NearIntegrability and Recurrence in FPU Chains with Alternating Masses
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Abstract
An important aspect of understanding FPU chains is the existence of invariant manifolds (called “bushes”) in FPU chains. We will focus on the classical periodic FPU chain and on the FPU chain with alternating masses where we show that also in the alternating case nested manifolds (related to bushes) exist. The use of symmetries leads to the emergence of systems of n particles as invariant manifolds of systems with a multiple of n particles. This analysis is followed by examples of existence and stability of special invariant manifolds and phasespace dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles. It turns out that in the alternating case the dynamics is strongly affected by the choice of the alternating mass m. Normal form calculations help to identify quasitrapping regions leading to delay of recurrence. The results suggest that equipartition of energy near stable equilibrium is improbable.
Keywords
Alternating FPU Invariant manifolds Symmetry Stability QuasitrappingMathematics Subject Classification
70H07 70H12 34E10 37J151 Introduction
The Fermi–Pasta–Ulam (FPU) problem has been the subject of many papers since its formulation in 1955. For a recent study of localization, recurrence and references see Christodoulidi et al. (2010). An important aspect of understanding FPU chains is the existence of invariant manifolds (called “bushes”) in classical FPU chains; basic results are found by in Chechin and Sakhnenko (1998), Chechin et al. (2002) and Chechin et al. (2005). In these papers grouptheoretical methods were used for coupled oscillator systems with special attention to \(\alpha \) and \(\beta \) FPU chains. See for the existence of invariant manifolds for \(\beta \)chains also (Rink 2001). We will focus on the FPU chain with alternating masses (which includes the classical periodic FPU chain) to show that also in the alternating case nested manifolds (bushes) exist. This is followed by examples of existence and stability of invariant manifolds and phasespace dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles.
The alternating case was studied by Galgani et al. (1992) for a FPU chain with fixed end points using mainly numerical tools for special cases to obtain insight into the equipartition of energy, in particular between the low (acoustical) frequency and the high (optical) frequency part. The terminology acoustical–optical derives from physics. The spectrum of the linearized system scales with the alternating mass m (or its inverse) producing a natural split of the frequency spectrum. See Sect. 3.3. A preliminary but important conclusion by Galgani et al. (1992) is that for the masses considered and on long timescales no equipartition takes place; the evidence is numerical. Inspired by these results we will study the periodic FPU problem in the case of alternating masses. The emphasis will be on invariant manifolds, periodic solutions, integrability of the normal forms (nearintegrability of the original system), chaos and recurrence phenomena; for recurrence see also Verhulst (2016, 2017a) for chains of FPU cells.
The periodic Fermi–Pasta–Ulam problem with alternating masses presents complicated dynamics. For the monoatomic case of the original periodic FPU problem with all masses equal it was shown by Rink and Verhulst (2000) for up to six degreesoffreedom (dof) and for an arbitrary number of dof by Rink (2001), that the corresponding normal forms are governed by 1 : 1 resonances and that these Hamiltonian normal forms are integrable. This explains the recurrence phenomena near stable equilibrium for long intervals of time.
Consider the periodic FPU chain with an even number (2n) of alternating masses 1, m, 1, m, \( \ldots ,\), with \(m>0\), which is to some extent related to the formulation in Galgani et al. (1992) where the emphasis is on energy partition for large systems (\(n=6, \ldots , 200 \)) with fixed end points and \(0 < m \le 1\). The mass ratio m : 1 is the important parameter; we choose \(a= 1/m\), \(0 < a \le 1\). If \(a=1\) we have the classical case of equal masses. By a symmetry argument we will leave out the case \(a>1\).
The eigenvalue spectrum of system (2) will be indicated by \(\lambda _i,\,i= 1, \ldots , 2n\); the corresponding frequencies of the linear normal modes are \(\omega _i = \sqrt{\lambda _i}\). The numerical value of \(H_2\) (for given initial conditions) is indicated by \(E_0\); near stable equilibrium and assuming that m is not very large, we can rescale \(p \rightarrow \varepsilon p,\, q \rightarrow \varepsilon q\) with \(\varepsilon \) a small positive parameter. Dividing by \(\varepsilon ^2\), we obtain from Hamiltonian (1) the Hamiltonian \(H_2+ \varepsilon \alpha H_3 + \varepsilon ^2 \beta H_4\) with \(H_3\) a cubic polynomial in (p, q), \(H_4\) quartic in (p, q). We have clearly for the energy \(E= H_2_{t=0} + O(\varepsilon )\) for all time.
The shortperiodic solutions of the equations of motion linearized near the origin are called the linear normal modes of the system. One of the questions that arise are whether the linear normal modes can be continued for the system with nonlinear interactions or not. The transformation to quasiharmonic form is natural, but may cause confusion. In position and impulse space we have the basis vectors \(e_1, e_3, \ldots ,e_{2n1}\) corresponding to particles with mass 1, and basis vectors with even index corresponding to the heavier particles with mass \(m=1/a\). After symplectic transformation we have n eigenvalues that are O(a), giving the “acoustic frequencies,” and n eigenvalues that are of order 1, giving the “optical frequencies.” The description of each eigenvector (corresponding with a socalled normal mode) involves a mix of particles of mass 1 and of mass m. The behavior of the solutions within the two sets of particles cannot in a simple way be identified with the normal mode (quasiharmonic) equations corresponding with the optical and acoustical part of the spectrum.
Note that according to Weinstein (1973) a n dof Hamiltonian system near stable equilibrium contains at least n families of periodic solutions parametrized by the energy. We will keep this in mind when looking for periodic solutions in particular systems.
In the sections on chains with 4 or 8 particles the analysis by averaging normal forms is a basic tool. For the normal form theory and results in the case of Hamiltonian systems see Sanders et al. (2007) ch. 10. In the analysis resonances in the frequency spectrum of the linearized equations of motion, generated by the quadratic part of the Hamiltonian \(H_2\), play a fundamental part. The cubic part \(H_3\) and if necessary the quartic part \(H_4\) will be normalized to \({\bar{H}}_3, {\bar{H}}_4\).
The general theory and proofs of averaging for ODEs involve spatial variables and time with bounds (independent of \(\varepsilon \)) on the spatial domain and the initial conditions. It is important to note that in the case of a Hamiltonian system near stable equilibrium we have dynamics on a bounded energy manifold. The implication is that the normal form analysis and its error estimates are valid for an arbitrary number of dof if the initial energy is bounded independent of \(\varepsilon \).
The numerical experiments were carried out by matcont under Matlab with ode code 78, for instance for Euclidean distances in phase space. The precision was increased until the picture did not change anymore with typical relative error \(e^{15}\), absolute error \(e^{15}\). Basic normal form computations were carried out using Mathematica.
In Sects. 2 and 3 we present results on periodic solutions and invariant manifolds for alternating periodic FPU chains with an even number of particles. Section 4 summarizes results for 4 particles, features that will be found again in periodic FPU chains with 4n particles. Section 5 contains the formulation of the system for 8 particles, where we find 3 invariant manifolds and study their stability. The normal form analysis of Sect. 6 produces interesting phenomena. At firstorder normalization only the cases \(m=2\) and \(m=4/3\) give nontrivial results. In particular the analysis of the dynamics near unstable invariant manifolds for \(m=4/3\) yields insight into the presence of resonance zones, recurrence and quasitrapping phenomena. These features will be found again in periodic FPU chains with 8n particles.
2 The 2n Particles FPU Chain, Three Families of Periodic Solutions
 Solutions determined byproducing the equations of motion (harmonic for an \(\alpha \)chain)$$\begin{aligned} q_{2j}(t) =0, j=1, \ldots , 2n; q_1(t)= q_3(t)=q_5(t)=q_7(t)= \cdots = q_{4n1}(t), \end{aligned}$$For a \(\beta \)chain we add a cubic term to the equations of motion. On a given energy manifold this represents a oneparameter family of periodic solutions.$$\begin{aligned} \ddot{q}_{2j1} + 2 q_{2j1} =0,\, j=1, 2, \ldots , 2n. \end{aligned}$$(6)
 Solutions determined byproducing the equations of motion (harmonic for an \(\alpha \)chain)$$\begin{aligned} q_{2j1}(t) =0, j=1, \ldots , 2n; q_2(t)= q_4(t)=q_6(t)=q_8(t)= \cdots = q_{4n}(t), \end{aligned}$$For a \(\beta \)chain we add a cubic term to the equations of motion. On a given energy manifold this represents a oneparameter family of periodic solutions.$$\begin{aligned} \ddot{q}_{2j} + 2a q_{2j} =0,\, j=1, 2, \ldots , 2n. \end{aligned}$$(7)
 Because of the symmetry of the equations we expect that solutions exist of the form:For an \(\alpha \)chain this leads to the linear system:$$\begin{aligned} q_1(t)=q_3(t)= \cdots = q_{4n1},\,\, q_2(t)=q_4(t)= \cdots = q_{4n}(t). \end{aligned}$$The eigenvalues are 0 and \(2(1+a)\); the solutions exist, are periodic (harmonic) and will have frequency \(\sqrt{2(1+a)}\). For a \(\beta \)chain we find:$$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{q}_1 + 2q_1 = &{} 2q_2,\\ \ddot{q}_2 + 2aq_2 = &{} 2aq_1. \end{array}\right. } \end{aligned}$$We conclude that \(a \ddot{q}_1 + \ddot{q}_2 =0\) so that we can eliminate \(q_2\). For \(q_1\) we find the equation:$$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{q}_1 + 2q_1 = &{} 2q_2  2 \varepsilon (q_1q_2)^3,\\ \ddot{q}_2 + 2aq_2 = &{} 2aq_1 + 2a \varepsilon (q_1q_2)^3. \end{array}\right. } \end{aligned}$$The solutions exist and are elliptic periodic functions.$$\begin{aligned} \ddot{q}_1 + 2(1+a)q_1 =  2 \varepsilon (1+a)^3q_1^3. \end{aligned}$$
3 The 2n Particles FPU Chain, Invariant Manifolds
We consider the general case of the alternating periodic FPU chain with 2n particles. We will give a relation between the system with 2n particles and an invariant submanifold of the system with 2k particles for each multiple k of n.
3.1 Earlier Results
In a seminal paper Chechin and Sakhnenko (1998) grouptheoretical methods were used for systems with certain symmetries. From irreducible representations of the symmetry group the authors conclude to the existence of specific dynamical regimes (called bushes) of essentially lower dimension than the dimension of the original systems. The theory was developed in Chechin and Sakhnenko (1998) both for Hamiltonian and nonHamiltonian systems with a number of physical applications.
Independently, in Poggi and Ruffo (1997) the periodic FPU \(\beta \)chain was considered for which special periodic solutions and twomode invariant manifolds were found; the analysis in Poggi and Ruffo (1997) is by inspection of equations and not by exploiting symmetries. The paper Chechin and Sakhnenko (1998) was continued in Chechin et al. (2002) to find invariant manifolds (bushes) in classical periodic, monatomic FPU chains. The analysis leads to the existence of a wide range of multimode invariant manifolds that include the results in Poggi and Ruffo (1997). Exploiting symmetries, a number of new invariant manifolds for the classical, periodic FPU chain were given in Rink (2001).
In this section we will exploit the symmetries of the periodic FPU chain with alternating masses to obtain a great many invariant manifolds in Theorem 3.1. The proof also leads to the determination of the eigenvalues, in Proposition 3.2.
3.2 Invariant Manifolds
Theorem 3.1
Let \(\alpha ,\beta \in {\mathbb C}\), \(a\in {\mathbb C}{\setminus }\{0\}\), \(n\ge 2\), and let k be a multiple of n. There is a 2ndimensional subspace \(M_{k,n}\subset {\mathbb C}^{2k}\) such that the restriction of the system \(\mathrm {FPU}_{2k}(a;\alpha ,\beta )\) to \(M_{k,n}\) is equivalent to the system \(\mathrm {FPU}_{2n}(a;\alpha ,\beta )\).
By equivalence we mean that there is a bijective map \(\Phi : {\mathbb C}^{2n}\rightarrow M_{k,n}\) such that the image \(t\mapsto \Phi q(t)\) of the solution \(t\mapsto q(t)\) with initial values \((q(0),q'(0))\in {\mathbb C}^{2n}\) is the solution in \({\mathbb C}^{2k}\) with initial values \(\left( \Phi q(0), \Phi q'(0)\right) \).
The cyclic group of order n acts naturally on the system \(\mathrm {FPU}_{2n}(a;\alpha ,\beta )\). It is generated by the shift over two particles (\(j\mapsto j+2\) modulo 2n). This is the group of symmetries used in the proof of the theorem. It enters via the dual group \(U_n\) of nth roots of unity.
Proof of Theorem 3.1
Now let k be a multiple of n. We form the linear map \(\Phi _{n,k}:{\mathbb C}^{2n}\rightarrow {\mathbb C}^{2k}\) determined by \(\Phi _{n,k} v_{2n}(\zeta ) = v_{2k}(\zeta )\), \(\Phi _{n,k}w_{2n}(\zeta ) = w_{2k}(\zeta )\). The image of \(\Phi _{n,k}\) is a linear subspace \(M_{k,n}\) of \({\mathbb C}^{2k}\). This subspace is determined by the condition that \(X_{2k}(\zeta ) \subset M_{kn}\) if \(\zeta \in U_n\subset U_k\) and \(X_{2k}(\zeta ) \cap M_{k,m}=\{0\}\) if \(\zeta \in U_k{\setminus }U_n\).
Suppose \(x\in M_{k,n}\). If \( \zeta \in U_k{\setminus }U_n\), then \(x(\zeta )=0\), and all factors in the righthand side vanish as well, since at least one of the \(\zeta _j\) has to be in \(U_k{\setminus }U_n\). If \(\zeta \in U_n\), then the form of the righthand side in (16) is the same as in the system with 2n particles. If some \(\zeta _j\) is not in \(U_n\), then the term \(T(\zeta _1,\zeta _2)\), respectively, \(S(\zeta _1,\zeta _2,\zeta _3)\) is zero. So the descriptions of the system of differential equations with 2n particles and the restriction to \(M_{k,n}\) of the system with 2k particles are exactly equal. \(\square \)
Bushes of Modes. This theorem can be seen as an example of the theory of Chechin and Sakhenko (1998). See also Chechin et al. (2002), where this theory is applied to the classical FPU chain with equal masses. For each divisor n of k the space \(M_{k,n}\) is the “bush” associated with the subgroup \(U_n \) of \(U_k\).
For the \(\beta \)chain the total group of symmetries is larger than the cyclic group used in the theorem. It is the dihedral group generated by the shift \(j\mapsto j+2\bmod 2n\) and the reflection \(j\mapsto 2j \bmod 2n\). The reflection interchanges \(X(\zeta )\) and \(X(\zeta ^{1})\).
Real Systems. Let \(a>0\), \(\alpha ,\beta \in {\mathbb R}\). The system (9) is a system of secondorder differential equations in \({\mathbb R}^{2n}\). For roots of unity \(\zeta \) that are not real the vectors \(v_{2n}(\zeta )\) and \(w_{2n}(\zeta )\) are not in \({\mathbb R}^n\), and the decomposition (13) does not respect the real structure. To get a real basis we have to consider \(X_{2n}(\zeta ) \oplus X_{2n}(\bar{\zeta })\), which intersects \({\mathbb R}^{2n}\) in a space with real dimension 4. That space is invariant under the operators \(A_{2n}C_{2n}\), \(D_{2n,+}\cdots \).
Periodic Solutions. In Sect. 2 we mentioned periodic solutions. We now recognize them as solutions of the form \(t\mapsto f(t) \, v_{2n}(1)\) and \(t\mapsto f(t)\, w_{2n}(1)\) living in \(X_{2n}(1)\).
There are also solutions in the space \(X_{2n}(1)\). One of them is the solution \(f(t) \left( v_{2n}(1)+ w_{2n}(1)\right) \) with \(f'=0\). It is in the translational component of the system. The third type of solution in Sect. 2 with eigenvalue \(2(a+1)\) is of the form \(t\mapsto f(t) \left( v_{2n}(1)a\, w_{2n}(1) \right) \).
Invariant Manifolds. The space \(M_{k,n}\subset {\mathbb C}^{2k}\) is a 2ndimensional linear subspace of the position space \({\mathbb C}^{2k}\) of the system \(\mathrm {FPU}_{2k}(a;\alpha ,\beta )\). In the context of Theorem 3.1 that seems a natural approach, since the symmetric group acts in the same way on positions and impulses of particles. The corresponding invariant manifold is a 4ndimensional linear subspace of the phase space \({\mathbb C}^{4k}\).
3.3 Eigenvalues
Proposition 3.2
Proof

From \(\zeta =1\) we have the eigenvalues 0 and \(2(a+1)\).

If n is even (4n particles), \(\zeta =1\) is one of the possible roots leading to eigenvalues 2 and 2a.

The other roots \(\zeta ^j\) and \(\zeta ^{nj}\) with \(1 \le j < n/2\) produce eigenvalues with multiplicity 2.
 The eigenvalues with \(0 \le j \le n1\) behave for \(a \downarrow 0\) asSo, for large mass values (\(a \downarrow 0\)) the eigenvalue spectrum consists of two groups, one with size \(2 + O(a)\) (the socalled optical group) and the second one with size O(a) (the socalled acoustical group).$$\begin{aligned} 2+a\left( 1 + \cos \frac{2 \pi j}{n} \right) + O(a^2) \,\, \mathrm{and}\,\, a\left( 1 \cos \frac{2 \pi j}{n}\right) + O(a^2). \end{aligned}$$
4 Four Alternating Masses, a Summary
This case has been analyzed in Bruggeman and Verhulst (2017b). The dynamics of this case will be found again in systems with 8 particles, in general 4n particles.
For the \(\alpha \) and \(\beta \)chain we find no three dof firstorder resonances in a cell with four particles. There are two dof resonances of which the normal form is integrable to a high order.
5 A Periodic Chain with Eight Particles
One can find a number of prominent resonances, but, as we will show below, of special interest are the cases \(a = \frac{1}{2}\), \(a = \frac{3}{4}\) and a near zero. The calculation of the normal forms \(H_3\) and \(H_4\) shows that these values of a correspond with effective resonances. The eigenvalue \(\lambda =0\) corresponds with the momentum integral \(\sum _1^8 m_i{\dot{q}}_i=\) constant (see 8) producing families of translations. We will use this integral to reduce the system to seven dof. For all choices of a and after this reduction there will be at least seven families of periodic solutions (Weinstein 1973).
The increased complexity of the eight particle system can be handled, but we restrict most results to \(\alpha \)chains.
5.1 Symplectic Transformation
The coefficients of \(H_3\), expressed in the eigencoordinates \(x_j\) where \(d_{\alpha \beta \gamma }\) is the coefficient of \(x_\alpha x_\beta x_\gamma \)
\(d_{\alpha \beta \gamma }\)  \(x_\alpha x_\beta x_\gamma \)  \(d_{\alpha \beta \gamma }\)  \(x_\alpha x_\beta x_\gamma \) 

\(\frac{\sqrt{a} \left( \sqrt{a^2+1}+a+1\right) }{2 \sqrt{a^2+1}}\)  \(x_2^2 \ x_5\)  \(\frac{\sqrt{a} \left( \sqrt{a^2+1}+a+1\right) }{2 \sqrt{a^2+1}}\)  \(x_3^2 \ x_5\) 
\(\frac{\sqrt{a} \left( \sqrt{a^2+1}+a+1\right) }{2 \sqrt{a^2+1}}\)  \(x_5 \ x_6^2\)  \(\frac{\sqrt{a} \left( \sqrt{a^2+1}a1\right) }{2 \sqrt{a^2+1}}\)  \(x_5 \ x_7^2\) 
\(\sqrt{2} \sqrt{a} \sqrt{a+1}\)  \(x_1 x_2 x_7 \)  \(\sqrt{2} \sqrt{a} \sqrt{a+1}\)  \(x_1 x_3 x_6\) 
\(\frac{a \left( \sqrt{a^2+1}+a+1\right) }{\sqrt{a^2+1}}\)  \(x_2 x_3 x_4\)  \(2 \sqrt{a} \sqrt{a+1}\)  \(x_1 x_4 x_5\) 
\(\frac{\sqrt{2} \sqrt{a}}{\sqrt{a^2+1}}\)  \(x_3 x_4 x_6\)  \( \frac{\sqrt{2} a^2}{\sqrt{a^2+1}}\)  \(x_2 x_5 x_6\) 
\(\frac{\sqrt{2} \sqrt{a}}{\sqrt{a^2+1}}\)  \(x_2 x_4 x_7\)  \( \frac{\sqrt{2} a^2}{\sqrt{a^2+1}}\)  \(x_3 x_5 x_7\) 
\(\frac{a \left( \sqrt{a^2+1}+a+1\right) }{\sqrt{a^2+1}}\)  \(x_4 x_6 x_7\) 
For \(H_4\) we find an expression with 49 terms of the form \(e_{i,j,k,l}x_ix_jx_kx_l\), among which occur all fourth powers \(x_j^4\), all products \(x_i^2 x_j^2\), and 21 other terms. The coefficients \(d_{ijk}\) and \(e_{ijkl}\) are algebraic functions of the parameter a. The coefficients \(d_{ijk}\) of \(H_3\) and \(e_{ijkl}\) of \(H_4\) are all nonzero for \(0<a\le 1\); the expressions for the \(d_{ijk}\) are given in Table 1.
 1.Manifold \(M_{145}\) composed by the \(x_1, x_4\) and \(x_5\) modes, associated with the eigenvalues \(2(1+a), 2\) and 2a. The dynamics is described by:For the \(\alpha \)chain this system was treated already in the system with four particles; see Eq. (18). This is an example of Theorem 3.1 showing how lowerdimensional invariant manifolds emerge as invariant manifolds in systems with more particles; see Sect. 3. In the case of the \(\alpha \)chain the \(x_1, x_4, x_5\) normal modes are harmonic solutions.$$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{x}_1 + 2(1+a) x_1 &{}=  \varepsilon x_4x_5 d_{145}, \\ \ddot{x}_4 + 2 x_4 &{}=  \varepsilon x_1 x_5 d_{145}, \\ \ddot{x}_5 + 2a x_5 &{}=  \varepsilon x_1 x_4 d_{145}. \end{array}\right. } \end{aligned}$$(24)
 2.A second 6dimensional invariant manifold \(M_{256}\) is composed from the \(x_2, x_5\) and \(x_6\) modes. The equations of motion are:The \(x_5, {\dot{x}}_5\) coordinate plane contains harmonic solutions.$$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{x}_2 + \lambda _2 x_2 &{}=  \varepsilon (2x_2 x_5 d_{225}+x_5 x_6 d_{256}), \\ \ddot{x}_5 + 2a x_5 &{}=  \varepsilon (x_2^2 d_{225}+x_6 x_2 d_{256}+x_6^2 d_{665}), \\ \ddot{x}_6 + \lambda _6 x_6 &{}=  \varepsilon (x_2 x_5 d_{256}+2 x_5 x_6 d_{665}). \end{array}\right. } \end{aligned}$$(25)
 3.A third 6dimensional invariant manifold \(M_{357}\) is composed from the \(x_3, x_5\) and \(x_7\) modes. The equations of motion are:The \(x_5, {\dot{x}}_5\) coordinate plane contains harmonic solutions.$$\begin{aligned} {\left\{ \begin{array}{ll} \ddot{x}_3 + \lambda _3 x_3 &{}=  \varepsilon (2 x_3 x_5 d_{335}+x_5 x_7 d_{357}), \\ \ddot{x}_5 + 2a x_5 &{}=  \varepsilon ( x_3^2 d_{335} +x_7^2 d_{775} +x_3 x_7 d_{357}), \\ \ddot{x}_7 + \lambda _7 x_7 &{}=  \varepsilon (x_3 x_5 d_{357}+2 x_5 x_7 d_{775}). \end{array}\right. } \end{aligned}$$(26)
5.2 Stability of the Invariant Manifolds
We turn to the question whether the invariant manifolds \(M_{145}\), \(M_{256}\) and \(M_{357}\) are stable or not on the energy manifold in 14dimensional phase space. For Hamiltonian systems the stability of solutions or manifolds is studied by linearization near the solution or manifold. The set of eigenvalues of the linearization is invariant under \(\lambda \mapsto  \lambda \) and \(\lambda \mapsto {\bar{\lambda }}\). There are at least two eigenvalues 0. The analysis is simple in the generic cases of two dof; the presence of eigenvalues with nonzero real part means instability, and stability occurs if the real parts of all eigenvalues are zero. For three or more dof the presence of eigenvalues with nonzero real part means instability because of the reflection invariance, but the fact that all real parts of eigenvalues are zero is inconclusive for stability. Higherorder terms may destroy the linear stability. Another complication is that if firstorder normal terms vanish, we should consider higherorder resonances as instability may arise at longer timescales. Both the cases of purely imaginary eigenvalues and vanishing firstorder normal forms will be indicated by quasistability or spectral stability. For a number of values of the mass ratio a quasi or spectral stability of \(M_{145}\), \(M_{256}\) and \(M_{357}\) is the rule. An important exception is that for the value \(a=\frac{3}{4}\) the three invariant manifolds are unstable.
Perturbation around mode 4 gives an unstable case \(a= 0.75\) as the matrix \(M_{4,3/4}\) has two positive eigenvalues \(\frac{3p}{20\sqrt{2}}\), two negative eigenvalues and the eigenvalue 0 with multiplicity 10. So the system is unstable at the eigenmode 4. Inspection of the eigenvectors for the positive eigenvalues shows that the unstability is in the direction of the variables \(x_6, x_7,y_6,y_7\). These directions are not in the invariant manifold \(M_{145}\). Hence the manifold is unstable. The eigenmode \(k=4\) is outside the invariant manifolds \(M_{256}\) and \(M_{357}\). The instability results are illustrated in Figs. 2, 3, and 4.
Perturbation around normal mode 5 is not conclusive as the matrix \(M_{5,1/2}\) has eight purely imaginary nonzero eigenvalues and the eigenvalue 0 with multiplicity 6. From this spectral stability we cannot draw definite conclusions concerning stability.
The stability analysis thus far is restricted to linearization around the modes 1, 4, 5, it produces nontrivial results for \(a= 0.5\) and 0.75 only. In the next section we obtain additional results by computing firstorder normal forms for system (23).
6 Eight Particles, Normalization of the \(\alpha \)Chain
Consider \(0<a<1\). We rescale \(x_i \rightarrow \varepsilon x_i, i = 1, \ldots , 7\) and divide the Hamiltonian by \(\varepsilon ^2\). The resulting Hamiltonian \(H_2 + \varepsilon H_3\) determines the equation of motion described by the vectors x en y in \({\mathbb R}^7\) (we can replace \(\alpha \) by \(\varepsilon \)); the reduction from 8 dof to 7 dof has been obtained by using the momentum integral.
6.1 The Case \(a \in (0, 1]\) with \(a \ne 0.5\) and \(a \ne 0.75\)
We exclude \(O(\varepsilon )\)neighborhoods of \(a=0.5\) and \(a= 0.75\), and a small interval near \(a=0\). Averaging–normalization, see Sanders et al. (2007) chapter 4, using Mathematica produces that the normal form \({\bar{H}}_3=0\) which means that to firstorder normalization the amplitudes \(r_j\) and phases \(\phi _j\) are constant. The firstorder normal form in these cases is trivially integrable. To study the structural stability of this result we would have to compute higherorder normal forms.
We present a numerical example of recurrence in the FPU system induced by Hamiltonian (1) for mass ratio \(1{:}8 (a= 0.125)\) in Fig. 1. A nonzero initial value is chosen only for the first mass of the chain, and the recurrence is demonstrated by computing the Euclidean distance d in 16dimensional phase space. The regularity of the recurrence agrees with the integrability of the normal form of \(H_2 + \varepsilon H_3\). However, the heavier particles at positions 2, 4, 6, 8 have zero initial values but the Euclidean distance \(d_2\) of the more massive part of the chain shows variation between 0 and 0.85. This suggests excitations and interactions that are not described by the firstorder normalization of \(H_3\). Other choices of the mass ratio a produce similar results.
6.2 The Detuned Case \(a = 0.5\)
We can continue the periodic and quasiperiodic solutions in the detuned case for c small enough.
6.3 Summary Invariant Manifolds \(M_{145}, M_{256}\) and \(M_{357}\) at Exact Resonance \(a=0.5\)

\(M_{145}\): The firstorder normal form vanishes and is trivially integrable.

\(M_{256}\): The normal form Eq. (29) produces 3 normal modes, 3 integrals and 2 general position periodic solutions.

\(M_{357}\): The normal form Eq. (29) produces 3 normal modes, 3 integrals and 2 general position periodic solutions.
6.4 The Detuned Case \(a = 0.75\)
The normal form system has the six independent integrals \(r_1,r_2,r_3,r_5,2r_4^2+r_6^2=E_1 ,2r_4^2+r_7^2=E_2\) and in addition the normal form \({\bar{H}}_3\) or equivalently \(H_2+ \varepsilon {\bar{H}}_3\). We conclude that the normal form \(H_2+{\bar{H}}_3\) of the \(\alpha \)chain for \(a= 0.75\) is integrable.
The quantity R in (33) describes the resonance manifold by \(R=0\) in the case \(a=\frac{3}{4}\)
In the case of \(a= 0.5\) we have two sum resonances, in the case \(a= 0.75\) we have the 2 : 1 : 1 resonance between modes 4, 6 and 7. Note that we start with zero initial value for mode 7. The dynamics in the resonance zone is illustrated in Fig. 7. First the 2 : 1 resonance of the \(x_4, x_6\) modes dominates (500 timesteps), then in the timesteps for \(t \ge 500\) the mode 7 is excited. The recurrent motion on tori around the periodic solution will produce exchanges of energy between the modes. In Fig. 8 we present the time series of the actions of the three modes.
7 Conclusions
 1.
The dynamics of 4 particles in a periodic FPU chain with alternating masses can be identified in a submanifold of a FPU chain with 8 particles. As is shown in Sect. 3, this is a general feature of FPU chains with 4n particles. There exist bushes (in the terminology of Chechin et al. 2002), families of invariant manifolds, for arbitrary large chains of this type with 4n particles. This also holds for the classical case \(m=1\).
 2.
In the cases of 4, 8 particles, incidental resonances emerge producing periodic solutions that, in the case of stability, are associated with invariant tori. These resonances and tori occur in systems with 4n and 8n particles as well.
 3.
Quasitrapping as formulated in Zaslavsky (2007) and demonstrated in our analysis, produces a significant delay of recurrence. The trapping regions are often associated with the primary resonances 1 : 2 or 1 : 1. The normal form approximations give precise estimates of validity for a long but finite interval of time. However, this is enough to demonstrate the phenomenon as the recurrence theorem applies to these quasitrapping regions, the flow will return an infinite number of times arbitrarily close to the regions.
 4.
Near stable equilibrium we did not find energy equipartition in a periodic FPU chain with alternating masses. The systematic presence of nested invariant manifolds (bushes) makes equipartition less probable.
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