Lagrangian descriptors: The shearless curve and the shearless attractor.

Hamiltonian systems with a nonmonotonic frequency profile are called nontwist. One of the key properties of such systems, depending on adjustable parameters, is the presence of a robust transport barrier in the phase space called the shearless curve, which becomes the equally robust shearless attractor when dissipation is introduced. We consider the standard nontwist map with and without dissipation. We derive analytical expressions for the Lagrangian descriptor (LD) for the unperturbed map and show how they are related to the rotation number profile. We show how the LDs can reconstruct finite segments of the invariant manifolds for the perturbed map. In the conservative case, we demonstrate how the LDs distinguish the chaotic seas from regular structures. The LDs also provide a remarkable tool to identify when the shearless curve is destroyed: we present a fractal boundary, in the parameter space, for the existence or not of the shearless torus. In the dissipative case, we show how the LDs can be used to localize point attractors and the shearless attractor and distinguish their basins of attraction.

Chaotic saddles and interior crises in a dissipative nontwist system.

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems; they are responsible for chaotic transients, fractal basin boundaries, and chaotic scattering, and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increases the transient times and we analyze the phenomenon of crisis induced intermittency.

Nonlinear photoassociation through exotic orbits.

We investigate the effects of a particular kind of orbits, which we call exotic orbits, on the process of classical molecular photoassociation. As a starting model system, we consider the process described by the Morse potential with a time-dependent perturbation consisting of the interaction of an external laser field with the molecular dipole. When the external perturbation is turned off, the bound molecular states are classically represented by librational motion, whereas the unbound, the collisional states, are represented by unbound motion, and in both cases, the energy is a constant of motion. When the perturbation is turned on, the total energy is no longer a constant of motion and initial conditions in the unbound region can reach the bound region, and vice versa, through chaotic orbits. Alternatively, we have found that the connection between the bound and unbound sectors can be achieved through exotic orbits, which are comprised by librationlike parts, a localized chaotic region, and an unbounded constant-energy part. Thus, if a colliding atomic pair is in an exotic orbit, it penetrates a chaotic region coming from the unbound sector, subsequently performing librationlike motion, during which the molecule with constant bound energy is formed. Afterwards, the molecule returns to the chaotic region and from this region, it can either access a distinct bound energy or dissociate. We call this phenomenon, in which a metastable molecule is formed, intermittent photoassociation. We show that the key for the emergence of exotic orbits is the relatively short range of the dipole as compared to the interacting potential range. In order to further verify our results, we have considered realistic forms for the potentials and dipole functions of several molecules and found the emergence of exotic orbits, and consequently of intermittent photoassociation, for the MgLi and SrLi molecular parameters.

Nonchaotic laser pulse dissociation through deformed tori.

We consider the nonlinear classical dynamics of a diatomic molecule under the action of a laser field in the framework of the driven Morse oscillator model. We investigate the influence of the dipole function and the laser field on the deformations of the surviving, invariant tori. For intense and high-frequency fields, some invariant tori traverse the separatrix of motion, visiting both the bound and unbound regions of the interatomic potential. Based on this fact, we propose the use of appropriately designed laser pulses to induce dissociation of trajectories on such invariant tori. This mechanism constitutes a controlled nonchaotic route for dissociation, which is an alternative to chaotic multiphoton dissociation and to chirped pulse dissociation.

Transport barriers with shearless attractors.

We present a mechanism to generate a sequence of shearless curves or attractors to form a band of transport barriers. We consider the labyrinthic nontwist standard map to prepare a scenario with three shearless curves. Dissipation is introduced and three shearless attractors coexist, very close to each other. In both cases a collective transport barrier is formed.

Dynamics of a light beam suffering the influence of a dispersing mechanism with tunable refraction index.

The dynamics of a monochromatic light beam is studied inside the oval billiard with an inner scatter circle, which can be interpreted as a cross section of a long optical fiber. The outer oval boundary acts as a perfect reflector for the light beam while the scatter circle encloses a medium with changeable refraction index. The light beam refracts when it enters inside this circle and some drastic changes in the phase space are observed. The increase of the refractive index destroys the center of stability in the phase space leading to a spread of scattered light and the islands of traps for the light propagation become more pulverized. Numerical results are presented and discussed.

Multifractality, stickiness, and recurrence-time statistics.

We identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics (RTS), which is based on the concept of Poincaré recurrences. The projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near-integrable system, the annular billiard. These microstructures interfere in the effective transport of a particle in the phase space, which can be observed through RTS. This technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea.

Role of the range of the dipole function in the classical dynamics of molecular dissociation.

The dissociation dynamics of heteronuclear diatomic molecules induced by infrared laser pulses is investigated within the framework of the classical driven Morse oscillator. The interaction between the molecule and the laser field described in the dipole formulation is given by the product of a time-dependent external field with a position-dependent permanent dipole function. The effects of changing the spatial range of the dipole function in the classical dissociation dynamics of large ensembles of trajectories are studied. Numerical calculations have been performed for distinct amplitudes and carrier frequencies of the external pulses and also for ensembles with different initial energies. It is found that there exist a set of values of the dipole range for which the dissociation probability can be completely suppressed. The dependence of the dissociation on the dipole range is explained through the examination of the Fourier series coefficients of the dipole function in the angle variable of the free system. In particular, the suppression of dissociation corresponds to dipole ranges for which the Fourier coefficients associated with nonlinear resonances are null and the chaotic region in the phase space is reduced to thin layers. In this context, it is shown that the suppression of dissociation of heteronuclear molecules for certain frequencies of the external field is a consequence of the finite range of the corresponding permanent dipole.

Energy gain induced by boundary crisis.

We consider a nonlinear system and show the unexpected and surprising result that, even for high dissipation, the mean energy of a particle can attain higher values than when there is no dissipation in the system. We reconsider the time-dependent annular billiard in the presence of inelastic collisions with the boundaries. For some magnitudes of dissipation, we observe the phenomenon of boundary crisis, which drives the particles to an asymptotic attractive fixed point located at a value of energy that is higher than the mean energy of the nondissipative case and so much higher than the mean energy just before the crisis. We should emphasize that the unexpected results presented here reveal the importance of a nonlinear dynamics analysis to explain the paradoxical strategy of introducing dissipation in the system in order to gain energy.

Fermi acceleration on the annular billiard.

We study the phenomenon of unlimited energy growth for a classical particle moving in the annular billiard. The model is considered under two different geometrical situations: static and breathing boundaries. We show that when the dynamics is chaotic for the static case, the introduction of a time-dependent perturbation allows that the particle experiences the phenomenon of Fermi acceleration even when the oscillations are periodic.