ABSTRACT. We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.

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We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis. [12,13]. While the concept of PR originates in the linear world [14], PRs deeply impact also the behavior of nonlinear conservative systems. However, the full nonlinear dynamics of PRs is relatively well understood only for lowdimensional Hamiltonian systems [2,15,16]. Conversely, the analysis of extended systems described by PDEs with periodicity in the evolution variable [17] is essentially limited to determine the region of parametric instability (Arnold tongues) via Floquet analysis [18][19][20][21], while the nonlinear stage of PR past the linearized growth of the unstable modes remains mostly unexplored.In this letter, taking the periodic defocusing nonlinear Schrödinger equation (NLSE) as a widespread example describing, e.g. periodic management of atom condensates [12,19,20,22], optical beam propagation in layered media [21], or optical fibers with periodic dispersion [18,23,24], we show that the PR gives rise to quasi-periodic evolutions which exhibit on average FermiPasta-Ulam (FPU) recurrence [25] with a remarkably complex (but ordered) underlying phase-plane structure. Such structure describes the continuation into the nonlinear regime of the modulational instability (MI) of a background solution, uniquely due to the parametric forcing (with zero forcing the defocusing NLSE is stable). A byproduct of this structure is the existence of breatherlike solutions [26]. This fact suggests the intriguing possibility of observing rogue waves [27], which are commonly associated with breathers [28], in the defocusing NLSE [29]. On the other hand, the richness of such structure allows us to remarkably predict that optimal parametric amplification occurs at a critical frequency where the system lies off-resonance (outside the PR bandwidth). Our approach retains its validity in the most interesting regime of strong parametric driving, where the system is found to exhibit a remarkably ordered structure despite its broken translational symmetry and integrability. In this sense the physics differs from other integrable models exhibiting a complex nonlinear dynamics of MI already in the undriven regime (e.g. focusing NLSE [26,[30][31][32][33][34][35][36]), around which chaos can develop under weak periodic perturbations [31,37,38].We consider the following periodic NLSEreferring, without loss of genera...