Vol. 4, No. 1 (2015): Regular Issue

Conventional synchronization-based chaotic communication is vulnerable to synchronization attacks enable to recuperate system parameters. However, it is possible to make these attacks inefficient. The simple way to resist synchronization attacks is to change a parameter of the master system faster than the time needed for the system to synchronize. To verify this idea we construct a hybrid communication system composed of two chaotic R¨ossler oscillators and the chaotic logistic map. The latter is used for fast variation of the most sensitive system parameter when the R¨ossler oscillators synchronize. The algorithm is robust to noise in the communication channel.

Lattice models for the second-order strain-gradient models of elasticity theory are discussed. To combine the advantageous properties of two classes of second-gradient models, we suggest a new lattice model that can be considered as a discrete microstructural basis for gradient continuum models. It was proved that two classes of the second-gradient models (with positive and negative sign in front the gradient) can have a general lattice model as a microstructural basis. To obtain the second-gradient continuum models we consider a lattice model with the nearest-neighbor and next-nearestneighbor interactions with two different coupling constants. The suggested lattice model gives unified description of the second-gradient models with positive and negative signs of the strain gradient terms. The sign in front the gradient is determined by the relation of the coupling constants of the nearest-neighbor and next-nearest-neighbor interactions.

A weakly nonlinear climate damage function used by Nordhaus in his DICE model and a strongly nonlinear climate damage function recently proposed by Weitzman are incorporated in the Kellie-Smith—Cox Integrated Assessment model of coupled climate—conomic dynamics, the initial version of which contained only linear climate damages. Long-term projections of the dynamics of coupled climate—economic system are computed with the modified versions of the model. Simulation results for Weitzman function demonstrate pronounced nonlinear emergent dynamics and suggest extremely dramatic long-term economic scenarios for high values of background economic growth rates.

With in the frame work of extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3 and py = p2 + ix4, we investigate the exact invariants for a coupled harmonic oscillator along with PT-symmetric version in two dimensions. For this purpose rationalization method is employed and the invariants obtained in this work play an important role to study the complex trajectories of the concerned classical system.

We consider mechanisms with impact deformations such that colliding parts are deformable and the Newton’s coefficient of restitution is variable. It is shown how a system with impact deformations can replace the KelvinVoigt viscoelastic model in analysis of a mechanism with contact motion. The suggested impact deformations are compared with the experimental data. By applying deformable surfaces of contacts and non-constant coeffi- cients of restitution, we suppress the chattering in two different mechanical models. We have investigated the existence and stability of periodic solutions in mechanisms with contacts. To actualize the theoretical results, extended examples with simulations are presented.

Detailed calculations reveal that the sensitivity of the dependence of the result of coin toss (head or tail) on the initial state in the phase space is not only very inhomogeneous but also fractal. In the most sensitive regions, the number of turns of the coin is still fluctuating as the initial height changes within the atomic scale. Thus, the predictability of the toss of a real coin fails in these regions. The portion of such unpredictable regions with sub-atomic sensitivity becomes dominant in the phase space with the improvement of the elasticity of the surface. This offers one unique example in the macroscopic physics that macroscopic determinism fails due to the extreme sensitivity. This also helps to understand why the long-time accurate weather report is not possible.

A central problem of dynamical systems theory is to identify a reduced description of the dynamical process one can deal easier. In this paper we present a systematic method of identifying those closed sub-systems of a given discrete time dynamical system in the frame of operator theory. It is shown that this problem is closely related to finding invariant sigma algebras of the dynamics.