Starting from a chaotic regime in the dynamics of a Duffing-Holmes oscillator, we show how it is possible, by means of a small parametric perturbation of suitable frequency, to bring the system to a regular regime. This situation is studied from the analytic point of view using the Melnikov method and from the numerical point of view computing Lyapunov exponents. The corresponding bounds for the perturbation are compared. Noting that the time, measured along the original unperturbed separatrix, that elapses between two successive homoclinic intersections grows when we approach the resonance, we propose a possible scenario for this type of regularization of the dynamics.

The Fermi-Pasta-Ulam β model has been studied by integrating numerically the equations of motion for a system of N nonlinearly coupled oscillators with N ranging from 64 to 512. Multimode excitations have been considered as initial conditions; the number Δn of initially excited modes is such that the ratio Δn/N is kept constant. We can consider the system as a gas of weakly coupled phonons (normal modes), so that if we keep the ratio Δn/N constant we find an analogy with the thermodynamical limit of statistical mechanics where the ratio M/V is constant when both the volume V and the number of particles M are increased up to infinity. The relaxation towards stationary states is followed through the time evolution of a suitably defined ‘‘spectral entropy’’ which depends on the shape of the space Fourier spectrum; this spectral entropy is a good equipartition indicator: Strong evidence is reported in favor of the existence of an equipartition threshold. Its persistence at very different values of N is also clearly shown. The main result concerns the occurrence of the threshold at the same value of the energy density (i.e., of the ‘‘control parameter’’) when the number of degrees of freedom is changed. More general initial conditions are also considered and the same result is found using as a control parameter a pseudo-Reynolds-number R: The threshold occurs at the same critical value $R_c$ when N is varied. It turns out that a fully chaotic regime (equipartition) is obtained with an ‘‘average nonlinearity’’ of the system of about 3%.

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 η; for a wide class of initial conditions, it follows a stretched exponential law η(t)∼exp[-(t/${\tau }_0$ ${\left)\right]}^{\xi }$, ξ<1, up to times t<${\tau }_R$, and η(t)=const for t≥${\tau }_R$. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value ${\varepsilon }_c$ (model dependent) of the energy density ɛ, the relaxation time ${\tau }_R$ is found to follow a scaling with ɛ, which is compatible with a ‘‘Nekhoroshev-like’’ law, i.e., ${\tau }_R$=${\tau }_0$exp(${\varepsilon }_0$/ɛ${)}^{\gamma }$, for both the Fermi-Pasta-Ulam (FPU) β model and the classical lattice ${\phi }^4$ model; a remarkable difference with respect to Nekhoroshev’s theorem (where the exponent γ scales as 1/$N^2$) is the N independence of numerical experiment results.

A nonvanishing Lyapunov exponent ${\lambda }_1$ provides the very definition of deterministic chaos in the solutions of a dynamical system; however, no theoretical mean of predicting its value exists. This paper copes with the problem of analytically computing the largest Lyapunov exponent ${\lambda }_1$ for many degrees of freedom Hamiltonian systems as a function of $\varepsilon =\frac{E}{N}$, the energy per degree of freedom. The functional dependence ${\lambda }_1\left(\varepsilon \right)$ is of great interest because, among other reasons, it detects the existence of weakly and strongly chaotic regimes. This aim, the analytic computation of ${\lambda }_1\left(\varepsilon \right)$, is successfully reached within a theoretical framework that makes use of a geometrization of Newtonian dynamics in the language of Riemannian differential geometry. An alternative point of view about the origin of chaos in these systems is obtained independently of the standard explanation based on homoclinic intersections. Dynamical instability (chaos) is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of the Jacobi-Levi-Civita equation (JLCE) for geodesic spread. In this paper it is shown how to derive from the JLCE an effective stability equation. Under general conditions, this effective equation formally describes a stochastic oscillator; an analytic formula for the instability growth rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam $\beta$ model and to a chain of coupled rotators. Excellent agreement is found between the theoretical prediction and numeric values of ${\lambda }_1\left(\varepsilon \right)$ for both models.