BIFURCATIONS OF DYNAMICAL SYSTEMS

Theory of bifurcations of dynamic systems on a plane. Wiley, New York. Jordan, D. W. and Smith, P. (1977). Nonlinear ordinary differential equations. Clarendon Press, Oxford. 

A. Andronov, E. Leontovich, I. Gordon, and A. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program of Sci. Translations, Jerusalem, 1971. 

In the following chapters we present the theory of bifurcations of dynamical systems with simple dynamics. It is difficult to over-emphasize the role of bifurcation theory in nonlinear dynamics the reason is quite simple the methods of the theory of bifurcations comprise a working tool kit for the study of dynamical models. Besides, bifurcation theory provides a universal language to communicate and exchange ideas for researchers from different scientific fields, and to understand each other in interdisciplinary discussions. 

The modern theory of bifurcations of dynamical system is directly linked to the notion of non-roughness, or structural instability of a system. The main motivation is that the analysis of a system will be rather incomplete if we restrict our consideration to only the regions of structural stability of the system. Indeed, by changing parameters we can move from one structurally stable system to another, a qualitatively different system, upon crossing some boundaries in the parameter space that correspond to non-rough systems. 

Bykov, V. V. [1978] On the structure of a neighborhood of a separatrix contour with a saddle-focus, in Methods of Qualitative Theory of Differential Equation (Gorky Gorky State University), 3-32 [1980] On bifurcations of dynamical systems with a separatrix contour containing a saddle-focus, ibid. 44-72 [1988] On the birth of a non-trivial hyperbolic set from a contour formed by separatrices of a saddle, ibid. 22-32. 

Lukyanov, V. [1982] On bifurcations of dynamical systems with a sep-aratrix loop to a saddle-node, Diff. Eq. Russian) 58, 1493-1506. 

Sotomayor, J. [1971] Generic bifurcations of dynamical systems, in Dynamical Systems Proc. Symp. Univ. Bahia, Salvador, 561-582, Zbl.296.58007. 

Turaev, D. V. [1991] On bifurcations of dynamical systems with two homoclinic curves of the saddle, Ph.D. Thesis, Nizhny Novgorod State University. 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

To interpret the problem under discussion concerning slow relaxations in chemistry, it was necessary to clarify what must be regarded as slow relaxations of dynamic systems (i.e. to introduce some reasonable definition). In addition, it was necessary to find connections of slow relaxations with bifurcations and other dynamic peculiarities. This has been done by Gorban et al. [13-19]. 

A.N. Gorban and V.M. Cheresiz, Slow Relaxations of Dynamical Systems and Bifurcations of co-Limit Sets, Prepr., Computer Centre, Krasnoyarsk, 1980 (in Russian). 

Classification of catastrophes will be preceded by the centre manifold theorem which is a counterpart to the splitting lemma in elementary catastrophe theory. It will turn out that in the catastrophe theory of dynamical systems such notions of elementary catastrophe theory as the catastrophe manifold, bifurcation set, sensitive state, splitting lemma, codimension, universal unfolding and structural stability are retained. 

Owing to the very simple and intuitively clear definition of the equation of state, the topology of the vapor-liquid-phase transition and critical point is examined easily using the methods of dynamic system and bifurcation theory. 

Certain mathematical-physical considerations and the subsequent fitting of f (p T) allow us to conclude that the coexistence envelope diameter point (pd(T), pa(T)) is an (orbitally unstable) improper node, i.e. that all solution paths leaving (pD(T), pequilibrium points (pG(T), p (T)), (pD(T), P (T)), and (pL(T), Pa(T)) converge to the critical point (1, 1). This multiple equilibrium point is an orbitally stable, but structurally (topologically) unstable, multiple node. The parameter T thus can be considered as a bifurcation parameter, and T = 1 as a bifurcation value of dynamic System 3. 

Eventually, aU of them are based on the methods of general qualitative theory of differential equations developed by Poincare more than a century ago [47]. This theory was essentially developed by Andronov in 1930s [48] and, finally, after Hopf s theorem on bifurcation appeared in 1942 [49] it became a self-consistent branch of mathematics. This subject is currently known luider several names Poincare-Andronov s general theory of dynamic systems theory of non-linear systems theory of bifurcation in dynamic systems. Although the first notion is, in our opinion, the most exact one, we will use the term bifurcation theory , or BT, for the sake of brevity. 

Andronov A. A., Leontovich, J. A., Gordon, 1.1. Mayer, A. G. (1967). Bifurcation theory of dynamic systems on the plane. Nauka, Moscow (in Russian). 

The generation of self-sustained oseillations is a particular case of bifurcation. The term bifurcation is often used in connection with the mathematical study of dynamical systems. It denotes a sudden qualitative ehange in the behavior of a system upon the smooth variation of a parameter, the so-eaUed bifureation parameter, and is applied to the point of the fundamental reeonstmetion of the phase portrait where the bifurcation parameter attains its critical value. The simplest examples of bifurcation are the appearance of a new rest point in the phase space, the loss of the rest-point stability, and the appearance of a new limit cycle. Bifurcation relates to physicochemical phenomena such as ignition and extinction, that is, a jump-like transition from one steady state to another one, the appearance of oscillations, or a chaotic regime, and so on. 

Gorban, A.N., Cheresiz, V.M., 1981. Slow relaxations of dynamical systems and bifurcations of <0-Umit sets. Etokl. Akad. Nauk SSSR 261, 1050-1054. 

The transition from vortex motion to turbulence is accompanied by a corresponding sequence of instabilities or bifurcations, which results in a qualitative change of the domain pattern. Prom the theory of dynamical systems [127] the instability called Andronov-Hopf bifurcation [58] is well known. In this case, an oscillating regime appeared between two or more states having almost equal free energy. 

The splitting of an SL of the quantum mechanical cmrent densities J and at a branching point in can be smdied within the framework of bifurcation theory of dynamical systems, see [94—96] for an introduction to the subject. Accordingly, let us consider a system of first-order differential equations in matrix form... 

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