Cite this article as:

Kuznetsov S. P., Kuptsov P. V. Lorenz Attractor in a System with Delay: an Example of Pseudogyperbolic Chaos. Izvestiya of Saratov University. New series. Series Physics, 2018, vol. 18, iss. 3, pp. 162-176. DOI:


Lorenz Attractor in a System with Delay: an Example of Pseudogyperbolic Chaos


Background and Objectives: The work contributes to a research direction aimed at search for and construction of physically realizable systems, which could fill the mathematical theory of pseudo-hyperbolic dynamics with physical content. Chaotic attractors belonging to this class generate genuine chaos that does not degrade under small variations of parameters and functions in dynamical equations.

Materials and Methods: The methodological apparatus of the study uses numerical methods for integrating differential equations with time-delay, methods for calculating Lyapunov exponents, and special methods for testing the absence of tangencies of subspaces of vectors of small perturbations of orbits on the attractors, that is an essential condition of pseudohyperbolicity according to the definition.

Results: An example of a system is introduced which is described by differential equations with retarded argument, in the infinite-dimensional phase space of which there occurs a chaotic attractor similar in properties to the classic Lorenz attractor. Presented and tested is a mathematical toolkit needed to identify and test the pseudo-hyperbolic nature of chaos. The scheme of the electronic generator governed by the proposed equations is presented, and its dynamics is simulated using the Multisim software environment, in particular, the oscilloscope traces and spectra of chaotic oscillations generated by the system are shown. Conclusion: The concept of pseudo-hyperbolic dynamics, which clearly is of interdisciplinary significance, deserves attention, particularly, in the frame of application to the design of electronic generators of robust chaos that survives variations in parameters and details of the construction, and therefore is of interest for possible applications of chaos.


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