On localized solutions of chains of oscillators with cubic nonlinearity

Transcription

1 On localized solutions of chains of oscillators with cubic nonlinearity Francesco Romeo, Giuseppe Rega Dipartimento di Ingegneria Strutturale e Geotecnica, SAPIENZA Università di Roma, Italia francesco.romeo@uniroma1.it, giuseppe.rega@uniroma1.it Keywords: localization, wave propagation, nonlinear maps. SUMMARY. Stationary localized solutions arising from the dynamics of chains of linearly coupled mechanical oscillators characterized by on site cubic nonlinearity are addressed aiming to explore their relationship with the underlying nonlinear wave propagation regions. 1 INTRODUCTION In this study the dynamics of chains of linearly coupled mechanical oscillators characterized by on site cubic nonlinearity is addressed. It is well known that such nonlinear periodic systems possess propagation and attenuation regions in the frequency domain. Nonlinear travelling wave solutions exist inside the propagation zones the boundaries of which can be analytically determined through a map approach when dealing with a discrete lattice structure [1]. Given the nature of the considered maps, namely reversible area preserving maps, periodic orbits can be also identified by resorting to their symmetry lines, i.e. fixed points of the involutions with which the map can be factorized. The analytical description of both, propagation regions boundaries and periodic orbits, enables to predict the nature of the periodic orbits and thereby to distinguish between hyperbolic and elliptic fixed points. Such a distinction plays a crucial role for the onset of stationary localized solutions in the nonlinear discrete lattice under investigation. A number of studies have indeed recently confirmed the ability of nonlinear lattices to support time-periodic and spatially localized solutions (discrete breathers) by linking them to the map homoclinic and heteroclinic orbits [2]. According to this geometric approach discrete breathers are identified with sequences of homoclinic or heteroclinic intersection points of the mapping so that the unstable fixed points govern their existence since stable and unstable manifolds emanate from them. The cases in which the trivial solution is hyperbolic or elliptic, namely outside or inside the linear passing band, respectively, were extensively investigated to show the existence of a variety of breather solutions. Simple, soliton-like solutions were identified with sequences of homoclinic or heteroclinic primary intersection points. Less insight is available for the other unstable fixed points belonging to the symmetry lines; in this respect, information drawn by both nonlinear propagation regions and internal bifurcation thresholds, merged with the higher order symmetry lines, are useful to locate and classify the relevant localized solutions, along with their features. In Section 1 the equations of motion of the mechanical system are introduced and the main aspects of its Hamiltonian dynamics described by a planar map are recalled. In particular, the emphasis is placed on the chain propagation regions showing how they affect the nature of the fixed point of the underlying map. In Section 2 the symmetry lines of the nonlinear map are obtained by means of recurrence formulas; these lines are used to identify the periodic orbits and to give more insight on the map phase portraits. In Section 3, the nonlinear propagation regions are used to locate and classify different types of localized solutions given by sequences of homoclinic or heteroclinic intersection points of the mapping. Some remarks and future developments are discussed in the concluding section. 1

2 2 THE OSCILLATORY CHAIN MODEL A simple mass and spring chain is considered to analyze the regions of existence of nonlinear localization phenomenon and their connection with the underlying nonlinear propagation regions. Discrete models similar to the studied one are often used to describe the nonlinear dynamics of micromechanical arrays [3, 4]. In this context the oscillation of a single cantilever is modeled following the Euler-Bernoulli beam theory, characterized by the cubic hardening nonlinearity. In essence through a Galerkin approach, in which the beam displacement is expanded in terms of the first linear mode, a sdof Duffing type equation is obtained, where the oscillator displacements describe the tip displacement amplitude of the n-th cantilever. Moreover, in the simplest model, the driven cantilever array accounts for nearest neighbor interaction only. In this study we consider a chain of linearly coupled nonlinear oscillators described by the following equations of motion m q n + k 1 q n + k 3 q 3 n + k(2q n q n 1 q n+1 )=0 (1) Time periodic solutions of equation (1) are sought for by assuming the harmonic solution q n = a n cos(ωt). Equating coefficients of cos(ωt) and setting α = mω2 k 1 k 2 and β = 3k3 4k the following second-order difference equation for the stationary amplitude is obtained (α + βa 2 n )a n + a n+1 + a n 1 =0 (2) In the frequency dependent parameter α in (2), k is the linear coupling stiffness while m and k 1 represent mass and linear stiffness of the single oscillator, respectively. In the nonlinear parameter β, k 3 represents the softening (β >0) or hardening (β <0) oscillator cubic stiffness. Equation (2), relating the amplitudes a in adjacent chain sites n 1, n and n +1, can be rewritten in matrix form as a n+1 = T(a n )a n, where a n+1 =(a n+1,a n ) and a n =(a n,a n 1 ); T(a n ) is the frequency dependent nonlinear transfer matrix belonging to the class of area preserving maps such that det(dt(a n )) = 1, where DT is the Jacobian or tangent map with reciprocal eigenvalues. Therefore, in order to study the stationary wave transmission properties of the one-dimensional nonlinear chain (1) of length N, it is convenient to rely on the eigenvalues of the linearized map equations in the neighborhood of an orbit ranging from (a 0,a 1 ) to (a N 1,a N ). Indeed, according to [5], by considering the introduced map as a dynamical system where the chain index n plays the role of discrete time n, the analysis of the transmission properties is equivalent to the stability analysis of the orbits. The interest lies in the linear stability of spatially periodic orbits a n+m = a n with cycle length m. The eigenvalues of DT are determined by its trace. If the eigenvalues lie on the unit circle, then stable elliptic periodic cycles or oscillating solutions occur; if the eigenvalues are real, then unstable hyperbolic periodic cycles or exponentially increasing solutions occur. By setting a n+1 = x n+1 and a n = y n+1, the map T can be written as T : x n+1 = y n x n (βx 2 n + α) y n+1 = x n (3) The stability of period-m orbits is governed by the condition tr(dt m ) =2; so, for m =1, the stability condition gives r := {(x, α, β) x 2 + α +2=0} s := {(x, α, β) x 2 + α 2=0} (4) 2

3 Table 1: Hyperbolic (r) and reflection hyperbolic (s) thresholds obtained from tr(dt m ) =2and m-periodic orbits p. m r s p 1 ± 2 α x = y = ± β 2 2 α ±i α x = y = 2 α β 3 4 ± 2 α 1 α ± 1 α ±i α 2 α ± 2 α 2 α x = y = 1 α β x = ±y = α β and the curves r and s represent hyperbolic (λ 1 = λ 2 =1) and reflection hyperbolic (λ 1 = λ 2 = 1) thresholds, respectively [1]. For period-m orbits, with m =1, 2, 3, 4, the latter curves are reported in Table 1. 3 SYMMETRY LINES AND PERIODIC ORBITS In this section the special properties of the map (3) are exploited to define and compute its symmetry lines with a twofold goal: to identify the periodic orbits and to give more insight on the map phase portraits. We begin by considering that the planar map (3) is reversible since it satisfies the condition R T R = T 1 where denotes the composition of two maps and R is called a reversing operator of T [6]; reversible maps have a number of special properties and occur often in applications. In particular, T is an area preserving map (4) for which the reversor R is an involution I 0, namely I 2 0 = 1. The higher order involutions are defined as I m := T m I 0, m = ±1, ±2,... The fixed points of the involutions with which the map can be factorized provide the symmetry lines, defined as γ m = Fix(I m )={ξ I m (ξ) =(ξ), ξ R} (5) The set γ m of I m is called the m-th order symmetry line satisfying the recursion relation (l = 0, 1, 2,...) [6] The main symmetry lines of the map (3) are summarized in Table 2, where the change of notation = n+1, = n is introduced. By defining the spatially m-periodic orbits by p m := {ξ T m (ξ) =(ξ)} the following relation 3

4 T l (γ m )=γ 2l+m m =0 T l (γ 0 )=γ 2l even symmetry line m =1 T l (γ 1 )=γ 2l+1 odd symmetry line Table 2: Main symmetry lines of the map (3) for β>0 and β<0. β>0 I 0 : I 1 : x = y y = x x = x y = y x(α + βx 2 ) γ 0 : y = x γ 1 : y = 1 2 x(βx2 + α) β<0 I 0 I 2 : x = y y = x x = x(α + βx 2 )+y y = x x(α + βx 2 ) 2 y(α + βx 2 ) γ 0 : y = x γ 2 : y = x( 1+α + βx 2 ) between periodic orbits and symmetry lines holds γ m γ l p m l (6) The symmetry lines and the relation (6) can be used to find spatially periodic solution of any order; in the last column of Table 1 they are reported for m =1, 2, 3, 4. In Figure 1 and Figure 2 the first 10 symmetry lines, generated by the involutions reported in Table 2, are superimposed on the phase portraits of the map (3) for two values of α. The periodic solutions in the figures are discussed against the T propagation region in Fig. 3a. In Figure 1, where α = 0.5 and β =1.0, the elliptic fixed point at (0, 0) as well as the period-4 orbits, located at x = ±y = ± 0.5 (both inside the pass band in Fig. 3a), are clearly highlighted by the relevant symmetry lines intersections obeying to (6); the quasi-periodic orbits scenario surrounding these points is also clearly related to the symmetry lines (Figure 1a). In Figure 1b, the symmetry lines are shown on a wider x, y range showing a large number of intersections (i.e. periodic solutions) confined in the range 2.5 x 2.5 (light gray); these boundaries correspond to hyperbolic period-2 orbits. Figure 2 refers to α = 3.0 and β =1.0; in Figure 2a the phase portrait refers to orbits corresponding to positive initial conditions x 0,y 0. The period-1 orbit at x = y =1undergoes a quadrupling bifurcation and the arrangement of the associated quasi-periodic orbits is governed by the main symmetry lines. The global view in Figure 2b shows that the scenario is mirrored with respect to both the main diagonals and that the symmetry lines intersections (periodic solutions) occur only within two separated regions given by 5 x 1 and 1 x 5. These regions are bounded below by the elliptic period-1 orbits and above by the hyperbolic period-2 ones. The analytical description of both, propagation regions boundaries and periodic orbits, enables to predict the nature of the periodic orbits and thereby to distinguish between hyperbolic and elliptic 4

5 Figure 1: Phase portrait of the map for α = 0.5, β =1.0 and the first 10 symmetry lines generated by the involutions I 0, I 1 in Table 2. solutions. As it will be shown in the next section, such a distinction plays a crucial role for the onset of stationary localized solutions in the nonlinear discrete lattice under investigation. 4 LOCALIZED SOLUTIONS AGAINST PROPAGATION REGIONS According to [7] the discrete breathers, i.e. time-periodic spatially localized solutions, are solutions of (1) with period T b =2πΩ b that can be expanded in a Fourier series as follows q n (t) = k A kn e ikω bt with A k, n 0 (7) where the assumption of Fourier coefficients localized in space is introduced. Following the approach proposed in [8] and [9] for hardening and softening type of nonlinearity, respectively, we look for localized states by relating them to the intersections of the invariant manifolds emanating from unstable fixed points of the map (3). As known, hyperbolic fixed points P possess stable W s (P ) and unstable W u (P ) manifolds such that a point belonging to W s (P )(W u (P )) approaches P under map iteration T n for n (n ). A homoclinic point Q is defined as Q W s W u ; thus, any Q belongs to both the stable and unstable manifold of P so, when iterated either forward or backward, it will converge to P and it corresponds to a breather solution. Due to the entangled pattern of the manifolds, additional intersection points are generated implying an infinite number of different breathers and multi-breathers, where more than one site exhibits large amplitude oscillations [7]. In Figure 3 the regions of existence of localized solutions are shown on the (α, x) plane for β =1.0 (Figure 3a) and β = 1.0 (Figure 3b); these regions are classified according to the nature of the map T fixed points, namely the trivial solution (x = y =0) and the period-1 solutions (x = y = β ). For the sake of completeness, the period-2 orbits are also reported as they play a twofold role. As shown in Figures 1b and 2b, they represent the upper boundary of all the periodic solutions; moreover, their location on the x = y symmetry line affects the topology of the invariant manifolds. Regions I +,I correspond to discrete breathers associated to the homoclinic connections emanating 5

6 Figure 2: Phase portrait of the map for α = 3.0, β =1.0 and the first 10 symmetry lines generated by the involutions I 0, I 1 in Table 2. from the trivial solutions for β>0 and β<0, respectively. Region II corresponds to nested homoclinic connections arising from the period doubling bifurcations series of period-1 orbits beginning at the intersection with the propagation region boundary s; for β =1, the latter bifurcation series occurs in the range α 4.0. As shown in [10], each period doubling bifurcation can be identified by the intersections of the newborn elliptic solutions and the relevant s-type higher order internal thresholds. Region III refers to the heteroclinic connections between the hyperbolic period-1 orbits. To validate the analytical predictions, numerical simulations pertaining to localized solutions belonging to regions I +, II and III are reported in the following. Figure 3: Projection on the (α, x) plane of the regions of existence of localized solutions against the T propagation regions. The dashed and solid lines correspond to hyperbolic and elliptic points, respectively; a) β = 1.0 (softening nonlinearity); b) β = 1.0 (hardening nonlinearity). In Figure 4 the localization scenario corresponding to the parameters α = 2.8, β = 1 be- 6

7 longing to region I + is illustrated. In Figure 4a stable (blue) and unstable (red) manifolds of (0, 0) surrounding the elliptic period-1 points (green) located at x = y = ±0.894 are shown; in Figure 4b a closer view of the manifolds is displayed with two families of homoclinic orbits; the ensuing discrete breathers are also shown in Figures 4c,d. Figure 4: Stable (blue) and unstable (red) manifolds of (0, 0) at α = 2.8, β =1(region I + ) and two homoclinic orbits; a) global view; b) closer view with intersections forming two families of homoclinic orbits; c) discrete breather corresponding to black dots in b); discrete breather corresponding to white dots in b). Figure 5 refers to region II localization, characterized by nested homoclinic orbits at α = 4.05). As shown by the global view of the invariant manifolds (Figure 5a), besides the global homoclinic connection emanating from the trivial solution, two more homoclinic tangles arise. They stem from new born period-1 hyperbolic branch following the period doubling bifurcations occurring at (± 2.0, ± 2.0) for α = 4.0. The closer view in Figure 5b shows one of the two homoclinic connections in which the newborn elliptic points are located along the γ 1 (see Table 2) symmetry line (dashed line). The associated homoclinic orbit (black dots) gives rise to the multibreather displayed in Figure 5d, whereas Figure 5c refers to a breather corresponding to a global homoclinic orbit. The numerical results presented in Figure 6 are related to the localized solutions associated with the heteroclinic connections occurring in region III. In particular, the invariant manifolds of ( 2, 2) and ( 2, 2) shown in Figure 6a are obtained for the parameters α =0.0, β = 1. The black 7

8 Figure 5: Stable (blue) and unstable (red) manifolds of (0, 0) and (± 2.05, ± 2.05) at α = 4.05, β =1and homoclinic orbits; a) global view; b) closer view of the manifolds of ( 2.05, 2.05); c) discrete breather corresponding to a homoclinic orbit of (0, 0); discrete multibreather corresponding to a homoclinic orbit of ( 2.05, 2.05). dots correspond to the discrete breather depicted in Figure 6b. 5 CONCLUSIONS The nonlinear propagation regions of chains of oscillators with cubic nonlinearity are used to identify the regions of existence of discrete breathers, i.e. time periodic and spatially localized solutions, and to guide their analysis. By describing the map symmetry lines through recurrence formulas, the phase portrait topological features and the periodic orbits arrangement have been discussed. Moreover, by merging the analytical descriptions of the map symmetry lines and propagation bands, the regions where the spatially periodic orbits are hyperbolic (saddle) or elliptic are determined. Since discrete breathers correspond to sequences of homoclinic or heteroclinic intersection points, the unstable fixed points govern their existence. Besides the known regions corresponding to localized solutions stemming from the trivial fixed points, located either outside (homoclinic connections) or inside (heteroclinic connections) the linear passing band, a further region has been identified. It corresponds to nested homoclinic orbits, emanating from period-1 unstable solutions, giving rise to multibreathers. In agreement with recent literature findings, the larger region of existence of 8

9 Figure 6: Stable (blue) and unstable (red) manifolds of ( 2, 2) and ( 2, 2) at α =0.0, β = 1; a) heteroclinic orbits; b) discrete breather corresponding to the heteroclinic orbit highlighted in a). breathers for the hardening cubic nonlinearity with respect to the softening one has been confirmed. Further studies will address the stability analysis in time of the different families of localized solutions. References [1] Romeo, F., Rega, G., Wave propagation properties of chains of oscillators with cubic nonlinearities via nonlinear map approach, Chaos, Solitons and Fractals, 27, (2006). [2] Hennig, D., Rasmussen, K.., Gabriel, H. and Blow, A., Soliton-like solutions of the generalized discrete nonlinear Schrödinger equation, Phys. Rev. E, 54, (1996). [3] Sato, M., Hubbard, B.E., Sievers, A.J., Nonlinear energy localization and its manipulation in micromechanical oscillator arrays, Rev. Modern Phys., 78, , (2006). [4] Sato, M., Sievers, A.J., Driven localized excitations in the acoustic spectrum of small nonlinear macroscopic and microscopic lattices, Phys. Rev. Lett., 98, , (2007). [5] Hennig, D., Tsironis, G.P., Wave transmission in nonlinear lattices, Physics Reports, 307, , (1999). [6] Shi, Y.G., Chen, L., Reversible maps and their symmetry lines, Commun Nonlinear Sci Numer Simulat., 16, (2011). [7] Flach, S., Gorbach, A.V., Discrete breathers Advances in theory and applications, Phys. Reports, 467, (2008). [8] Bountis, T., Capel, H.W., Kollmann, M., Ross, J.C., Bergamin, J.M., van der Weele J.P., Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices, Phys. Lett. A, 268, 50-60, (2000). 9

10 [9] Panagopoulos, P., Bountis, T., Skokos, C., Existence and stability of localized oscillations in 1-dimensional lattices with soft-spring and hard-spring potentials, J. of Vib. and Acoustics, 126, , (2004). [10] Romeo, F., Rega, G., Propagation properties of bi-coupled nonlinear oscillatory chains: analytical prediction and numerical validation, Int. J. of Bif. and Chaos, 18, , (2008). 10