The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie-Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function psi there is an unstable point (called the X point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the "nodal-point- X -point complex." We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point- X -point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point- X -point complex. It is also an inverse power of the distance of a trajectory from the X point's stable manifold far from the complex. This distance plays the role of an effective "impact parameter." The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point- X -point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.

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We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are (a) a proof of the existence of bounded domains in configuration space which are devoid of nodal points, (b) an analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability, (c) a detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point-X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.

We focus on two approaches that have been proposed in recent years for the explanation of the so-called Fermi-Pasta-Ulam (FPU) paradox, i.e., the persistence of energy localization in the "low-q " Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of "natural packets" exhibiting exponential stability, while in the second, emphasis is placed on the existence of "q breathers," i.e., periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of " q-tori" representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.

The possibility that giant extrasolar planets could have small trojan co-orbital companions has been examined in the literature from both viewpoints of the origin and dynamical stability of such a configuration. Here we aim to investigate the dynamics of hypothetical small trojan exoplanets in domains of secondary resonances embedded within the tadpole domain of motion. To this end, we consider the limit of a massless trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of additional planets (hereafter referred to as the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the dynamics of the trojan body. The paper contains a theoretical and a numerical part. In the theoretical part, we develop a Hamiltonian formalism in action-angle variables, which allows to treat in a unified way resonant dynamics and secular effects on the trojan body in both the ERTBP or the RMPP. In both cases, our formalism leads to a decomposition of the Hamiltonian in two parts, H = H b + H sec . H b , called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom, while H sec contains only terms depending trigonometrically on slow (secular) angles. H b is formally identical in the ERTBP and the RMPP, apart from a re-definition of some angular variables. An important physical consequence of this analysis is that the slow chaotic diffusion along resonances proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism. We found that this is best approximated by the paradigm of 'modulational diffusion'. In the paper's numerical part, we then focus on the ERTBP in order to make a detailed numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we first provide a survey of the resonant web for characteristic mass parameter values of the primary, in which the secondary resonances from 1:5 to 1:12 (ratio of the short over the synodic period), as well as their transverse resonant multiplets, appear. We give numerical examples of diffusion of weakly chaotic orbits in the resonant web. We finally make a statistics of the escaping times in the resonant domain, and find power-law tails of the distribution of the escaping times for the slowly diffusing chaotic orbits. Implications of resonant dynamics in the search for trojan exoplanets are discussed.