**5. Conclusion**

170 Nonlinearity, Bifurcation and Chaos – Theory and Applications

neighborhood of the center of the second spiral wave.

spiral waves (b), (c).

**Figure 28.** Spiral waves in the plane (,) *x y* at 1*c* 0.5, <sup>2</sup>*c* -0.68 (a) and projection of the section 1 1 *ux y* (,)0 of four-dimensional subspace of phase space near center of bottom spiral wave.

At the reduction of negative values of parameter 2*c* there is a complication of structure of spiral waves and solutions corresponding to them in phase space of a boundary value problem (57). In Fig. 29a the picture of spiral waves on a plane (,) *x y* is shown at value 2*c* 0.7 , and in Fig. 29b the projection of one of two parts of section 1 1 *ux y* (,)0 on a plane of coordinates 11 22 ( ( , ), ( , )) *vx y ux y* for two points from a neighborhood of the center of a spiral wave of the greatest radius from Fig. 29a is shown in the increased scale. It is visible, that in phase space of solutions complex two-dimensional torus of the period three from Sharkovskii subharmonic cascade corresponds to a neighborhood of the center of this spiral wave. In Fig. 29c the projection of section 1 1 *ux y* (,)0 in a neighborhood of the other spiral wave located in a right bottom corner in Fig. 29a is presented. The projection represents the shaded ring area. But the second section by the plane 2 2 *ux y* ( , ) 28 of three-dimensional space of points received after carrying out the first section, gives in coordinates 11 22 ( ( , ), ( , )) *vx y vx y* two closed curves. These curves testify the existence of three-dimensional torus in phase subspace of solutions in a

**Figure 29.** Spiral waves in the plane (,) *x y* at 1*c* 0.5, <sup>2</sup>*c* -0.7 (a) and projections of parts of sections of four-dimensional subspace of phase space of solutions of the problem (57) in neighborhoods of two

In the chapter it is proved and illustrated with numerous analytical and numerical examples that there exists a uniform universal bifurcation mechanism of transition to dynamical chaos in all kinds of nonlinear systems of differential equations including dissipative and conservative, ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments. This mechanism is working for all nonlinear continuous models describing both natural and social phenomena of a macrocosm surrounding us, including various physical, chemical, biological, medical, economic and sociological processes and laws. And this universal mechanism is described by the Feigenbaum-Sharkovskii-Magnitskii theory - the theory of development of complexity in nonlinear systems through subharmonic and homoclinic cascades of bifurcations of stable limit cycles or stable two-dimensional or manydimensional invariant tori.

Notice, that theory FSM is also applicable for solutions of Navier-Stokes equations, i.e. it solves a problem of turbulence describing various bifurcation scenarios of transition from laminar to turbulent regimes in spatially three-dimensional problem of motion of a viscous incompressible liquid (Evstigneev *et al*., 2009a,b; Evstigneev *et al*., 2010; Evstigneev & Magnitskii, 2010). The solution of this super complex problem is presented in the separate chapter in the present book. Similar scenarios with classical Feigenbaum scenario and

Sharkovskii windows of periodicity where recently found also in (Awrejcewicz *et al*., 2012} for initial-boundary value problems in continuous mechanical systems such as flexible plates and shallow shells. As to the processes occurring in a microcosm they, in opinion of the author, also can be successfully described by nonlinear systems of differential equations and their bifurcations. The first results in this direction are received by the author in (Magnitskii, 2010b; Magnitskii, 2011b; Magnitskii, 2012) where the basic equations and formulas of classical electrodynamics, quantum field theory and theory of gravitation are deduced from the nonlinear equations of dynamics of physical vacuum (ether).

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