Strong Stochasticity Threshold Research Articles

Vortex revivals and Fermi-Pasta-Ulam-Tsingou recurrence.

We study the self-trapped vortex-ring eigenstates of the two-dimensional Schrödinger equation with focusing Poisson and cubic nonlinearities. For each value of the topological charge l, there is a family of solutions depending on a parameter that can be understood as the relative importance of the cubic term. We analyze the perturbative stability of the solutions and simulate the fate of the unstable ones. For l=1 and l=2, there is an interval of the family of eigenstates for which the initial profile breaks apart but subsequently reconstructs itself, a process that can be interpreted as a nontrivial nonlinear oscillation between the vortex and an azimuthon. This revival provides a compelling realization of a recurrence of the Fermi-Pasta-Ulam-Tsingou type. Outside this interval, the vortices can be stable (for small cubic terms) or unstable and nonrecurrent (for large cubic terms). We argue that there is a crossover between these regimes that resembles a strong stochasticity threshold. For l≥3, all solutions are unstable and nonrecurrent. Finally, we comment on the possible experimental implementation of this phenomenon in the context of nonlinear laser beam propagation in thermo-optical media.

Variational approach to renormalized phonon in momentum-nonconserving nonlinear lattices

In this letter, we extend a previously proposed variational approach to systematically investigate general momentum-nonconserving nonlinear lattices. Two intrinsic identities characterizing optimal reference systems are firstly revealed, which enables us to derive explicit expressions for optimal variational parameters. The resulting optimal harmonic reference systems provide information for the band gap as well as the dispersion of renormalized phonons in momentum-nonconserving nonlinear lattices. As a demonstration, we consider the one-dimensional lattice. We show that the phonon band gap endows a simple power-law temperature dependence in the weak stochasticity regime where predicted dispersion is reliable by comparing with numerical results. In addition, an exact relation between ensemble averages of the lattice in the whole temperature range is found, regardless of the existence of the strong stochasticity threshold.

A stochasticity threshold in holography and the instability of AdS

We give strong numerical evidence that a self-interacting probe scalar field in AdS, with only a few modes turned on initially, will undergo fast thermalization only if it is above a certain energetic threshold. Below the threshold the energy stays close to constant in a few modes for a very long time instead of cascading quickly. This indicates the existence of a Strong Stochasticity Threshold (SST) in holography. The idea of SST is familiar from certain statistical mechanical systems, and we suggest that it exists also in AdS gravity. This would naturally reconcile the generic nonlinear instability of AdS observed by Bizon and Rostworowski, with the Fermi–Pasta–Ulam–Tsingou-like quasiperiodicity noticed recently for some classes of initial conditions. We show that our simple setup captures many of the relevant features of the full gravity-scalar system.

Covariant hydrodynamic Lyapunov modes and strong stochasticity threshold in Hamiltonian lattices

We scrutinize the reliability of covariant and Gram-Schmidt Lyapunov vectors for capturing hydrodynamic Lyapunov modes (HLMs) in one-dimensional Hamiltonian lattices. We show that, in contrast with previous claims, HLMs do exist for any energy density, so that strong chaos is not essential for the appearance of genuine (covariant) HLMs. In contrast, Gram-Schmidt Lyapunov vectors lead to misleading results concerning the existence of HLMs in the case of weak chaos.

Efficient time-series detection of the strong stochasticity threshold in Fermi-Pasta-Ulam oscillator lattices

In this work we study the possibility of detecting the so-called strong stochasticity threshold (i.e., the transition between weak and strong chaos as the energy density of the system is increased) in anharmonic oscillator chains by means of the 0-1 test for chaos. We compare the result of the aforementioned methodology with the scaling behavior of the largest Lyapunov exponent computed by means of tangent space dynamics, which has, so far, been the most reliable method available to detect the strong stochasticity threshold. We find that indeed the 0-1 test can perform the detection in the range of energy density values studied. Furthermore, we determined that conventional nonlinear time series analysis methods fail to properly compute the largest Lyapounov exponent even for very large data sets, whereas the computational effort of the 0-1 test remains the same in the whole range of values of the energy density considered with moderate size time series. Therefore, our results show that, for a qualitative probing of phase space, the 0-1 test can be an effective tool if its limitations are properly taken into account.

Hydrodynamic Lyapunov modes and strong stochasticity threshold in the dynamicXYmodel: An alternative scenario

Crossover from weak to strong chaos in high-dimensional Hamiltonian systems at the strong stochasticity threshold (SST) was anticipated to indicate a global transition in the geometric structure of phase space. Our recent study of Fermi-Pasta-Ulam models showed that corresponding to this transition the energy density dependence of all Lyapunov exponents is identical apart from a scaling factor. The current investigation of the dynamic XY model discovers an alternative scenario for the energy dependence of the system dynamics at SSTs. Though similar in tendency, the Lyapunov exponents now show individually different energy dependencies except in the near-harmonic regime. Such a finding restricts the use of indices such as the largest Lyapunov exponent and the Ricci curvatures to characterize the global transition in the dynamics of high-dimensional Hamiltonian systems. These observations are consistent with our conjecture that the quasi-isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities. Moreover, numerical simulations demonstrate the existence of hydrodynamical Lyapunov modes (HLMs) in the dynamic XY model and show that corresponding to the crossover in the Lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of HLMs. In particular, our numerical results confirm that strong chaos is essential for the appearance of HLMs.

Hydrodynamic Lyapunov modes and strong stochasticity threshold in Fermi-Pasta-Ulam models

The existence of a strong stochasticity threshold (SST) has been detected in many Hamiltonian lattice systems, including the Fermi-Pasta-Ulam (FPU) model, which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density epsilon. Correspondingly, the relaxation time to energy equipartition and the largest Lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value. In this paper, we attempt to go one step further in this direction to explore further changes in the energy density dependence of other Lyapunov exponents and of hydrodynamic Lyapunov modes (HLMs). In particular, we find that for the FPU-beta and FPU-alpha(beta) models the scalings of the energy density dependence of all Lyapunov exponents experience a similar change at the SST as that of the largest Lyapunov exponent. In addition, the threshold values of the crossover of all Lyapunov exponents are nearly identical. These facts lend support to the point of view that the crossover in the system dynamics at the SST manifests a global change in the geometric structure of phase space. They also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi-isotropic works quite well in the analytical calculation of the largest Lyapunov exponent. Furthermore, the FPU-beta model is used as an example to show that HLMs exist in Hamiltonian lattice models with continuous symmetries. Some measures are defined to indicate the significance of HLMs. Numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in Lyapunov exponents at the SST. In particular, our numerical results indicate that strong chaos is essential for the appearance of HLMs and those modes become more significant with increasing degree of chaoticity.

Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.

Macroscopic detection of the strong stochasticity threshold in Fermi-Pasta-Ulam chains of oscillators.

The largest Lyapunov exponent of a system composed by a heavy impurity embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators is numerically computed for various values of the impurity mass M. A crossover between weak and strong chaos is obtained at the same value epsilon(T) of the energy density epsilon (energy per degree of freedom) for all the considered values of the impurity mass M. The threshold epsilon(T) coincides with the value of the energy density epsilon at which a change of scaling of the relaxation time of the momentum autocorrelation function of the impurity occurs and that was obtained in a previous work [Phys. Rev. E 65, 036228 (2002)]]. The complete Lyapunov spectrum does not depend significantly on the impurity mass M. These results suggest that the impurity does not contribute significantly to the dynamical instability (chaos) of the chain and can be considered as a probe for the dynamics of the system to which the impurity is coupled. Finally, it is shown that the Kolmogorov-Sinai entropy of the chain has a crossover from weak to strong chaos at the same value of the energy density as the crossover value epsilon(T) of largest Lyapunov exponent. Implications of this result are discussed.

Phase space geometry and stochasticity thresholds in hamiltonian dynamics

Results of numerical computations of the largest Lyapunov exponent lambda(1)(varepsilon,N) as a function of the energy density varepsilon and the number of particles N are here reported for a Fermi-Pasta-Ulam alpha+beta model. These results show the coexistence at large N of two thresholds: a stochasticity threshold, found before for the alpha model alone, and a strong stochasticity threshold (SST), found before for the beta model alone. Although this coexistence may seem at first sight plausible, it is not obvious a priori that the alpha+beta model superimposes properties of the alpha and beta models independently. The main point of this paper, however, is a geometric characterization of the SST via the mean curvature of the constant energy hypersurfaces in the phase space of the model and the characteristic decay time of its time autocorrelation function tau(c)(varepsilon,N), which correlates with that of lambda(1)(varepsilon,N) for fixed N. This appears to provide important information on the very complicated geometry of the phase space of this simple solidlike model.

Dynamical instability and statistical behaviour of N-body systems

In this paper, we argue about a synthetic characterization of the qualitative properties of generic many-degrees-of-freedom (mdf) dynamical systems (DS's) by means of a geometric description of the dynamics [Geometro-Dynamical Approach (GDA)]. We exhaustively describe the mathematical framework needed to link geometry and dynamical (in)stability, discussing in particular which geometrical quantity is actually related to instability and why some others cannot give, in general, any indication of the occurrence of chaos. The relevance of the Schur theorem to select such Geometrodynamic Indicators (GDI) of instability is then emphasized, as its implications seem to have been underestimated in some of the previous works. We then compare the analytical and numerical results obtained by us and by Pettini and coworkers concerning the FPU chain, verifying a complete agreement between the outcomes of averaging the relevant GDI's over phase space (Casetti and Pettini, 1995) and our findings (Cipriani, 1993), obtained in a more conservative way, time-averaging along geodesics. Along with the check of the ergodic properties of GDI's, these results confirm that the mechanism responsible for chaos in realistic DS's largely depends on the fluctuations of curvatures rather than on their negative values, whose occurrence is very unlikely. On these grounds we emphasize the importance of the virialization process, which separates two different regimes of instability. This evolutionary path, predicted on the basis of analytical estimates, receives clear support from numerical simulations, which, at the same time, confirm also the features of the evolution of the GDI's along with their dependence on the number of degrees of freedom, N , and on the other relevant parameters of the system, pointing out the scarce relevance of negative curvature (for N ⪢ 1) as a source of instability. The general arguments outlined above, are then concretely applied to two specific N-body problems, obtaining some new insights into known outcomes and also some new results The comparative analysis of the FPU chain and the gravitational N-body system allows us to suggest a new definition of strong stochasticity, for any DS. The generalization of the concept of dynamical time-scale, t D, is at the basis of this new criterion. We derive for both the mdf systems considered the ( N , ε)-dependence of t D (ε being the specific energy) of the system. In light of this, the results obtained (Cerruti-Sola and Pettini, 1995), indeed turn out to be reliable, the perplexity there raised originating from the neglected N -dependence of t D, and not to an excessive degree of approximation in the averaged equations used. This points out also the peculiarities of gravitationally bound systems, which are always in a regime of strong instability; the dimensionless quantity L 1 = γ 1 · t D [γ 1 is the maximal Lyapunov Characteristic Number (LCN)] being always positive and independent of ε, as it happens for the FPU chain only above the strong stochasticity threshold (SST). The numerical checks on the analytical estimates about the ( N , ε)-dependence of GDI's, allow us to single out their scaling laws, which support our claim that, for N ⪢ 1, the probability of finding a negative value of Ricci curvature is practically negligible, always for the FPU chain, whereas in the case of the Gravitational N-body system, this is certainly true when the virial equilibrium has been attained. The strong stochasticity of the latter DS is clearly due to the large amplitude of curvature fluctuations. To prove the positivity of Ricci curvature, we need to discuss the pathologies of mathematical Newtonian interaction, which have some implications also on the ergodicity of the GDI's for this DS. We discuss the Statistical Mechanical properties of gravity, arguing how they are related to its long range nature rather than to its short scale divergencies. The N -scaling behaviour of the single terms entering the Ricci curvature show that the dominant contribution comes from the Laplacian of the potential energy, whose singularity is reflected on the issue of equality between time and static averages. However, we find that the physical N-body system is actually ergodic where the GDI's are concerned, and that the Ricci curvature associated is indeed almost everywhere (and then almost always) positive, as long as N ⪢ 1 and the system is gravitationally bound and virialized. On these grounds the equality among the above mentioned averages is restored, and the GDA to instability of gravitating systems gives fully reliable and understandable results. Finally, as a by-product of the numerical simulations performed, for both the DS's considered, it emerges that the time averages of GDI's quickly approach the corresponding canonical ones, even in the quasi-integrable limit, whereas, as expected, their fluctuations relax on much longer timescales, in particular below the SST.

Ergodic Properties of Microcanonical Observables

The problem of the existence of a strong stochasticity threshold in the FPU-β model is reconsidered, using suitable microcanonical observables of thermodynamic nature, like the temperature and the specific heat. Explicit expressions for these observables are obtained by exploiting rigorous methods of differential geometry. Measurements of the corresponding temporal autocorrelation functions locate the threshold at a finite value of the energy density, which is independent of the number of degrees of freedom.

Modulational estimate for the maximal Lyapunov exponent in Fermi-Pasta-Ulam chains

In the framework of the Fermi-Pasta-Ulam (FPU) model, we show a simple method to give an accurate analytical estimation of the maximal Lyapunov exponent at high energy density. The method is based on the computation of the mean value of the modulational instability growth rates associated to unstable modes. Moreover, we show that the strong stochasticity threshold found in the $\ensuremath{\beta}\ensuremath{-}\mathrm{F}\mathrm{P}\mathrm{U}$ system is closely related to a transition in tangent space, the Lyapunov eigenvector being more localized in space at high energy.

Nonlinear dynamics of classical Heisenberg chains

The maximum Lyapunov exponent (MLE) is evaluated as a function of temperature in the isotropic Heisenberg chain. At low temperatures the MLE varies almost quadratically with temperature; it corresponds closely with ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$, the rate at which the local self-correlation decays to half its initial value. At higher temperatures, the MLE saturates at a value close to 1/2 in the limit of large chain lengths. The strong stochasticity threshold (defined by the change of slope of the MLE) parallels closely the transition from predominantly ballistic to predominantly diffusive behavior of the self-correlation and the concomitant steep increase in ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$. The complete Lyapunov spectrum has been derived for a chain of 18 spins; deviations from linearity occurring at infinite temperature suggest that the chaoticity of the system is incomplete. Finally, it is suggested that a systematic study of finite-size effects might be useful in deciding the issue of anomalous versus conventional spin diffusion.

Strong stochasticity threshold in some anharmonic lattices

Numerical studies have been done on some one-dimensional anharmonic lattices with nearest neighbor interaction. The spectral entropy η and the largest Lyapunov exponent λ 1 were calculated. The diatomic Toda lattice, the Lennard-Jones lattice, and the lattice interacting with 1 r repulsive potential were employed. The present paper aims at investigating how the dynamics of the models is affected by the energy density, paying particular attention to whether the strong stochasticity threshold (SST), as was observed in other models, exits with sufficient generality or not. The results show that the qualitative dependence of the dynamics on the energy density is quite same between the above models. The dynamics is distinguished by two different qualitative behaviors. The dynamics is weakly chaotic in low energy density regime, while it is strongly chaotic in high energy density regime. Evidence for the existence of a rather sharp transition between the two different chaoticity behaviors at a certain critical energy density is provided. Hence, the SST can be defined for all the models employed.

Analytic computation of the strong stochasticity threshold in Hamiltonian dynamics using Riemannian geometry.

This paper concerns the study of the strong stochasticity threshold (SST) in Hamiltonian systems with many degrees of freedom, and more specifically the stability problem of this threshold in the thermodynamic limit (N\ensuremath{\rightarrow}\ensuremath{\infty}). The investigation is based on a recently proposed differential geometrical description of Hamiltonian chaos. The mathematical framework is given by Eisenhart's formulation of Newtonian mechanics in a suitably enlarged configuration space-time: a Riemannian manifold equipped with an affine metric. Using the Jacobi--Levi-Civita equation for geodesic spread, we establish a relation between curvature properties of the ambient manifold and the stability---or instability---of the dynamics. The use of the Eisenhart metric makes it clearly evident that a dominating source of chaoticity in Hamiltonian flows of phyical interest is represented by parametric resonance induced by curvature fluctuations along the geodesics and not by negativeness of some curvature property. Here only Ricci curvature is involved because the scalar curvature vanishes identically with this metric. Thus a geometric quantity relevant to the study of chaos is the degree of bumpiness of the ambient manifold, i.e., the integral of the Ricci curvature carried over the whole manifold with the constraint of energy constancy. This quantity, considered as a function of the energy density \ensuremath{\varepsilon}, clearly marks the SST. A simple sufficient criterion is given to identify integrable systems and is here applied to a chain of linear oscillators as well as to the Toda lattice. In the case of the Fermi-Pasta-Ulam \ensuremath{\beta} model we have worked out analytically the \ensuremath{\varepsilon} dependence of the degree of bumpiness of the ambient manifold. This allowed us to prove that the SST is stable in the N\ensuremath{\rightarrow}\ensuremath{\infty} limit. An operational definition of the critical energy density is also provided and shown to yield predictions in excellent agreement with previous results based on Lyapunov exponents.

Geometrical hints for a nonperturbative approach to Hamiltonian dynamics.

In this paper it is shown that elementary tools of Riemannian differential geometry can be successfully used to explain the origin of Hamiltonian chaos beyond the usual picture of homoclinic intersections. This approach stems out of first principles of mechanics and fundamental tools of Riemannian geometry. Natural motions of Hamiltonian systems can be viewed as geodesics of the configuration-space manifold M equipped with a suitable metric ${\mathit{g}}_{\mathit{J}}$, and the stability properties of such geodesics can be investigated by means of the Jacobi--Levi-Civita equation for geodesic spread. The study of the relationship between chaos and the curvature properties of the configuration-space manifold is the main concern of the present paper and is carried out by numerical simulations. Two different mechanisms for chaotic instability are found: (i) the trajectories are ``scattered'' by random encounters of regions of negative curvature (either scalar or Ricci curvature---it depends on the averaging procedure adopted); (ii) the ``bumpiness'' of (M,${\mathit{g}}_{\mathit{J}}$) yields oscillations of the Ricci curvature along the geodesics so that parametric resonance makes them unstable also in regions of positive curvature.The geometric approach is intrinsically nonperturbative because everything is well defined at any energy, i.e., quasi-integrability is not required as in the case of classical perturbation theory. Therefore this approach is fit to describe the existence of the strong-stochasticity threshold (SST) in high-dimensional Hamiltonian flows. This threshold refers to a transition between weak and strong chaoticity of the dynamics and, correspondingly, between slow and fast mixing in phase space. In view of applications to equilibrium and nonequilibrium statistical mechanics, the SST appears as the transition feature of high dimensional Hamiltonian flows with the greatest physical significance. As the SST concerns chaotic dynamical behaviors, its existence cannot be understood within the framework of classical perturbation theory, whereas it is shown that the SST can be related to some major geometrical change of the constant-energy surfaces of phase space. A clear-cut distinction can be made between integrable and nonintegrable systems. Finally, a new meaning is given to the standard algorithm to compute numerical Lyapunov exponents, and it is shown that Oseledets multiplicative theorem is not necessary to justify it.

Strong stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times.

The dynamics of high-dimensional Hamiltonian flows is extensively investigated by means of numerical simulations in the case of the Fermi-Pasta-Ulam (FPU) \ensuremath{\beta} model and classical lattice ${\mathit{cphi}}^{4}$ model; both are considered at N=512 degrees of freedom. This work aims at investigating the major consequences on the dynamical phenomenology of the existence of a strong stochasticity threshold. This threshold corresponds to a transition from two different diffusion regimes in phase space: slow diffusion (along resonances) at low-energy density and fast diffusion (across resonances) at high-energy density. Wave packets are initially excited. The relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$ toward equipartition of energy is measured following the time behavior of spectral entropy. A systematic study of ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$=${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$(\ensuremath{\varepsilon},n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$) is reported, where \ensuremath{\varepsilon} is the energy per degree of freedom and n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$ is the average wave number of the initially excited packet.In the FPU case, it is found that below the strong stochasticity threshold ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$, the equipartition time is an increasing function of n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$, i.e., high-frequency modes tend to freeze compared to low-frequency modes. This is in qualitative agreement with the predictions of a so-called narrow-packet approximation in which the FPU model is approximated by a nonlinear Schr\odinger equation. However, above ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$, the situation is reversed, and initial excitation of high-frequency modes yields quicker mixing. Also, this is in qualitative agreement with some analytical predictions. In the ${\mathit{cphi}}^{4}$ case, at \ensuremath{\varepsilon}g${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$ the excitation of high-frequency modes results in exponentially increasing ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$ as a function of n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$. At \ensuremath{\varepsilon}${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$, above some critical n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$, ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$ is apparently divergent. It is also shown that the crossover in the scaling behavior ${\ensuremath{\lambda}}_{1}$(\ensuremath{\varepsilon}) of the largest Lyapunov exponent occurs always at ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$ independent of the initial conditions, thus providing a good intrinsic probe of the strong stochasticity threshold.

Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics.

The time evolution toward equipartition of energy is numerically investigated in nonlinear Hamiltonian systems with a large number N of degrees of freedom. The relaxation process is studied by computing the spectral entropy \ensuremath{\eta}; for a wide class of initial conditions, it follows a stretched exponential law \ensuremath{\eta}(t)\ensuremath{\sim}exp[-(t/${\ensuremath{\tau}}_{0}$${)]}^{\ensuremath{\xi}}$, \ensuremath{\xi}1, up to times t${\ensuremath{\tau}}_{R}$, and \ensuremath{\eta}(t)=const for t\ensuremath{\ge}${\ensuremath{\tau}}_{R}$. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value ${\ensuremath{\varepsilon}}_{c}$ (model dependent) of the energy density \ensuremath{\varepsilon}, the relaxation time ${\ensuremath{\tau}}_{R}$ is found to follow a scaling with \ensuremath{\varepsilon}, which is compatible with a ``Nekhoroshev-like'' law, i.e., ${\ensuremath{\tau}}_{R}$=${\ensuremath{\tau}}_{0}$exp(${\ensuremath{\varepsilon}}_{0}$/\ensuremath{\varepsilon}${)}^{\ensuremath{\gamma}}$, for both the Fermi-Pasta-Ulam (FPU) \ensuremath{\beta} model and the classical lattice ${\ensuremath{\varphi}}^{4}$ model; a remarkable difference with respect to Nekhoroshev's theorem (where the exponent \ensuremath{\gamma} scales as 1/${N}^{2}$) is the N independence of numerical experiment results.An important consequence of this fact is the existence of nonequilibrium states of arbitrary lifetimes also at large N values. On the other hand, at high-energy densities (\ensuremath{\varepsilon}g${\ensuremath{\varepsilon}}_{c}$) ${\ensuremath{\tau}}_{R}$ is almost independent of \ensuremath{\varepsilon}. The maximum Lyapunov characteristic exponent ${\ensuremath{\lambda}}_{1}$ is measured in both models as a function of the energy density. The striking result is a change in the scaling ${\ensuremath{\lambda}}_{1}$(\ensuremath{\varepsilon}) occurring at the same \ensuremath{\varepsilon} (=${\ensuremath{\varepsilon}}_{c}$) at which ${\ensuremath{\tau}}_{R}$ has its crossover. At \ensuremath{\varepsilon}g${\ensuremath{\varepsilon}}_{c}$, the scaling ${\ensuremath{\lambda}}_{1}$\ensuremath{\sim}${\ensuremath{\varepsilon}}^{2/3}$ is found for both the FPU and the ${\ensuremath{\varphi}}^{4}$ models; this is in agreement with a random matrix approximation of the tangent dynamics, which means that the dynamics itself mimics a random process. Below ${\ensuremath{\varepsilon}}_{c}$, steeper and model-dependent scalings are found. Hence, evidence for the existence of a rather sharp transition of the phase-space structure is provided, and a strong stochasticity threshold (SST) can be defined. A preliminary result, suggesting the stability of the SST in the limit of large N, is also reported.