**6. References**

26 Will-be-set-by-IN-TECH

**Figure 21.** Triple period stable cycle at point, *Pr* <sup>=</sup> 1.354839, *Ra* <sup>=</sup> 1.308 · 106; Stream lines in <sup>Ω</sup> for the

The following chapter covers five year work that has been conducted in the Chaotic Dynamics Laboratory in the Institute for Systems Analysis of Russian Academy of Sci., lead by professor N.A. Magnitskii. Our attempt in using standard open source or commercial software for this kind of analysis failed so we had to consider a specially constructed accurate and trustworthy numerical solution code for Navier-Stokes equations, partly described here. The results of numerical solution for initial-boundary value problems considered confirmed that laminar-turbulent transition undergoing the bifurcation process and has different scenarios. It is interesting to point out that similar scenarios with classical Feigenbaum scenario and Sharkovskiy windows of periodicity where recently found in [66] for initial-boundary value problems in continuous mechanical systems such as flexible plates and shallow shells. However, the universal FSM scenario [7] is found in all problems considered, despite the difference between problems. Recently we found that Boltzmann equations in hydrodynamic limit with BGK collision integral [63–65] also exhibit FSM scenario for laminar-turbulent transition process. It is likely that all hydrodynamic type chaotic solutions for PDEs are undergoing the FSM scenario in various modifications. The work continues and now we are considering compressible fluid dynamics (transonic and supersonic turbulence) and

3. FSM scenario with subharmonic cascade of bifurcations of stable limit cycles.

magnetohydrodynamics as well as other initial-boundary value problems for (1).

*Laboratory 11-3, Chaotic Dynamical Systems, Institute for Systems Analysis of RAS, Russian*

same *Pr* and *Ra*.

**5. Conclusion**

**Author details**

*Federation*

Nikolai Magnitskii and Nikolay Evstigneev

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© 2012 Podlevskyi, 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,

**Numerical Algorithms of Finding the Branching** 

*A*(,) , *f f*

*f* nonlinearly depends both on the parameter

*f* , the formalistic approach, which is based on linearization, is applied. The application of this approach shows, that the branching points of equation can be only those values of

> *A*( ) *f f*

with the operator-valued function *A* : () *C XH* ( *X H*( ) is a set of linear operators,

the branching points of initial equation. In a general case the curves of eigenvalues

appears and then the branching points will be those values of parameter

*A*() (), 

The theory of branching solutions of nonlinear equations arose in close connection with applied problems and development of its ever-regulated by the new applied problems.

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

 *f f*

, i.e. *Af f*

, then its eigenvalues will be

) is the eigenvalue of the corresponding linearized

and the function

*C*

> ( )

of the

. If the linearized

**Lines and Bifurcation Points of Solutions for** 

**One Class of Nonlinear Integral Equations** 

B. M. Podlevskyi

**1. Introduction** 

parameter

problem

for which

 ( ) 1.

http://dx.doi.org/10.5772/48735

where the operator *A*(,)

equation (see, eg, [20])

Additional information is available at the end of the chapter

When investigating the nonlinear equations of the form

, for which unit ( 1

equation linearly depends on the parameter

is the spectral parameter), nonlinearly depending on the parameter
