APPLICATION OF THE SUBHARMONIC MELNIKOV METHOD TO PIECEWISE-SMOOTH SYSTEMS

. We extend a reﬁned version of the subharmonic Melnikov method to piecewise-smooth systems and demonstrate the theory for bi- and trilinear oscillators. Fundamental results for approximating solutions of piecewise- smooth systems by those of smooth systems are given and used to obtain the main result. Special attention is paid to degenerate resonance behavior, and analytical results are illustrated by numerical ones.


1.
Introduction. Greenspan and Holmes [11] developed a perturbation method for analyzing (homoclinic and) subharmonic orbits in time-periodic perturbations of planar Hamiltonian systems. It is now called "Melnikov's method" since its original idea was found in [13]. Existence of periodic orbits and their saddle-node bifurcations were analyzed but some difficulty arose in determination of their stability. Subsequently, some extensions have been made for the stability of subharmonics and their Hopf (Neimark-Sacker) and Bogdanov-Takens bifurcations in [17,20,21]. Especially, simple formulas for determining their stability and these bifurcations were given, and degenerate resonance behavior, in which the derivative of the unperturbed frequency with respect to the Hamiltonian energy disappears, was appropriately treated. Piecewise-smooth systems can be studied by using the extended approaches, as demonstrated below.
Piecewise-smooth systems also naturally arise in many engineering and physical applications due to impact, friction, collision and switching, and have attracted much attention for more than several decades. Modern theories in dynamical systems have been applied and developed to uncover and understand nonlinear phenomena such as bifurcations and chaos in these systems. The nonsmoothness of systems yields many interesting behaviors which never occur in smooth system. See, e.g., [6] and references therein for more details. Traditional techniques such as the averaging and multiple-scale methods [14,15], which were developed for smooth systems, have also been used to study these piecewise-smooth systems (see, x α (t) Figure 1. Unperturbed phase plane e.g., [2,4]). These analytical results are helpful to understand some behaviors in piecewise-smooth systems even though they are not mathematically rigorous. In fact, in some cases, such analyses were carried out rigorously [3,8].
In this paper, we extend the refined version of the subharmonic Melnikov method [17,21] to piecewise-smooth systems and demonstrate the theory for bi-and trilinear oscillators. Here we mainly treat two-dimensional systems, for which the subharmonic Melnikov method has been well developed although it was also extended to a special class of four-dimensional systems in [5,19]. Fundamental results for approximating solutions of piecewise-smooth systems by those of smooth systems are given and used to obtain the main result. Special attention is paid to degenerate resonance behavior and analytical results are also illustrated by numerical ones via the computer software AUTO97 [7]. Similar analyses were given earlier for a mathematical model of vibrating microcantilevers in tapping mode atomic force microscopy in [22,23]. The analytical results succeeded in explaining many nonlinear behaviors which were experimentally and numerically observed in the microcantilevers. Our result gives a mathematical basis for the theoretical analyses.
The outline of this paper is as follows: In Section 2 we review the extended version of the subharmonic Melnikov method. The existence, stability and saddlenode bifurcation results for periodic orbits are stated and degenerate resonances are briefly discussed. In Section 3 we give the fundamental results in general settings and our main result for piecewise-smooth systems. We apply the theory to bi-and trilinear oscillators in Sections 4 and 5, respectively. Discontinuous and continuous cases are, respectively, considered for bi-and trilinear oscillators. Moreover, the analytical results are compared with numerical ones via AUTO97 for both oscillators. Finally, we give a summary and some comments in Section 6.

Subharmonic Melnikov method.
2.1. Setup. We consider systems of the forṁ is C r and 2π-periodic in θ = ωt, and J is the 2 × 2 symplectic matrix We often drop out the dependence of (1) on the parameter µ below. When = 0, which is a planar Hamiltonian system with a Hamiltonian H(x). Thus, Eq. (1) is a time-periodic perturbation of the Hamiltonian system (2). We make the following assumption on the unperturbed system (2): There is a one-parameter family of periodic orbits, x α (t), α ∈ [α , α u ] ⊂ R, with period T α . See Fig. 1. The family x α (t) is also C r with respect to α.
Note that x α (t) is automatically C r with respect to t since the vector field of (2) is C r .

2.2.
Outline of the approaches. Here we give an outline of the approaches used to obtain the results below in this section. See Section 2 of [17] for the proofs and technical details. We first introduce action-angle coordinates (I, φ) ∈ R + × S 1 for the unperturbed system (2), where R + is the set of non-negative real numbers and S 1 = R/2π is the circle of length 2π. The action variable I is defined as for the periodic orbit x α (t) (e.g., [1]). Note that the value of I depends only on α in (3). The Hamiltonian is constant on x α (t), so that it can be regarded as a function of only I: H = H(I). By a well-known result on integrable Hamiltonian systems (see, e.g., Section 50 of [1]), we define the angle variable φ such that Eq. (2) is rewritten asİ where Ω(I) = dH dI (I).
Hence, we see that the symplectic transformation (I, φ) → x is given by if the angle variable φ is chosen such that φ(0) = 0, where α(I) represents the inverse of the relation (3). See p. 1723 of [17] for the proof that the transformation (6) is actually symplectic. We next transform (1) into the action-angle coordinates (I, φ) to obtaiṅ where where "·" represents the inner product. Define the Poincaré map P for (7) as where (I, φ) = (I(t), φ(t)) is a solution of (7) and T = 2π/ω. We also write P 0 for P when = 0, so that P 0 represents the Poincaré map for (4) and yields a simple rotation. The mth iterate of P is estimated up to O( ) as where We reduce the study of mth-order subharmonic orbits in (1) to that of fixed points of P m . Let I α denote the value of I for x α (t) and let Ω α = Ω(I α ). We define the subharmonic Melnikov functions as In general, the function N m/n is difficult to estimate directly, compared with M m/n and L m/n , but we have the following result.
See Appendix A of [17] for the proof but a small typographical error exists in the formula (A.2) of that paper.
For m, n ∈ N relatively prime, let α m/n and I m/n be, respectively, the value of α and I α such that nT α = mT (mΩ α = nω).
The mth iterate P m 0 has fixed points on I 0 = I m/n . We can show that the firstorder terms Q m , R m in (8) are expressed by the Melnikov functions M m/n , N m/n on I 0 = I m/n (see Proposition 2.2 of [17]). Using this fact, we can analyze fixed points of P m near I 0 = I m/n in (8) to obtain existence, stability and bifurcation results for mth-order subharmonic orbits in (1), as given below. We also definê M m/n (θ; µ) =M m/n (α m/n , θ; µ) = mT 0 DH(x α m/n (t)) · g x α m/n (t), ωt + θ; µ dt andL m/n (θ; µ) = L m/n (α m/n , θ; µ).

2.3.
Existence, stability and saddle-node bifurcations. We begin with existence and stability theorems for subharmonics in (1) (see Section 3.1 of [17] for the proof). In this subsection we assume the nondegenerate condition Theorem 2.2. Suppose thatM (θ) has a simple zero at θ = θ 0 , i.e., Then for > 0 sufficiently small there exists a subharmonic orbit of period mT near x α m/n (t − θ 0 /ω). Moreover, it is of a saddle type if and of a sink type (resp. of a source type) if dΩ dI (I m/n ) dM m/n dθ (θ 0 ) > 0 andL m/n (θ 0 ) < 0 (resp. > 0).
The first part of Theorem 2.2 is standard in the subharmonic Melnikov method [11,12,16] but the second one is not and was originally obtained in [17].
We next state a saddle-node bifurcation theorem for subharmonics in (1) (see Section 4.1 of [17] for the proof).
This result is basically standard but the formula (11) was not given in [11,12,16].
We have the following result in the situation of (12).
Theorem 2.4. Suppose that at some point (θ 0 , µ 0 ) the following conditions hold: Then in the (ν, µ)-parameter plane there exist three saddle-node bifurcation curves for subharmonics of period mT in a neighborhood of (ν 0 , µ 0 ), where as shown in Fig. 3, depending on whetherΩ is positive or negative.

2.5.
Example. Consider general single-degree-of-freedom nonlinear oscillators of the formẋ as an example to illustrate the above theories, where γ, δ are positive constants, ϕ : R → R is C r , ϕ(0) = 0 and ϕ (0) > 0. When = 0, Eq. (14) is Hamiltonian with a Hamiltonian function and has a one-parameter family of periodic orbits around a center-type equilibrium at the origin. Moreover, the periodic orbits can be chosen such that x α 1 (t) and x α 2 (t), respectively, are even and odd functions of t.
3. Main result. The subharmonic Melnikov method described above is originally applicable only to smooth systems. In this section we extend the method to piecewise-smooth systems. To this end, we give fundamental results to approximate solutions of piecewise-smooth systems by those of smooth systems in general settings. As stated in Section 1, the dynamics of piecewise-smooth systems are often different from those of smooth systems. These results guarantee that such a difference does not occur under some conditions.
For simplicity we assume that h(x) is independent of the parameter µ. Consider piecewise-smooth systems of the forṁ We assume that an initial value problem for (18) has a unique solution which is C r outside Π and continuous on Π. Again, we drop out the dependence of (18) on the parameter µ below. Suppose that a function f ρ (x, t) is C r in (x, t), and satisfies and . If x(t) passes through Π only finitely many times on [0, T ] and the left-and right-hand limits are nonzero and have the same sign as (20) at any This theorem is very fundamental but has been previously unpublished at least, to the author's knowledge.
Proof. For simplicity we assume that x(t) passes through Π once. It is obvious that the general case follows from this case.
Let (t 1 , t 2 ) be an interval on which x(t) stays in U ρ and let u ± be the left-and right-hand limits of (22). Specifically, we assume that u ± > 0, so that h(x(t)) < 0 for t < t 1 and h(x(t)) > 0 for t > t 2 since (d/dt)h(x(t)) = f (x(t), t) · Dh(x(t)). Let so that by Gronwall's inequality (see, e.g., Lemma 4.1.2 of [12]) where L 1 is the Lipschitz constant of f in D 1 . On the other hand, let which is positive by assumption. We estimate the length of the time interval (t 1 , t 1 ) as so that by Gronwall's inequality Since x(t 1 ) is O(ρ)-close to the boundary of U ρ , we estimate t 1 − t 1 = O(ρ) to obtain (24) for t 1 < t ≤ t 1 , although x ρ (t) ∈ U ρ . Replacing x(t) and x ρ (t) in the above arguments, we prove (24) generally when x(t) or x ρ (t) ∈ D 1 \ U ρ . Let 0 < t 1 < t 1 < t 2 < t 2 and suppose that x ρ (t) stays in U ρ for t 1 ≤ t ≤ t 2 and in D 2 \ U ρ for t 2 < t ≤ t 2 . We easily see that t 2 − t 1 = O(ρ). Hence, for t 1 < t ≤ t 2 we still have (23), which yields (24). For t > t 2 we have   (18) is continuous, then for the statement of Theorem 3.1 to hold, we only have to assume without the assumptions that x(t) passes Π finitely many times. Actually, we have The statement of Theorem 3.1 also holds even when f is only continuous in t.
We now consider the case in which the piecewise-smooth system (18) is periodic in t and depends on the parameter µ. Theorem 3.3. Let f (x, t; µ) be T -periodic in t and suppose that f ρ (x, t; µ) is Tperiodic in t and C r in (x, t, µ), and satisfies (19). If for ρ > 0 sufficiently small the smooth system (21) has a hyperbolic periodic orbitx ρ (t) satisfying the hypotheses of Theorem 3.1 with x(t) =x ρ (t) (see Remark 3.2(i)), then the piecewise-smooth system (18) has a hyperbolic periodic orbit of the same stability type asx ρ (t) in its O(ρ)-neighborhood. Moreover, if the periodic orbitx ρ (t) undergoes a local bifurcation at µ = µ 0 , then the same local bifurcation of the corresponding periodic orbit occurs in (18) at µ = µ 0 + O(ρ).
Proof. For simplicity we assume thatx ρ (t) passes through Π once as in the proof of Theorem 3.1. We first note that the period of x ρ (t) must be mT for some positive integer m (see, e.g., Exercise 1.5.3 of [12]). We apply Theorem 3.1 tox ρ (t) to obtain an orbitx(t) of (18), which may not be periodic, such thatx(t) =x Let x(t) and x ρ (t) be solutions of (18) and (21) with x(0) and x ρ (0) nearx(0) andx ρ (0), respectively. Define mT -time Poincaré mapsP andP ρ nearx(0) and x ρ (0), respectively, in a standard manner as By assumption, x(t) also passes through Π once, say, at t = t 0 , and it is C r on [0, t 0 ] and [t 0 , mT ] if only right-or left-hand higher-order continuous differentiability is required at the end of the intervals. Since it is a root of h(x(t 0 )) = 0, t 0 depends on x(0) in a C r manner. Hence, two maps are C r , so thatP =P 2P1 is also C r , although the system (18) is not smooth.
Sincex ρ (0) is a hyperbolic fixed point ofP ρ andP =P ρ + O(ρ) by Theorem 3.1, it follows from the persistence of hyperbolic fixed points [12] thatP has a hyperbolic fixed point of the same stability type asx ρ (0) in its O(ρ)-neighborhood. This means the first part. We use a standard result for smooth diffeomorphisms to prove the second part.
Remark 3.4. Obviously, we have the same result as Theorem 3.3 even when the system (18) is autonomous.

3.2.
Applications for the subharmonic Melnikov analyses. Now we reconsider the system (1) but only assume that H(x) is C r+1 , g(x, t; µ) is C r in x ∈ R 2 \ Π and they are nonsmooth for x ∈ Π while g(x, t; µ) is C r in t and µ, where Π = {x ∈ R 2 | h(x) = 0} and h : R 2 → R is C r . We also modify Assumption (A) as follows: (A1) There is a one-parameter family of periodic orbits, x α (t), α ∈ [α , α u ], with period T α . They are C r about t and α in R 2 \ Π but only continuous on Π. (A2) The left-and right-hand limits lim t→t0±0 JDH(x α (t), t) · Dh(x α (t)) are nonzero and have the same sign at any t = t 0 ∈ [0, T α ) such that x α (t 0 ) ∈ Π. Note that T α and I α are C r if Assumption (A1) holds since the sum of C r functions is also C r (see Eq. (3)). Letting Ω α = 2π/T α , we can also define the Melnikov functions using (9) for the piecewise-smooth case.
Using Theorem 3.3, we can prove the following result. Proof. We first approximate solutions of (1) up to O(ρ) by those of a smooth systeṁ where H ρ is C r+1 and g ρ is C r . Here f ρ = JDH ρ and g ρ satisfy (19) for f = JDH and g, respectively. Note that H ρ and g ρ can be chosen independently of . Hence, we see that the small constant ρ can be taken uniformly in [0, 0 ] for 0 > 0 sufficiently small. Moreover, the continuous family x α (t) are approximated by a smooth family of periodic orbits x α ρ (t), via Theorem 3.3 and Remark 3.4. The periodic orbits x α ρ (t) also satisfy Assumption (A2) with H = H ρ for ρ > 0 sufficiently small.
We apply Theorems 2.2-2.4 to the smooth system (25). Note that the Melnikov functions for (25) can also be approximated by those for the piecewise-smooth system (1). For instance, where M m/n denotes the Melnikov function estimated for the piecewise-smooth system (1). The following result gives a key to the proof of Theorem 3.5 and guarantees that the size of for which the statements of Theorems 2.2-2.4 hold does not tend to zero as ρ → 0.  Proof. As in Section 2.2, consider the Poincaré map P ,ρ for the action-angle coordinates (I, φ) in (25). We also define P ,0 = lim ρ→0 P ,ρ for ∈ [0, 1 ] with some small 1 > 0.
Recall that x α (t) is C r in (t, α) except on Π by Assumption (A1) and I α is C r in α. Hence, the symplectic transformation (6) is C r if x ∈ Π. Noting this fact and using the argument in the proof of Theorem 3.3, we take P ,ρ to be C r not only in (I, φ) but also in ∈ [0, 1 ] for ρ ∈ [0, ρ 0 ] . For ρ ∈ (0, ρ 0 ] we also estimate its mth iterate as (8)  For > 0 sufficiently small, periodic orbits detected by the Melnikov method in (25) for ρ ∈ (0, ρ 0 ) satisfy the hypotheses of Theorem 3.1 (they pass through Π only finitely many times and the left-and right-hand limits (19) are nonzero and have the same sign when they do) like the unperturbed orbits x α ρ (t). From the proof of Lemma 3.6 we see that P ,0 is the Poincaré map for the action-angle coordinates (I, φ) in (1). Using Lemma 3.6 and taking the limit ρ → 0, we complete the proof by Theorem 3.3.
We see that there are four saddle-node bifurcation curves and two cusp bifurcation points in the (ω, γ)-parameter space of Fig. 9(a). A fairly good agreement between the theoretical prediction and numerical computation is found not near the degenerate resonance point. In comparison with the prediction of Fig. 3(a), additional bifurcation curve and cusp exist. Such behavior typically occur in weakly forced nonlinear oscillators and overlooked by the version of the subharmonic Melnikov method described here: We have to rely on the subharmonic Melnikov method for weakly nonlinear systems (see Section 5 of [17]) or the averaging method (see, e.g., [18]). Note that is taken to be very small here, compared with the numerical example of Section 4. This is the reason why similar behavior was not observed in the numerical example of Section 4. Moreover, in the bifurcation diagram of Fig. 9(b) four saddle-node bifurcations are observed. Three of them correspond to the bifurcation curves passing through the line γ = 0.05 in Fig. 9(a), and the other one near (ω, γ) = (0.95, 0.01) is due to the nonsmoothness of the system. Such a nonsmooth saddle-node or fold bifurcation is typical in nonsmooth systems (see, e.g., [6]). Again, one of the saddle-node bifurcation curves suddenly disappears when it collides with the border collision curve in Fig. 9(a). 6. Conclusions. In this paper, we have extended the refined version of the subharmonic Melnikov method developed in [17,21] and demonstrated its usefulness and validity for bi-and trilinear oscillators. Fundamental results for approximating solutions of piecewise-smooth systems by those of smooth systems were given and used to obtain the main result. Special attention was paid to degenerate resonance behavior, and analytical results were illustrated by numerical ones via the computer software AUTO97 [7]. In particular, we showed that the degenerate resonance behavior can occur in continuous trilinear oscillators as well as in discontinuous bilinear oscillators.
Although demonstrated only for piecewise-linear systems, our theory is also valid for more general piecewise-smooth nonlinear systems, as shown for a mathematical model of vibrating microcantilevers in tapping-mode atomic force microscopy in [22,23]. Especially, a degenerate bifurcation behavior can also occur in such systems (cf. [23]). Theorems 3.1 and 3.3 also give a mathematical basis for applying other techniques such as the averaging and multiple-scale methods [14,15], which were developed for smooth systems, to piecewise-smooth systems. Moreover, by referring to the results of [9,10], our result may be extended to the case in which the unperturbed periodic orbits have jumps, i.e., are discontinuous.