is view as the perturbed system subject to a periodic resonant external input provided by (u(t), v(t))⊤ , τ ∈ Z is the resonance factor. The perturbation admits the Harmonic Balance description Ã u(t) v(t) ! ! N Ã X Uk i k ω t e = Vk k=−N which leads to the Harmonic Balance system ( i k ω Uk − ´Vk = 0 ³ 2 2 k ω + i k δ ω − 1 Uk − δ Gk = 0 where 1 Gk = 2π Z π 2 N N −i k ω t X X i n ω t i m ω t e d(ωt). Vn e Um e −π m=−N n=−N The unperturbed system has a stable limit cycle whose Harmonic Balance description is obtained solving ( i k ω Xk − τ´ Yk = 0 ³ 2 2 k ω + i k δ τ ω − τ Xk − τ δ F k = 0 τ where 1 Fk = 2π π Z N 2 N X X Xm ei m ω t Yn ei n ω t e−i k ω t d(ω t). −π m=−N n=−N The Harmonic Balance approximation of the limit cycle is showed in figure (6) where is compared to the solution obtained through numerical integration of the state equations. 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 −2 −1 0 1 2 3 −3 −3 −2 −1 0 1 2 3 Figure 6: Accuracy of the Harmonic Balance Technique. The Van der Pol limit cycle obtained through numerical integration (dashed line) and through the Harmonic balance technique (solid line) using 5 harmonics (left figure) and 7 harmonics (right figure) respectively. In the right figure the two curves are almost coincident. When the Harmonic Balance approximation of the limit cycle is known, it is possible to determine subharmonic branch points searching zeroes of the bifurcation function (72). 24 M B Computing the terms appearing in (72) we obtain i2 ´ h ³ ÃZ t ³ ´ ! y20 + δ 1 − x02 y0 − x0 2 β(t) = τ δ 1 − x ds £ ¡ ¢ ¤ exp y2 + δ 1 − x 2 y − x 2 0 Z t ½ x2 + y2 + δ x3 y + δ x y ³2y2 − 1´ £ ¤¾ α(t) = −2τ − δx y + 1 β(s) ds £ ¡ ¢ ¤2 0 y2 + δ 1 − x 2 y − x Z t Rs 1 − 0 τ δ(1−x2 ) dr ½ ¾ M(γ0 ) = y u e ds h ³ i2 ´ 2 2 0 τ y0 + δ 1 − x 0 y0 − x 0 1 N(γ0 ) = τ Z t 0 ) ( ³ ´ α(s) 2 y ds ¢ ¤ δ 1−x y−x− £ ¡ β(s) y2 + δ 1 − x 2 y − x 2 u Unfortunately we are not able to solve the last three integrals by hand, however computing the Harmonic Balance approximations of x(t), y(t) and u(t) for every t0 ∈ (0, T ] and introducing such coefficients in the expression of the integrals above, the bifurcation function can be computed for every γ0 lying on the unperturbed limit cycle. Figure (7) shows the evolution of the bifurcation function (72) versus the initial condition for τ = 2. The two zeroes of B(γ0 ) identify two subharmonic branch points. Both their number and locations are in good agreement with numerical simulations carried out for small values of ε. B(x0) 0 0.25 0.5 0.75 T Figure 7: Bifurcation function B(γ0 ) as a function of the initial condition. 5. C Considering recent enhancements to the classical subharmonic Melnikov’s method, the effect of periodic perturbations on both non-hyperbolic and hyperbolic oscillators have been investigated. As a main contribution we have shown that Melnikov’s integrals can be accurately computed over the Harmonic Balance approximation of the unperturbed trajectories. The main advantage is that the integrability of the unperturbed system is not required anymore, and the applicability of the Melnikov’s method is considerably extended. 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