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[en] In this letter, we study the chaotic behaviors in the fractional order Chen system. We found that chaos exists in the fractional order Chen system with order less than 3. The lowest order we found to have chaos in this system is 2.1. Linear feedback control of chaos in this system is also studied
[en] This paper surveys some recent and classical investigations of geometric progressions of residues that generalize the little Fermat theorem, connect this topic with the theory of dynamical systems, and estimate the degree of chaotic behaviour of systems of residues forming a geometric progression and displaying a distinctive mutual repulsion. As an auxiliary tool, the graphs of squaring operations for the elements of finite groups and rings are studied. For commutative groups the connected components of these graphs turn out to be attracting cycles homogeneously equipped with products of binary rooted trees, the algebra of which is also described in the paper. The equipping with trees turns out to be homogeneous also for the graphs of symmetric groups of permutations, as well as for the groups of even permutations
[en] In a coupled laser system, the dynamics of the receiving laser is investigated when two separate transmitting lasers are injected into the receiving laser with different coupling strengths. It is shown that the phenomenon of preference of chaotic synchronization appears under appropriate coupling conditions. The receiving laser will entrain the pulses of either one or both transmitting lasers when the coupling is strong while it keeps its own dynamics when the coupling is weak.
[en] Transition from chaotic to quasi-periodic phase in modified Lorenz model is analyzed by performing the contact transformation such that the trajectory in R3 is projected on R2. The relative torsion number and the characteristics of the template are measured using the eigenvector of the Jacobian instead of vectors on moving frame along the closed trajectory. Application to the circulation of a fluid in a convection loop and oscillation of the electric field in single-mode laser system are performed. The time series of the eigenvalues of the Jacobian and the scatter plot of the trajectory in the transformed coordinate plane X-Z in the former and |X|-|Z| in the latter, allow to visualize characteristic pattern change at the transition from quasi-periodic to chaotic. In the case of single mode laser, we observe the correlation between the critical movement of the eigenvalues of the Jacobian in the complex plane and intermittency
[en] In this Letter, an observer-based chaotic synchronization scheme is proposed. Our method concerns chaotic systems having special triangular form. Using the sliding mode theory, the synchronization of the response system with the drive system is achieved in finite time. An application to secure chaotic communication is also proposed. To demonstrate the efficiency of the proposed scheme, two well-known chaotic systems: Lur'e-like system and Duffing equation are considered as illustrative examples
[en] A cubic discrete coupled logistic equation is proposed to model the predator-prey problem. The coupling depends on the population size of both species and on a positive constant λ, which could depend on the prey reproduction rate and on the predator hunting strategy. Different dynamical regimes are obtained when λ is modified. For small λ, the species become extinct. For a bigger λ, the preys survive but the predators extinguish. Only when the prey population reaches a critical value then predators can coexist with preys. For increasing λ, a bistable regime appears where the populations apart of being stabilized in fixed quantities can present periodic, quasiperiodic and chaotic oscillations. Finally, bistability is lost and the system settles down in a steady state, or, for the biggest permitted λ, in an invariant curve. We also present the basins for the different regimes. The use of the critical curves lets us determine the influence of the zones with different number of first rank preimages in the bifurcation mechanisms of those basins
[en] In this paper, we introduce a new practical method for distinguishing the chaotic, periodic and quasi-periodic orbits, and analysis the Hopf bifurcation using an analytic technique for the Lue system. As a result, we have further explored the dynamical behaviors
[en] This paper describes the security weakness of a recently proposed improved chaotic encryption method based on the modulation of a signal generated by a chaotic system with an appropriately chosen scalar signal. The aim of the improvement is to avoid the breaking of chaotic encryption schemes by means of the return map attack introduced by Perez and Cerdeira. A method of attack based on taking the absolute value of the ciphertext is presented, that allows for the cancellation of the modulation scalar signal and the determination of some system parameters that play the role of system key. The proposed improved method is shown to be compromised without any knowledge of the chaotic system parameter values and even without knowing the transmitter structure
[en] A linear array of N mutually coupled single-mode lasers is investigated. It is shown that the intensities of N lasers are chaotically synchronized when the coupling between lasers is relatively strong. The chaotic synchronization of intensities depends on the location of the lasers in the array. The chaotic synchronization appears between two outmost lasers, the second two outmost lasers, etc. There is no synchronization between nearest neighbors of the lasers. If the number of N is odd, the middle laser is never synchronized between any lasers. The chaotic synchronization of phases between nearest lasers in the array is examined by using the analytic signal and the Gaussian filter methods based on the peak of the power spectrum of the intensity. It can be seen that the message of chaotic intensity synchronization is conveyed through the phase synchronization.