Implementation of electronic devices manifesting different phenomena of nonlinear dynamics and their applications including analog modeling systems of different nature
Grant of Russian Science Foundation No 17-12-01008
The project is carried out in 2017-2019
Participants of the project:
The project aims at elaborating principles and implementing electronic devices, specifically directed to provide diverse phenomena of nonlinear dynamics, including chaos, quasi-periodic and strange nonchaotic dynamics, various types of synchronization and scenarios of emergence of the complex dynamics and bifurcations.
This also concerns to identifying areas for electronic analog simulation in application to systems of different nature (mechanical, biophysical, technical systems) and to information and communication applications (communication systems, radar systems, masking and suppressing signals, random number generation and generation of cryptographic keys).
Topicality and relevance of the work due to the need of mastering and development of material accumulated in the modern theory of dynamical systems in the physical and technical context as this material remains yet largely of abstract and mathematical character.
The main specificity of the project is that the object of study are electronic circuits built using modern element base and mathematical models of these systems targeted specifically at the implementation of definite phenomena of complex dynamics. It is supposed to consider different approaches to obtaining the rough chaos, including the class of uniformly hyperbolic, partially hyperbolic, singular hyperbolic (Lorenz type attractors), pseudo-hyperbolic attractors.
Exploiting numerical simulation and involving the principles and results of the modern theory, new systems of electronic nature will be proposed and investigated implementing chaos, quasi-periodic dynamics, and other types of behavior, and the possibility of analog modeling will be considered for systems of other physical nature.
For description of the dynamics mathematical models will be developed, the topography of parameter space device s will be studied for the models, which implement different types of behavior, bifurcation phenomena, chaos and synchronization. In the course of the research methods of computer studies for the phenomena of complex nonlinear dynamics will be developed and improved, including those relating to processing data of electronic experiments.
In a view of analog simulation and possible applications it is planned to put attention to such areas as parametric excitation of oscillations to control dynamics of micro and nano-scale systems, use of the chaos generators in complex with quasi-periodic oscillators for communications with encoding of the transmitted signals referring to radio and optical systems, analysis of possible significance of quasi-periodic oscillations, and chaos synchronization in model systems of biophysical and biomedical nature.
• Isaeva O. B., Obychev M. A., Savin D. V. Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map. Rus. J. Nonlin. Dyn., 2017, 13, No 3, 331–348. (In Russian.)
Main results 2017
A generalized model with bifurcations associated with the so-called blue sky catastrophe was introduced. Depending on the integer index m, various types of attractors arise, including those associated with quasiperiodic oscillations and hyperbolic chaos, and the hyperbolicity test is performed at the level of numerical calculations based on a statistical analysis of the intersection angles of stable and unstable manifolds.
A non-autonomous system with uniformly hyperbolic attractor of Smale-Williams type in the Poincaré section is proposed, in which the generation is performed on the basis of the oscillation death effect. The results of numerical study of the system are presented: iterative diagrams and portraits of the attractor in the Poincaré stroboscopic section, power density spectra, Lyapunov exponents and their dependence on parameters, chart of dynamical regimes. The hyperbolicity of the attractor is verified basing on the criterion of angles.
An abstract dynamical system is introduced by an implicit function of values of a variable in consecutive discrete time moments, so that the dynamics are ambiguously defined both in the reverse and in direct time. The investigated mapping allows to carry out a transition from the unambiguously defined in the direct time case to the implicit and, further, to a kind of "conservative" limit. Recorded on the basis of the complex Mandelbrot map, it demonstrates the transformation of the phenomena of complex analytical dynamics into "conservative" phenomena and makes it possible to identify the relationship between them.