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**Chapter 6** 

© 2012 Magnitskii, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Universality of Transition to Chaos** 

Additional information is available at the end of the chapter

numerous examples, illustrations and numerical calculations.

Magnitskii, 2007; Magnitskii, 2008; Magnitskii, 2008b; Magnitskii, 2010).

Nikolai A. Magnitskii

http://dx.doi.org/10.5772/48810

**1. Introduction** 

**in All Kinds of Nonlinear Differential Equations** 

The basically finished universal theory of dynamical chaos in all kinds of nonlinear differential equations including dissipative and conservative, nonautonomous and autonomous nonlinear systems of ordinary and partial differential equations and differential equations with delay arguments is shortly presented in the paper. Consequence of the theory is an existence of the uniform universal mechanism of self-organizing in the huge class of the mathematical models having the applications in many areas of science and techniques and describing the numerous physical, chemical, biological, economic and social both natural and public phenomena and processes. All theoretical positions and results are received within last several years by extremely author and his pupils and confirmed with

The basis of this theory consists of the Feigenbaum theory of period doubling bifurcations in one-dimensional mappings (Feigenbaum, 1978), the Sharkovskii theory of subharmonic bifurcations of stable cycles of an arbitrary period up to the cycle of period three in onedimensional mappings (Sharkovskii, 1964), the Magnitskii theory of homoclinic and heteroclinic bifurcations of stable cycles and tori in systems of differential equations and the Magnitskii theory of rotor type singular points of two-dimensional nonautonomous systems of differential equations with periodic coefficients of leading linear parts as a bridge between one-dimensional mappings and differential equations (Magnitskii & Sidorov, 2006;

It is shown that this universal Feigenbaum-Sharkovskii-Magnitskii (FSM) bifurcation theory of transition to dynamical chaos takes place in all classical three-dimensional chaotic dissipative systems of ordinary differential equations including Lorenz hydrodynamic system, Ressler chemical system, Chua electro technical system, Magnitskii macroeconomic system and many others. It takes place also in well-known two-dimensional non-

and reproduction in any medium, provided the original work is properly cited.
