A chaotic attractor containing unstable periodic orbits with different numbers of unstable directions is said to exhibit unstable dimension variability (UDV). We present general mechanisms for the progressive development of UDV in uni- and bidirectionally coupled systems of chaotic elements. Our results are applicable to systems of dissimilar elements without invariant manifolds. We also quanti...

We discuss the situation where two periodic signals compete to phase synchronize a chaotic attractor. Depending on the relative position of the periods with respect to the synchronization tongue for a single frequency signal, we distinguish several different cases. We find that, depending on parameters, it is possible that one or the other signal will entrain exclusively, or that they will entr...

Methods of harmonic linearization & describing function, numerical methods, and the applied bifurcation theory together discover new opportunities for analysis of periodic oscillations of control systems. In the present paper these opportunities are demonstrated. New analytical-numerical method based on the above-mentioned technique is discussed. Application of this technique for hidden attract...

The dynamics of two nonlinear Bloch systems is studied from the viewpoint of bifurcation and a particular parameter space has been explored for the stability analysis based on stability criterion. This enables the choice of the desired unstable periodic orbit from the numerous unstable ones present within the attractor through the process of closed return pairs. A generalized active control met...

We consider the stability of synchronized states (including equilibrium point, periodic orbit, or chaotic attractor) in arbitrarily coupled dynamical systems (maps or ordinary differential equations). We develop a general approach, based on the master stability function and Gershgörin disk theory, to yield constraints on the coupling strengths to ensure the stability of synchronized dynamics. S...

We describe an attracting piecewise rotation with two atoms with a self-similar structure of periodic domains on its attractor which resembles Sierpi nski's gasket. Besides its natural beauty, this example appears a return map in certain piecewise aane maps on the torus which have been studied by Adler, Kitchens, and Tresser, and by a number of other researchers advancing the theory of digital ...

A time series analysis of observed solar radio pulsations suggests that there must be a low-dimensional attractor. The power spectrum cannot be interpreted as a superposition of periodic components. Estimates of the maximum Lyapunov exponent and of the Kolmogorov entropy give some indications for a deterministic chaos. In order to study the limitations inherent in small data samples we include ...

Grebogi, Ott and Yorke (Phys. Rev. A 38(7), 1988) have investigated the effect of finite precision on average period length of chaotic maps. They showed that the average length of periodic orbits (T ) of a dynamical system scales as a function of computer precision (ε) and the correlation dimension (d) of the chaotic attractor: T ∼ ε. In this work, we are concerned with increasing the average p...

Using weighted L p {norms we derive new bounds on the long{time behavior of the solutions improving on the known results of the polynomial growth with respect to the instability parameter. These estimates are valid for quite arbitrary, possibly unbounded domains. We establish precise estimates on the maximal innuence of the boundaries on the dynamics in the interior. For instance, the attractor...

The work is devoted to the systematic research of the periodic, quasi-periodic or chaotic oscillations and the coexistence in the nonlinear Duffing-van der Pol type dynamical systems describing processes and phenomena in nature or engineering. The achieved result is the elaboration of the basic theory for searching nonlinear effects based on the concepts of periodic skeletons, bifurcation theor...