Chirikov Criterion Research Articles

Thermalization process in a one-dimensional lattice with two-dimensional motions: The role of angular momentum conservation.

We study the thermalization process in a one-dimensional lattice with two-dimensional motions. The phonon modes in such a lattice consist of two branches. Unlike in general nonlinear Hamiltonian systems, for which the only conserved quantity is the total energy, the total angular momentum J is also conserved in this system. Consequently, the intra- and interbranch energy transports behave significantly differently. For the intrabranch transport, all the existing rules for the one-dimensional systems including the Chirikov overlap criterion apply. As for the interbranch transport, some trivial processes in one-dimensional lattices become nontrivial. During these processes, all the conservation laws can be satisfied exactly; thus the Chirikov criterion does not apply. These processes provide some fast channels for the interbranch transport, although the thermalization cannot be reached through them alone. A system with nonzero-J initial state, however, can never be thermalized to an equipartition state having zero J. Quite counterintuitively, the corresponding asymptotic mode energy distribution greatly concentrates to a few lowest-frequency modes in one branch.

Surfing in the phase space of spin--orbit coupling in binary asteroid systems

ABSTRACTFor a satellite with an irregular shape, which is the common shape among asteroids, the well-known spin--orbit resonance problem could be changed to a spin--orbit coupling problem since a decoupled model does not accurately capture the dynamics of the system. In this paper, having provided a definition for close binary asteroid systems, we explore the structure of the phase space in a classical Hamiltonian model for spin--orbit coupling in a binary system. To map out the geography of resonances analytically and the cartography of resonances numerically, we reformulate a fourth-order gravitational potential function, in Poincare variables, via Stokes coefficients. For a binary system with a near-circular orbit, isolating the Hamiltonian near each resonance yields the pendulum model. Analysis of the results shows the geographical information, including the location and width of resonances, is modified due to the prominent role of the semimajor axis in the spin--orbit coupling model but not structurally altered. However, this resulted in modified Chirikov criterion to predict onset of large-scale chaos. For a binary system with arbitrary closed orbit, we thoroughly surf in the phase space via cartography of resonances created by fast Lyapunov indicator maps. The numerical study confirms the analytical results, provides insight into the spin--orbit coupling, and shows some bifurcations in the secondary resonances which can occur due to material transfer. Also, we take the (65803) Didymos binary asteroid as a case to show analytical and numerical results.

Electron heating in 2D: Combining Fermi–Ulam acceleration and magnetic-moment non-adiabaticity in a mirror-configuration plasma

We analyze a new mechanism for the creation and confinement of energetic electrons in a mirror-configuration plasma. A Fermi–Ulam-type process, driven by end-localized coherent electrostatic oscillations, provides axial acceleration, while a natural non-adiabaticity of μ provides phase decorrelation and energy isotropization. This novel 2D combination causes the electron energy distribution function, calculated with a diffusive-loss model, to assume a Maxwellian shape with the μ non-adiabaticity, reducing loss-cone escape and annulling the absolute-barrier energy-limiting Chirikov criterion of lower dimensional models. The theoretical predictions are compared with data from an experiment.

Quantum Chirikov criterion: Two particles in a box as a toy model for a quantum gas

We consider a toy model for emergence of chaos in a quantum many-body short-range-interacting system: two one-dimensional hard-core particles in a box, with a small mass defect as a perturbation over an integrable system, the latter represented by two equal mass particles. To that system, we apply a quantum generalization of Chirikov's criterion for the onset of chaos, i.e. the criterion of overlapping resonances. There, classical nonlinear resonances translate almost automatically to the quantum language. Quantum mechanics intervenes at a later stage: the resonances occupying less than one Hamiltonian eigenstate are excluded from the chaos criterion. Resonances appear as contiguous patches of low purity unperturbed eigenstates, separated by the groups of undestroyed states-the quantum analogues of the classical KAM tori.

Chaos in Non-Linear Planar Oscillation of a Satellite in an Elliptic Orbit under the Influence of Aerodynamic Torque

This article deals with the non-linear oscillation of a satellite in an elliptic orbit around the Earth under the influence of aerodynamic torque. It is assumed that the orbital plane of the satellite coincides with the equatorial plane of the Earth. The half width of the chaotic separatrix has been estimated by Chirikov criterion. Through surface of section method, it has been observed that the aerodynamic torque parameter (e), the eccentricity of the orbit (e) and the mass ratio (ω0) play an important role in changing the regular motion into the chaotic one. We studied the behaviour of trajectories due to change in parameters with Lyapunov exponents and time series plots. The theory is applied to Resourcesat1 an artiflcial satellite of the Earth.

FOXTAIL: Modeling the nonlinear interaction between Alfvén eigenmodes and energetic particles in tokamaks

FOXTAIL is a new hybrid magnetohydrodynamic-kinetic code used to describe interactions between energetic particles and Alfvén eigenmodes in tokamaks with realistic geometries. The code simulates the nonlinear dynamics of the amplitudes of individual eigenmodes and of a set of discrete markers in five-dimensional phase space representing the energetic particle distribution. Action–angle coordinates of the equilibrium system are used for efficient tracing of energetic particles, and the particle acceleration by the wave fields of the eigenmodes is Fourier decomposed in the same angles. The eigenmodes are described using temporally constant eigenfunctions with dynamic complex amplitudes. Possible applications of the code are presented, e.g., making a quantitative validity evaluation of the one-dimensional bump-on-tail approximation of the system. Expected effects of the fulfillment of the Chirikov criterion in two-mode scenarios have also been verified.

Dynamic Characteristics of Excited Atomic Systems

The dynamics of excited atom interactions with other atoms, which often lead to associative ionization, is largely governed by stochastic diffusion of the valence electron through Rydberg states prior to the ionization. Such processes are associated with random changes of the energy state of the highly excited electron, and they are likely to influence the nuclear dynamics, especially at subthermal collision energies. Possibilities of manipulation of the chaotic dynamics of Rydberg states require a detailed exploration. For an electron in a given Rydberg state moving in a microwave field, which can be generated via interaction with another atom or molecule, there exists critical field strength, above which motion of the electron in the energy space is chaotic. Recently a way to block the dynamic chaos regime was shown, if a given Rydberg state is located somewhat above the middle between the two other states with the orbital quantum number differing by one, whereby level shifts can be controlled by employing Stark/Zeeman shifts in external DC electric/magnetic fields. The stochastic effects in collisions involving Rydberg particles, in which the initial and final reaction channels are connected via intermediate highly excited collision complexes with multiple crossings of energy levels, can be treated using the dynamic chaos approach (Chirikov criterion, Standard and Keppler mapping of time evolution of the Rydberg electron, solution of the Fokker-Plank- and Langevin-type of equations, etc.). Such approach to obtaining dynamics characteristics is a natural choice, since the treatment of Rydberg electron dynamics as a kind of diffusion process allowing one to bypass the multi-level-crossing problem, which can hardly be solved by conventional quantum chemistry methods.

Uma introdução ao controle do caos em sistemas hamiltonianos quase integráveis

Sistemas Hamiltonianos quase integráveis são de grande interesse em diversos campos de pesquisa da física e da matemática. Nesses sistemas, o espaço de fase apresenta trajetórias regulares e caóticas. Essas trajetórias dependem, em parte, da amplitude da perturbação que quebra a integrabilidade do sistema. O valor da perturbação crítica responsável por esta transição é um elemento chave no controle do caos. No presente trabalho, exploramos um procedimento para o controle do caos em sistema hamiltoniano quase integrável baseado na alteração de um parâmetro (perturbação). Inicialmente, apresentamos as ferramentas básicas para este estudo: mapa hamiltoniano, linearização do mapa e critério de Chirikov. Posteriormente, investigamos o comportamento de uma interação do tipo onda-partícula frente a perturbação. Por fim, confrontamos os resultados analíticos com uma abordagem numérica (iteração do mapa), mostrando um bom acordo.

Rotational Oscillation of Satellite in Elliptic Orbit under Magnetic Torque

Magnetic disturbance torque results from interaction between spacecraft's residual magnetic field and geomagnetic field. The primary sources of magnetic disturbance torques are spacecraft magnetic moments, eddy currents and hysteresis. The instantaneous magnetic disturbance torque due to spacecraft effective magnetic moment is the product of geocentric magnetic flux density and sum of individual magnetic moments caused by permanent and induced magnetism and spacecraft generated current loops. Let us consider a rigid satellite S moving around the central body C and orbital plane of satellite coincides with equatorial plane of C. This paper deals with rotational oscillation of satellite under influence of magnetic torque in elliptic orbit. Non-linear planar equation of motion of model has been derived. Using BKM method, it is observed that resonances exist. The analysis regarding stability of stationary planar oscillation of satellite near resonance frequency shows that discontinuity occurs in amplitude of oscillation at frequency of e xternal periodic force which is less than frequency of the natural oscillation. The half width of chaotic separatrix has been estimated by Chirikov's criterion. Through surface of section method, it has been observed that magnetic torque parameter, eccentricity and mass ratio play an important role in changing regular motion into chaotic one.

Estimating the size of the regular region of a topographically trapped vortex

We formulate a dynamically consistent two-layer quasigeostrophic model of geophysical flow using the concept of background currents which are characterized by a constant potential vorticity minimizing the energy. An incident current over a delta-like isolated topographic feature generates a topographically trapped vortex in the bottom layer with a singular elliptic point, and one with a regular elliptic point in the upper layer. Such vortices are finite regions of recirculation which occur in the vicinity of isolated topographic features. The corresponding Hamiltonian equations of motion for a fluid particle are known to produce chaotic advection under the presence of the periodic incident current. When a periodic incident current is superimposed, fluid is entrained and detrained from the neighbourhood of the vortex and chaotic particle motion occurs. At a small amplitude periodic incident current we have a near-integrable case of weak chaotization in narrow stochastic layer near separatrix. At a finite amplitude periodic incident current we have the case of strong chaos, which will be investigated here. For the bottom layer of the flow, there is always a regular region (regular vortex core) around the singular elliptical point even in the case of a finite amplitude periodic incident current. Particles in this regular region have regular trajectories and will remain there permanently, whereas particles with chaotic trajectories will be emanated from the vortex by the incident current. Using the Chirikov criterion of resonance overlap, we estimate the radius of this regular vortex core and a range of the optimal frequencies, i.e. those frequencies of the incident current which provide a maximal possible stochastization of fluid particle trajectories.

Nonlinear dynamics, stability and control of the scan process in noncontacting atomic force microscopy

The nonlinear equations of motion for the scan process in noncontacting atomic force microscopy are consistently derived using the extended Hamilton’s principle. A modal dynamical system obtained from the continuum model reveals that scan control appears in the form of parametric excitation. The system is analyzed asymptotically and numerically to yield escape bounds limiting the noncontacting mode of operation. Approximate stability bounds are deduced from both a global Melnikov integral and a local Moon–Chirikov overlap criterion. The Melnikov–Holmes stability curve and the overlap criterion are found to be similar for small damping. However, for very small damping, typical of ultra-high vacuum conditions, where the Melnikov bound becomes trivial, the Moon–Chirikov criterion yields an improved stability threshold.

Fast wave current drive in the ion cyclotron frequency range from the single-particle point of view

The quasilinear model is commonly used to determine the heating rate and the current drive efficiency of the fast wave in the ion cyclotron range of frequency. The validity of this model at these frequencies is a controversial issue [Bécoulet et al., Phys. Plasmas 1, 2908 (1994)]. Here this problem is addressed from the single-particle point of view by deriving an approximated Hamiltonian which accounts also for the inhomogeneity of the confining magnetic field. Realistic values of the wave field are calculated by means of the toroidal full-wave code TORIC for typical scenarios of an ASDEX Upgrade [Gruber et al., Nucl. Fusion 41, 1369 (2001)] experiment, and apply the Chirikov criterion to determine the threshold for the development of the intrinsic stochasticity. These results are verified by numerically integrating the Hamiltonian equations. It is found that, due to the discreteness of the parallel phase velocity spectrum of a fast wave at these frequencies, for the parameters of ASDEX Upgrade the intrinsic stochasticity is marginal, especially close to integer and low-rational values of the safety factor. The failure of intrinsic stochasticity to guarantee the validity of the conventional quasilinear model, however, does not imply the onset of nonlinear saturation in the power absorption and in the driven current, since in all realistic situations Coulomb collisions suffice, by a large margin, to ensure the applicability of the linearized Vlasov equation. Under this condition, power and momentum transfer rates to resonant electrons are identical to those predicted by the quasilinear theory.

Chaos in the relativistic two-electron atom.

Chaotic autoionization of the relativistic two-electron atom is investigated. A theoretical analysis of chaotic dynamics of the relativistic outer electron under the periodic perturbation due to the inner electron, assumed to be on a circular orbit, based on the Chirikov criterion, is given. The diffusion coefficient, the ionization rate, and time are calculated. (c) 2002 American Institute of Physics.

Stochastic ionization through noble tori: renormalization results.

We find that chaos in the stochastic ionization problem develops through the breakup of a sequence of noble tori. In addition to being very accurate, our method of choice, the renormalization map, is ideally suited for analyzing properties at criticality. Our computations of chaos thresholds agree closely with the widely used empirical Chirikov criterion.

Stochastic autoionization of a relativistic two-electron atom

Stochastization of a relativistic two-electron atom, which leads to diffusion autoionization, is studied. Using the Chirikov criterion, the condition of stochastization of the outer relativistic electron under the action of the inner electron is obtained. The diffusion coefficient, the autoionization time, and the autoionization probability are calculated.

Analytical estimation of the dynamic apertures of circular accelerators

By considering delta function sextupole, octupole, and decapole perturbations and using difference action-angle variable equations, we find some useful analytical formulae for the estimation of the dynamic apertures of circular accelerators due to single sextupole, single octupole, single decapole (single 2 m pole in general); and their combined effects are derived based on the Chirikov criterion of the onset of stochastic motions. Comparisons with numerical simulations are made, and the agreement is quite satisfactory.

Accumulation of resonances in the Fermi Accelerator: a quantum route to chaos?

We discuss the applicability of the adiabatic theory to the quantum version of a classically chaotic system using as an example the Fermi Accelerator. The behavior of this system is analyzed for different values of the characteristic action of the system. We derive an analytical expression for the system resonances and observe that a transition between the adiabatic and diffusive regime takes place when the quantum resonances of the system accumulate. The situation is thus formally similar to the classical transition to chaos by resonance overlap, described by Chirikov's criterion.

ANALYTICAL PREDICTIONS OF CHAOS IN A NON-LINEAR ROD

The resonant and chaotic conditions for non-dampened, non-linear, planar rods are developed through the Chirikov criterion, and the subharmonic bifurcation conditions for weakly dampened, non-linear, planar rods are also presented through the Melnikov method. The analytical conditions are based on a simply supported, geometrically non-linear, planar rod model with a specific single-mode response, but these conditions are applicable to geometrically non-linear planar rods with different supports. Chaos and transient motion from chaos to periodic motion in the non-linear rod are simulated through the approximate rod model, and they are illustrated through the Poincare mapping section.

Diffusive ionization of a relativistic hydrogenlike atom

Stochastic ionization of a highly excited relativistic hydrogenlike atom in the monochromatic field is investigated. A theoretical analysis of chaotic dynamics of the relativistic electron based on the Chirikov criterion is given for the cases of one- and three-dimensional atoms. The critical value of the external field is evaluated analytically. The diffusion coefficient and ionization time are calculated.

Quantum and classical chaos in a thin film (heterostructure) in a tilted magnetic field

Classical and quantum dynamics of a spinless charged particle moving in a thin film placed in a tilted magnetic field is investigated. In the classical limit, nonlinear resonances and the condition of stochastization (Chirikov criterion) are studied. In the quantum case, a system of interacting resonances and an isolated resonance are analyzed. The energy spectrum is calculated numerically in the resonance approximation, and the statistics of the interlevel intervals is examined.