Multiscale dynamics in communities of phase oscillators

We investigate the dynamics of systems of many coupled phase oscillators with heterogeneous frequencies. We suppose that the oscillators occur in M groups. Each oscillator is connected to other oscillators in its group with "attractive" coupling, such that the coupling promotes synchronization within the group. The coupling between oscillators in different groups is "repulsive," i.e., their oscillation phases repel. To address this problem, we reduce the governing equations to a lower-dimensional form via the ansatz of Ott and Antonsen, Chaos 18, 037113 (2008). We first consider the symmetric case where all group parameters are the same, and the attractive and repulsive coupling are also the same for each of the M groups. We find a manifold L of neutrally stable equilibria, and we show that all other equilibria are unstable. For M?≥?3, L has dimension M?-?2, and for M?=?2, it has dimension 1. To address the general asymmetric case, we then introduce small deviations from symmetry in the group and coupling parameters. Doing a slow/fast timescale analysis, we obtain slow time evolution equations for the motion of the M groups on the manifold L. We use these equations to study the dynamics of the groups and compare the results with numerical simulations.     

Synchronization Regimes in an Ensemble of Phase Oscillators Coupled Through a Diffusion Field

Radiophysics and Quantum Electronics - We consider an ensemble of identical phase oscillators coupled through a common diffusion field. Using the Ott–Antonsen reduction, we develop dynamical...     

Breathing chimera in a system of phase oscillators

Chimera states consisting of synchronous and asynchronous domains in a medium of nonlinearly coupled phase oscillators have been considered. Stationary inhomogeneous solutions of the Ott–Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. The direct numerical simulation has shown that these structures under certain conditions are transformed to oscillatory (breathing) chimera regimes because of the development of instability.     

Order parameter analysis of synchronization transitions on star networks

The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe–Strogatz transformation, Ott–Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.     

Experimental bifurcations and chaos in a modified self-sustained macro electromechanical system

A class of self-sustained Macro ElectroMechanical (MaEMS) Systems is made up of a Rayleigh-Duffing oscillator actuating a mechanical arm through a magnetic coupling. In this paper, to avoid experimental constraints, an audio amplifier is added to the device. Quenching phenomenon, bifurcation and chaos are predicted and shown to occur in a device of this class of MaEMS. Especially by using linear stability analysis, the condition for the quenching phenomenon is derived. Chaos and bifurcation are predicted using Lyapunov exponent and bifurcation diagram. A prototype of device is designed and fabricated. Experimental results for this device that are consistent with results from theoretical investigations are presented and convincingly show quenching phenomenon, bifurcation and chaos.     

Controlling a leaky tap

We apply the Ott, Grebogy and Yorke mechanism for the control of chaos to the analytical oscillator model of a leaky tap obtaining good results. We exhibit the robustness of the control against both dynamical noise and measurement noise. A possible way of controlling experimentally a leaky tap using magnetic-field-produced variations in the viscosity of a magnetorheological fluid is suggested.     

Comment on "Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks" [Chaos 16, 023119 (2006)

This paper comments on a recent paper by Yu and Cao [Chaos, 16, 023119 (2006)]. We find that the theorem in this paper is incorrect by numerical simulations. The consequence of the incorrectness is analyzed as well.     

Comment on "Identification of low order manifolds: validating the algorithm of Maas and Pope" [Chaos 9, 108-123 (1999)

It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame 88, 239-264 (1992)] identifies the slow invariant manifold of a system of ordinary differential equations with time-scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore, Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] conjectured that away from a slow manifold, the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari, and Wiggins. In the first example, the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari, and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example, the projection algorithm of Maas and Pope leads to a manifold that lies in a region where no slow manifold exists at all. This rejects the conjecture of Rhodes, Morari, and Wiggins mentioned above.     

Comment on "Finding finite-time invariant manifolds in two-dimensional velocity fields" [Chaos 10, 99 (2000)

This note serves as a commentary of the paper of Haller [Chaos 10, 99 (2000)] on techniques for detecting invariant manifolds. Here we show that the criterion of Haller can be improved in two ways. First, by using the strain basis reference frame, a more efficient version of theorem 1 of Haller (2000) allows to better detect the manifolds. Second, we emphasize the need to nondimensionalize the estimate of hyperbolic persistence. These statements are illustrated by the example of the Kida ellipse. (c) 2001 American Institute of Physics.     

All-optical generation of states for "Encoding a qubit in an oscillator"

Most quantum computation schemes propose encoding qubits in two-level systems. Others exploit the use of an infinite-dimensional system. In "Encoding a qubit in an oscillator" [Phys. Rev. A 64, 012310 (2001)], Gottesman, Kitaev, and Preskill (GKP) combined these approaches when they proposed a fault-tolerant quantum computation scheme in which a qubit is encoded in the continuous position and momentum degrees of freedom of an oscillator. One advantage of this scheme is that it can be performed by use of relatively simple linear optical devices, squeezing, and homodyne detection. However, we lack a practical method to prepare the initial GKP states. Here we propose the generation of an approximate GKP state by using superpositions of optical coherent states (sometimes called "Schr?dinger cat states"), squeezing, linear optical devices, and homodyne detection.     

Chaos Generation in Opto-Mechanical Coupling System Assisted by Two-Level Atoms

Chaos is important in nonlinear science and promotes the development of fundamental studies, such as neural networks, extreme event statistics, and electron transport. In this paper, a theoretical scheme for generating dynamical chaos in a Fabry–Perot cavity with two-level atoms is investigated. Through the injection of atoms, controllable chaos of the cavity and mechanical oscillator is achieved simultaneously by the external laser field. The trajectory and the maximal Lyapunov exponent that characterize the properties of the chaos could be optimized by adjusting the coupling constant, driving field, injection of atoms, the frequency of the cavity, and the frequency of the mechanical oscillator. This study provides a new idea to manipulate the cavity and mechanical chaos assisted by two level atoms. It is hoped that the results presented can benefit controllable chaos generation and its applications.     

Synchronization of coupled van der pole and Kislov-Dmitriev self-oscillators

The problem of interaction of self-oscillating elements of different origin is considered for a coupled van der Pole oscillator and Kislov-Dmitriev generator. Domains with different types of dynamics in the space of parameters are indicated taking into account the possibility of broadband synchronization of the systems. The case of essentially different control parameters is considered. Chaos stabilization effects and the opposite effect (initiated chaos) are detected in the system under investigation for various values of parameters.     

The Coulomb-oscillator relation on <Emphasis Type="Italic">n</Emphasis>-dimensional spheres and hyperboloids

We establish a relation between Coulomb and oscillator systems on n-dimensional spheres and hyperboloids for n≥2. We show that, as in Euclidean space, the quasiradial equation for the (n+1)-dimensional Coulomb problem coincides with the 2n-dimensional quasiradial oscillator equation on spheres and hyperboloids. Using the solution of the Schrödinger equation for the oscillator system, we construct the energy spectrum and wave functions for the Coulomb problem.     

A linear path toward synchronization

Using a recently introduced linear reformulation of the Kuramoto model of self-synchronizing oscillator systems [1], we study a new class of analytically solvable oscillator systems defined by a particular coupling scheme. We show that these systems have a logarithmic scaling law in the vicinity of the critical point, which may be seen as anomalous with respect to the usual power-law behavior exhibited by the standard Kuramoto model.     

Photoabsorption cross section of the Thomas-Fermi atom

The photoabsorption cross section σ(ω) and the distribution of oscillator strengths df/dω [these values are related as σ=(2π²e²/mc)(df/dω)] were determined for an atom with a large Z value using the semiclassical approach. These values were found for low frequencies with the use of the Vlasov kinetic equations, which were numerically solved by the method of particles. The asymptotic behavior of the distribution of oscillator strengths at high frequencies was determined by semiclassical equations for the photoabsorption cross section of electron shells in a Coulomb potential. The asymptotic equations were used to suggest an interpolation equation for the distribution of oscillator strengths over the whole Thomas-Fermi frequency range 27 eV ? ?ω ? 27Z² eV. This equation was used to calculate the logarithmic mean excitation energy, which appears in problems of ionization loss of charged particles. The distribution of oscillator strengths in a neutral atom allows the radiative properties of dense matter to be determined.     

Lightweight secure image encryption scheme based on chaotic differential equation

The ultimate secure choice for block cryptosystem until now is advanced encryption standard (AES). It is very difficult to implement AES for the constrained situations such as sensor networks, image encryption and RFID tags. In this article, a chaotic oscillator generated by a second order differential equation is used to produce confusion and diffusion in the plaintext message to achieve the desired secrecy. The produced chaotic sequence of random numbers from dynamical system is utilized to scramble the pixels of an image to obtain an encrypted image. Chaos based encryption technique is found secure enough to tackle chosen plaintext attacks and brute force attacks. The specific attributes of chaotic system like, sensitivity to initial conditions, randomness and uncertainty make it suitable for the design of cryptosystem. The dominance of the proposed scheme is acknowledged due to the fact of better cryptographic properties when compared with the algorithms already developed in the literature.     

Distribution of the oscillator strengths in the 2<Emphasis Type="Italic">p</Emphasis> absorption spectra of 3<Emphasis Type="Italic">d</Emphasis> transition metal films

The absorption cross-sections in the region of the resonance structure of the 2p ionization threshold of metal films are originally measured for the total series of 3d metals from Sc to Cu. The partial 2p absorption cross-sections are found, and the integral oscillator strengths are determined. The linear dependence of the sum of oscillator strengths in the range from 0 to 80 eV above the L ₃ absorption edge on the number of unfilled 3d states is established. The dependence breaks down for Cr, Fe, and Ni.