Vol. 3, No. 4 (2014): Regular Issue

In this paper, we focus on the adaptive synchronization of delayed Chen chaotic systems with unknown parameters. An adaptive synchronization controller and the adaptive updating law are designed. At last, the numerical simulation is shown to prove the effectiveness of the proposed synchronization controller schemes.

We study the Ulam-Hyers-Rassias stability for linear and semilinear equations on Banach spaces from a functional analysis point of view with several illustrative examples. More precisely, surjective linear equations on Banach spaces, linear equations on Banach spaces with closed ranges and surjective semilinear equations between Banach spaces are investigated one by one.

Detailed analysis of trajectories reveals a filaments-nets structure of the phase space of coin tossing, leading to a simple and unified explanation for the extremely sensitive dependence of the outcome, head or tail, on the initial state, for the extremely complex geometry of the cross sections of basins of attraction for heads and tails, and for the big difference between the transitional region and the “completely random region”. A “GDGC” (Great Differentiation & Great Combination) condition is proposed for the stability of statistical regularity, which can also be summarized by the following “Compensation Principle”: The more sensitive, i.e. the more unstable, the deterministic process is, the more stable, i.e. the more insensitive, the associated statistical regularity would be.

Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.

Introducing a new 6-dimensional Lie algebra aims at generating a Lax pair whose compatibility condition gives rise to (1+1)-dimensional integrable hierarchy of equations which can reduce to the nonlinear Schr¨odinger equation and two sets of nonlinear integrable equations by taking various parameters. The Hamiltonian structure of the (1+1)-dimensional hierarchy is also obtained by using the trace identity. The reason for generating the above (1+1)-dimensional integrable hierarchy lies in obtaining (2+1)-dimensional equation hierarchy. That is to say, with the hep of the higher dimensional Lie algebra, we introduce two 4 × 4 matrix operators in an associative algebra A [ ξ ] for which a new (2+1)-dimensional hierarchy of equations is derived by using the TAH scheme and the Hamiltonian operator in the case of 1+1 dimensions , which generalizes the results presented by Tu, that is, the reduced case of the hierarchy obtained by us can be reduced to the Davey-Stewartson (DS) hierarchy. Finally, the Hamiltonian structure of the (2+1)-dimensional hierarchy is produced by the trace identity used for 2+1 dimensions, which was proposed by Tu. As we have known that there is no paper involving such the problem on generating expanding models of (2+1)-dimensional integrable hierarchy.

In this paper, we revisit the distribution function derived by making use of the adiabatic approximation, for electrons acted upon by a slowly varying electrostatic wave. This allows us to resolve an apparent paradoxical discrepancy between the results of two published papers, namely, V.B. Krapchev and A.K. Ram (1980), Phys. Rev. A 22, 1229-12242, and, D. B´enisti and L. Gremillet (2007), Phys. Plasmas, 14, 042304. We then briefly recall the relevance of the adiabatic approximation as well as its limitations, and further indicate how to go beyond this approximation to derive very accurate electron distribution functions.

The detailed analysis of stationary solutions of the reduced Gardner– Ostrovsky (GO) equation is presented. The GO equation (ut + c0ux + αuux + α1u2ux + βuxxx )x = γu is the popular model for the description of large-amplitude internal oceanic waves affected by Earth’s rotation. Its reduced version in which the small-scale dispersion is neglected ( β = 0 ) is used when very long internal waves are considered. The equation is also applicable to other types of nonlinear waves in various media (plasma, optical media, relaxing media, etc.) when the large-scale dispersion ∼ γ plays a dominant role in comparison with the small-scale dispersion ∼ β. Balancing the nonlinear effect such dispersion gives rise to existence of stationary waves, both periodic and non-periodic. It is shown that only smooth periodic waves make physical sense. Systematic analysis of stationary solutions to the GO equation and their categorisation is presented.

Linear second order elliptic equations containing the sum of the two Laplace operators with convection terms or a free Laplacian and a Laplacian with drift are considered in Rd. The corresponding operator L may be non Fredholm, such that solvability conditions for the equation Lu = f are unknown. We obtain solvability conditions in H2 (Rd ) for the non selfadjoint problem via relating it to a self-adjoint Schrödinger type operator, for which solvability relations are derived in our preceding work [16].