Chaotic behaviour in deformable models: the asymmetric doubly periodic oscillators

The motion of a particle in a one-dimensional perturbed asymmetric doubly periodic (ASDP) potential is investigated analytically and numerically. A simple physical model for calculating analytically the Melnikov function is proposed. The onset of chaos is studied through an analysis of the phase space, a construction of the bifurcation diagram and a computation of the Lyapunov exponent. Theory predicts the regions of chaotic behaviour of orbits in a good agreement with computer calculations.     

Unfolding homogenization in doubly periodic media and applications

We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium.     

Synchronous states in time-delay coupled periodic oscillators: A stability criterion

There are several works showing that nonzero time delay between nodes in an oscillator network can be responsible for several kinds of behavior as synchronization and chaos. Here, by using the Lyapunov linearizing method, in a system of two coupled oscillators derived as a particular case of the full connected network, it is shown that the time delay parameter has two sets of values: one that destabilizes the whole system and other that implies stability. Besides, there is a set of time delay values responsible for chaotic behaviors, even in a simple coupled oscillators system.     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

Quasi-periodic solutions and periodic bursters in quasiperiodically driven oscillators

In this paper, we propose a perturbation method to determine an approximation and conditions of existence of quasi-periodic (QP) solutions and bursting dynamics in a quasi-periodically driven system. The QP forcing consists of two periodic excitations, one with a very slow frequency and the other with a frequency of the same order of the proper frequency of the oscillator. A first averaging is done over the fast dynamics, then the quasi-static solutions of the modulation equations of amplitude and phase are determined and their stability analyzed. We show that a necessary condition for the occurrence of periodic bursters is that the slow excitation is parametric.     

Chaotic behavior in the 3-center problem

We consider the motion of a particle in a plane under the gravitational action of 3 fixed centers (the 3-center planar problem). As it is well known (Vestnik Moskov. Univ. Ser. 1 Matem. Mekh 6 (1984) 65; Prikl. Matem. i Mekhan. 48 (1984) 356; Classical Planar Scattering by Coulombic Potentials, Lecture Notes in Physics, Springer, Berlin, 1992) on the non-negative level sets of the energy E there do not exist non-constant analytic first integrals, and moreover the system has chaotic trajectories. These results were proved by variational methods.Here we investigate the problem in the domain of small negative values of E. Moreover, we assume that one of the centers is far away from the other two. Then we get a two-parameter singular perturbation of an integrable dynamical system: the 2-center problem on the zero-energy level. The main problem we deal with is to prove that the Poincaré-Melnikov theory applies in the limit case E→0.     

Multiple periodic solutions in a delay-coupled system of neural oscillators

In this paper, effects of the synaptic delay of signal transmissions on the pattern formation of nonlinear waves in a bidirectional ring of neural oscillators is studied. Firstly, the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Meanwhile, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, we discuss the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Finally, Hopf bifurcation directions and corresponding stabilities of bifurcating periodic orbits are derived by using the normal form approach and the center manifold theory. These theoretical results are significant to complement experimental and numerical observations made in living neuronal systems and artificial neural networks, in order to better understand the mechanisms underlying the system’s dynamics.     

Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

Science China Mathematics - This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in...     

On the existence of a stable periodic motion of two impacting oscillators

A system that consists of two impacting oscillators has been considered in this paper. A method of analytical determination of the existence of periodic solutions to the equations of motion and a method of investigation of the stability of these solutions have been presented. The results of the computations carried out by means of these methods have been illustrated by a few examples.     

Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach

In this work, the Hamiltonian approach is applied to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quintic nonlinear oscillator. The Hamiltonian approach is then extended to the second and third orders to find more precise results. The accuracy of the results obtained is examined through time histories and error analyses for different values for the initial conditions. Excellent agreement of the approximate frequencies and the exact solution is demonstrated. It is shown that this method is powerful and accurate for solving nonlinear conservative oscillatory systems.     

Open waveguides in doubly periodic junctions of domains with different limit dimensions

Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter ε ? 1) with nodes, which are sets of unit size, we show that by changing or removing one or several semi-infinite chains of nodes we can form additional spectral segments, the wave passage bands, in the essential spectrum of the original grid. The corresponding waveguide processes are localized in a neighborhood of the said chains, forming I-shaped, V-shaped, and L-shaped open waveguides. To derive the result, we use the asymptotic analysis of the eigenvalues of model problems on various periodicity cells.     

Population models in almost periodic environments

We establish the basic theory of almost periodic sequences on ?₊. Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models.     

Stochastic resonance with different periodic forces in overdamped two coupled anharmonic oscillators

We study the stochastic resonance phenomenon in the overdamped two coupled anharmonic oscillators with Gaussian noise and driven by different external periodic forces. We consider (i) sine, (ii) square, (iii) symmetric saw-tooth, (iv) asymmetric saw-tooth, (v) modulus of sine and (vi) rectified sinusoidal forces. The external periodic forces and Gaussian noise term are added to one of the two state variables of the system. The effect of each force is studied separately. In the absence of noise term, when the amplitude f of the applied periodic force is varied cross-well motion is realized above a critical value (f_c) of f. This is found for all the forces except the modulus of sine and rectified sinusoidal forces. For fixed values of angular frequency ω of the periodic forces, f_c is minimum for square wave and maximum for asymmetric saw-tooth wave. f_c is found to scale as Ae^0.75ω + B where A and B are constants. Stochastic resonance is observed in the presence of noise and periodic forces. The effect of different forces is compared. The stochastic resonance behaviour is quantized using power spectrum, signal-to-noise ratio, mean residence time and distribution of normalized residence times. The logarithmic plot of mean residence time τ_MR against 1/(D − D_c) where D is the intensity of the noise and D_c is the value of D at which cross-well motion is initiated shows a sharp knee-like structure for all the forces. Signal-to-noise ratio is found to be maximum at the noise intensity D = D_max at which mean residence time is half of the period of the driving force for the forces such as sine, square, symmetric saw-tooth and asymmetric saw-tooth waves. With modulus of sine wave and rectified sine wave, the SNR peaks at a value of D for which sum of τ_MR in two wells of the potential of the system is half of the period of the driving force. For the chosen values of f and ω, signal-to-noise ratio is found to be maximum for square wave while it is minimum for modulus of sine and rectified sinusoidal waves. The values of D_c at which cross-well behaviour is initiated and D_max are found to depend on the shape of the periodic forces.     

Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method

In this paper, the iteration perturbation method proposed by He [J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, 2006; J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005) 827–833] is used to generate periodic solutions of van der Pol oscillator with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. The comparison of the obtained results verifies its convenience and effectiveness.     

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation.     

Chaotic behavior for the secrete key of cryptographic system

In this article we want to prove that the cryptographic systems produce cryptograms that have a chaotic behavior. We can then, deduce that it is possible to put the chaos theory at the disposal of cryptology. This enables us to cipher the data and do a cryptanalysis on the basis of chaos. In other words, can we assume that cryptography is a particular case of the chaos?