The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.
- nonlinear systems
- dynamical chaos
- singular attractors
- FShM theory
Well-known, that chaotic dynamics is inherent practically in all nonlinear mappings and systems of differential equations having irregular attractors, distinct from stable fixed and singular points, limit cycles and tori. However, many years there was no clear understanding of that from itself represent irregular attractors and how they are formed. In this connection it was possible to find in the literature more than 20 various definitions of irregular attractors: stochastic, chaotic, strange, hyperbolic, quasiattractors, attractors of Lorenz, Ressler, Chua, Shilnikov, Chen, Sprott, Magnitskii and many others. It was considered that there are differences between attractors of autonomous and nonautonomous nonlinear systems, systems of ordinary differential equations and the equations with partial derivatives, and that the chaos in dissipative systems essentially differs from chaos in conservative and Hamiltonian systems. There was also an opinion which many outstanding scientists adhered, including Nobel prize winner I.R. Prigogine, that irregular attractors of complex nonlinear systems cannot be described by trajectory approach, that are systems of differential equations. And only in twenty-first century it has been proved and on numerous examples it was convincingly shown, that there is one universal bifurcation scenario of transition to chaos in nonlinear systems of mappings and differential equations: autonomous and nonautonomous, dissipative and conservative, ordinary, with private derivatives and with delay argument (see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9]). It is bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario, beginning with the Feigenbaum cascade of period-doubling bifurcations of stable cycles or tori and continuing from the Sharkovskii subharmonic cascade of bifurcations of stable cycles or tori of an arbitrary period up to the cycle or torus of the period three, and then proceeding to the Magnitskii homoclinic or heteroclinic cascade of bifurcations of stable cycles or tori. All irregular attractors born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories.
However, in the scientific literature many papers continue to appear in which authors, not understanding an essence of occurring processes, write about opened by them new attractors in nonlinear systems of differential equations. Such papers are, for example, papers [10, 11] which authors with surprise ascertain an existence of chaotic dynamics in nonlinear system of ordinary differential equations with one stable singular point and try to explain this phenomenon by presence in the system of Smale’s horseshoe. Numerous papers continue to be published also in which presence of chaotic attractor in the system of ordinary differential equations is connected with Lyapunov’s positive exponent found numerically, diffusion chaos in nonlinear system of equations with partial derivatives is explained by the Ruelle-Takens (RT) theory and is connected with birth of mythical strange attractor at destruction of three-dimensional torus, and presence of chaotic dynamics in Hamiltonian or conservative system is explained by the Kolmogorov-Arnold-Mozer (KAM) theory and is connected with consecutive destruction in the system of rational and mostly irrational tori of nonperturbed system.
The purpose of the present paper is once again to show on concrete new, not entered in [1, 2, 3, 4, 5, 6, 7, 8, 9], examples, that chaos in the system considered in Refs. [10, 11], and also chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under the FShM scenario. Thus, in any nonlinear system there can be an infinite number of various singular attractors, becoming complicated at change of bifurcation parameter in a direction of the cascade of bifurcations. Presence or absence in system of stable or unstable singular points, presence or absence of saddle-nodes or saddle-focuses, homoclinic or heteroclinic separatrix contours and Smale’s horseshoes and also positivity of the calculated senior Lyapunov’s exponent are not criteria of occurrence in system of chaotic dynamics. And the birth in the system of three-dimensional and even multi-dimensional stable torus leads not only to its destruction with birth of mythical strange attractor, but also to cascade of its period-doubling bifurcations along one of its frequencies or several frequencies simultaneously. Chaotic dynamics in Hamiltonian and conservative systems also is consequence of cascades of bifurcations of birth of new tori, instead of consequence of destruction of some already ostensibly existing mythical tori of nonperturbed system. Thus, for the analysis of chaotic dynamics of any nonlinear system, attempts of calculation of a positive Lyapunov’s exponent, application of КАМ and RT theories and the proof of existence of Smale’s horseshoes are absolutely senseless. Let us notice, that the results of Feigenbaum and Sharkovsky are received only for one-dimensional unimodal maps and then were transferred by Magnitskii at first on two-dimensional systems of differential equations with periodic coefficients, then on three-dimensional, multi-dimensional and infinitely dimensional dissipative and conservative autonomous systems of ordinary differential equations and then on systems of the equations with partial derivatives. Besides this, it is proved by Magnitskii, that the subharmonic cascade of Sharkovsky bifurcations can be continued by homoclinic or heteroclinic bifurcations cascade both in the differential equations, and in continuous one-dimensional unimodal mappings.
1.1. FShM-cascades of bifurcations of stable cycles and a birth of singular attractors in one-dimensional unimodal mappings
Let us give a summary of bifurcation FShM theory of chaos in one-dimensional continuous unimodal mappings. Detailed proof of statements of the present section can be found in Ref. .
1.1.1. FShM-cascade of bifurcations in logistic mapping
Studying the properties of logistic mapping
Feigenbaum proved that in this equation there is a cascade of period-doubling bifurcations of its cycles and found a sequence of values of the parameter
Considering the map (1) on an interval
Definition. Ordering in set of the natural numbers, looking like
is called as Sharkovsky’s order. Theorem of Sharkovsky approves, that if continuous unimodal map
It also follows from the Sharkovsky theorem, that at change of values of bifurcation parameter, stable cycles in one-dimensional unimodal continuous mappings are obliged to be born according to the order (2). And their births occur in pairs together with unstable cycles as a result of saddle-node (tangent) bifurcations. Each stable cycle of Sharkovsky cascade, which has born thus, undergoes then the cascade of period-doubling bifurcations, generating its own window of periodicity (Figure 1). A limit of such cascade is more complex singular attractor—nonperiodic almost stable trajectory any neighborhood of which contains the infinite number of unstable periodic trajectories. Hence, the cascade of Feigenbaum bifurcations is an initial stage of the full subharmonic cascade of bifurcations, described by Sharkovsky order. In the case of logistic mapping (1) cycle of the period three is born at value
Behind subharmonic Sharkovsky cascade, homoclinic (heteroclinic) cascade of bifurcations lays, opened by Magnitskii at first in nonlinear systems of ordinary differential equations, and then found out in logistic and other unimodal continuous mappings. Homoclinic (heteroclinic) cascade of bifurcations consists of a consecutive birth of stable homoclinic (heteroclinic) cycles of the period
2. Dynamical chaos in nonlinear dissipative systems of ordinary differential equations
Bases of the FShM theory with reference to nonlinear dissipative systems of ordinary differential equations are stated in Refs. [1, 2, 3, 7]. Thus in systems with strong dissipation it is realized both the full subharmonic cascade of Sharkovsky bifurcations, and full (or incomplete) homoclinic (or heteroclinic) cascade of Magnitskii bifurcations depending on, whether exists homoclinic (or heteroclinic) separatrix contour in the system. In systems with weak dissipation the FShM-order of bifurcations can be broken in its right part. Hence, attractors of such systems are regular attractors (stable singular points, stable cycles and stable tori of any dimension), or singular cyclic or toroidal attractors—limited nonperiodic almost stable trajectories or the toroidal manifolds, being limits of cascades of the period-doubling bifurcations of regular attractors (cycles, tori). In Refs. [1, 2, 3, 7] it is proved, that the FShM scenario of transition to chaos takes place in such classical two-dimensional dissipative systems with periodic coefficients, as systems of Duffing-Holmes, Mathieu, Croquette and Krasnoschekov; in three-dimensional autonomous dissipative systems, as systems of Lorenz, Ressler, Chua, Magnitskii, Vallis, Anishchenko-Astakhov, Rabinovich-Fabricant, Pikovskii-Rabinovich-Trakhtengertz, Sviregev, Volterra-Gause, Sprott, Chen, Rucklidge, Genezio-Tesi, Wiedlich-Trubetskov and many others; in multi-dimensional and infinitely dimensional autonomous dissipative systems, as systems of Rikitaki, Lorenz complex system, Mackey-Glass equation and many others. These systems describe processes and the phenomena in all areas of scientific researches. Lorenz system is a hydrodynamic system, Ressler system is a chemical system, Chua system describes the electro technical processes, Magnitskii system is a macroeconomic system, Widlich-Trubetskov system describes the social processes and phenomena, Mackey-Glass equation describes the processes of hematopoiesis.
2.1. Transition to chaos in the system with one stable singular point
In this chapter, let us consider the three-dimensional system of ordinary differential equations with one stable singular point which has been proposed in Ref. 
This system has the only stable singular point (0.25, 0.0625, −0.096) of stable focus type as Jacobian matrix in the singular point has eigenvalues (−0.96069, −0.01966 ± 0.50975
As bifurcation parameter we choose the parameter
3. Dynamical chaos in Hamiltonian and conservative systems
Conservative system saves its volume at movement along the trajectories and, hence, cannot have attractors. Therefore studying of dynamical chaos in Hamiltonian and, especially, simply conservative systems is more a difficult task in comparison with the analysis of chaotic dynamics in dissipative systems which can be described by universal bifurcation FShM theory. The main problem solved by the modern classical theory of Hamiltonian systems (the Kolmogorov-Arnold-Moser theory) is the problem of integrability of such a system, that is, the problem of its reduction to the “action-angle” variables by constructing some canonical transformation. It is assumed that in such variables the motion in a Hamiltonian system is periodic or quasiperiodic and occurs on the surface of an n-dimensional torus. In this formulation, any non-integrable Hamiltonian system is considered as a perturbation of the integrable system, and the analysis of the dynamics of the perturbed system reduces to studying the problem of the destruction of the tori of an unperturbed system with increasing values of the perturbation parameter. But numerous examples of Hamiltonian and simply conservative systems, considered by the author in [4, 5, 6, 7], deny existence such classical KAM-scenario of transition to chaos.
One of the most effective approaches to the decision of a problem of the analysis of chaotic dynamics in conservative systems is offered by the author in Ref.  (see also [5, 6, 7]). The approach assumes consideration of conservative system in the form of limiting transition from corresponding extended dissipative system (in which the dissipative member is added) to initial conservative system. This approach can be evidently shown by means of construction of two-parametrical bifurcation diagram which corresponds to transition from dissipative state to conservative state. Attractors (stable cycles, tori and singular attractors) of extended dissipative system can be found numerically with use of results of universal bifurcation FShM theory. Further transition to chaos in conservative (Hamiltonian) system is carried out through cascades of bifurcations of attractors of extended dissipative system when dissipation parameter tends to zero. Areas of stability of stable cycles of the extended system at zero dissipation turn into tori of conservative (Hamiltonian) system around of its elliptic cycles into which stable cycles transform. Thus tori of conservative (Hamiltonian) system touch through hyperbolic cycles into which saddle cycles of extended dissipative system transform. In [4, 5, 6, 7] the considered above approach is described in detail with reference to Hamiltonian systems with one and a half, two, two and a half and three degrees of freedom, and also to simply conservative systems of differential equations, including the conservative Croquette equation, the equation of a pendulum with oscillating point of fixing, the conservative generalized Mathieu equation, well-known Hamiltonian system of Henon-Heiles equations. In Refs. [12, 13] the given approach has been applied and strictly proved by continuation along parameter of solutions from dissipative into conservative areas by means of the Magnitskii method of stabilization of unstable periodic orbits  at research bifurcations and chaos in the Duffing-Holmes equation
and in the model of a space pendulum
Corresponding bifurcation diagrams in a plane (
3.1. Hamiltonian Yang-Mills-Higgs system with two degrees of freedom
In this chapter, let us illustrate Magnitskii approach by the example of Yang-Mills-Higgs system with two degrees of freedom and with Hamiltonian
passing into classical system of the Yang-Mills equations at
The system (8) has four sets of periodic solutions to which there correspond four basic cycles in phase space
where . Complication of solutions of Hamiltonian system (8) of the Yang-Mills-Higgs equations down to full chaotic dynamics occurs at
Further, the process continues with the birth of infinitely folded heteroclinic separatrix manifold, stretched over separatrix Feigenbaum tree, both in extended dissipative system (10), and in Hamiltonian system (8) close to it. Accordions of corresponding heteroclinic separatrix zigzag fill the whole phase space of the system, however the limited accuracy of numerical methods does not allow to track this process up to the value
4. Spatio-temporal chaos in nonlinear partial differential equations
Bases of FShM theory with reference to a wide class of nonlinear systems of partial differential equations are stated in Refs. [6, 7, 8, 9]. This class includes systems of the equations of reaction-diffusion type, describing various autowave oscillatory processes in chemical, biological, social and economic systems, including the well-known brusselator equations; the equations of FitzHugh-Nagumo type, describing processes of chemical and biological turbulence in excitable media; the equations of Kuramoto-Tsuzuki (or Time Dependent Ginzburg-Landau) type, describing complex autooscillating processes after loss of stability of a thermodynamic branch in reaction-diffusion systems; the systems of Navier-Stokes equations, describing laminar-turbulent transitions in hydrodynamical and gazodynamical mediums.
4.1. Diffusion chaos in reaction-diffusion systems
Wide class of physical, chemical, biological, ecological and economic processes and phenomena is described by reaction-diffusion systems of partial differential equations
depending on the scalar or vector parameter
The nonlinear processes occurring in so-called excitable media, are described by a special case of systems of the reaction-diffusion equations—FitzHugh-Nagumo type systems
Solutions of the system (12) are: switching waves, traveling waves and running impulses, dissipative stationary spatially inhomogeneous structures, and diffusion chaos—nonstationary nonperiodic inhomogeneous structures, sometimes called biological or chemical turbulence. All such solutions can be analyze on a line by replacement
where the derivative is taken over the variable
where derivative is taken with respect to the variable
At the further reduction of values of parameter
4.2. Spatio-temporal chaos in autooscillating mediums
It is well-known that any solution of the reaction-diffusion system (11) in a neighborhood
In this chapter, we consider the problem of research of nonlinear effects in model of surface plasmon-polyariton. The passage of an electromagnetic wave through a configuration from three various environments dielectric-metal-dielectric can be described by following system of the equations in partial derivatives in the complex variables, turning out of Maxwell equations (see ):
The system (17) represents two connected Ginzburg-Landau equations,
In the considered initial boundary-value problem it is possible to allocate a subclass of spatially homogeneous solutions, not dependent on a variable
Critical value of parameter is
Following bifurcation in the system (17) occurs in a range of parameter values
For the problem with Neumann’s homogeneous boundary conditions also it was possible to observe a non-homogeneous stable cycle at
4.3. Laminar-turbulent transition in Navier-Stokes equations
The problem of turbulence consists in explaining the nature of the disordered chaotic motion of a nonlinear continuous medium and in finding ways and methods of its adequate mathematical description. Originating more than a 100 years ago, the problem of turbulence is still one of the most complicated and most interesting problems in mathematical physics. It is in the list of seven mathematical millennium problems, named so by the Clay Institute of Mathematics . In addition, the turbulence problem is formulated in the list of S.Smale's 18 most important mathematical problems of the twenty-first century . The most important and interesting in the problem of turbulence is to elucidate the causes and mechanisms of chaos generation in a nonlinear continuous medium when passing from the laminar to the turbulent state. Currently, there are several mathematical models that claim to explain the mechanisms of generation of chaos and turbulence in nonlinear continuous media. The most famous among these models are: the Landau-Hopf model explaining turbulence by motion along an infinite-dimensional torus generated by an infinite cascade of Andronov-Hopf bifurcations; and the Ruelle-Takens model, which explains turbulence by moving along a strange attractor generated by the destruction of a three-dimensional torus. In recent years, the author and his pupils have proved (see [8, 9, 20, 21, 22]) that the universal bifurcation FShM mechanism for the transition to space-time chaos in nonlinear systems of partial differential equations through subharmonic cascades of bifurcations of stable cycles or two-dimensional and multi-dimensional tori also takes place in problems of laminar-turbulent transitions for Navier-Stokes equations
In this chapter, we consider shortly the bifurcation scenario in coupled Kelvin-Helmholtz and Rayleigh-Taylor problem. This problem is solved in detail in Ref. . We begin our consideration from the value of
We make some general remarks on the chaotic dynamics of nonlinear systems of differential equations, since the very publication of papers [10, 11] and many similar papers, even in prestigious refereed journals, attests to a complete lack of understanding of the mechanism of transition to chaos in nonlinear systems of differential equations. In this chapter, on numerous examples, it is convincingly demonstrated that there exists one universal FShM bifurcation scenario of transition to chaos in all systems of nonlinear differential equations without exception: autonomous and nonautonomous, dissipative and conservative, ordinary, with partial derivatives and with delayed argument. All irregular attractors that arise during the implementation of this scenario are exclusively singular attractors. Each nonlinear system can have infinitely many different structurally unstable singular attractors for different values of the bifurcation parameter, which can enter implicitly into the equations of the system. Thus, neither the presence or absence of stable or unstable singular points in the system, nor the presence or absence of saddle-nodes or saddle-focuses, as well as homoclinic or heteroclinic separatrix contours, is not a criterion for the appearance of chaotic dynamics in the system. Also, neither the positivity of the senior Lyapunov exponent, nor the proof of existence of Smale’s horseshoe, nor the KAM (Kolmogorov-Arnold-Moser) theory, nor the theory of RT (Ruelle-Takens), are such criteria either. The positivity of the Lyapunov exponent is purely a consequence of computational errors, because due to the presence of an everywhere dense set of nonperiodic trajectories, numerical motion is possible only over the whole region occupied by the trajectory of the singular attractor, and not along its trajectory itself. In addition, the Lyapunov exponent will also be positive when moving along a stable periodic trajectory of a large period in the vicinity of some singular attractor. The presence of Smale’s horseshoe in the system testifies to the complex dynamics of the solutions, however, even in the neighborhood of the separatrix loop of saddle-focus, where by Shilnikov’s theorem there exists an infinite number of Smale’s horseshoes, the dynamics of solutions are determined not by horseshoes, but by a much more complex infinite set of unstable periodic solutions generated at all stages of all three cascades of bifurcations of the FShM scenario, whose homoclinic cascade of cycles ends in the limit precisely with the separatrix loop of saddle-focus. The only method that allows establishing reliably the presence of chaotic dynamics in the system is the numerical finding of stable cycles or tori of the FSM-cascades of bifurcations.
Paper is supported by Russian Foundation for Basic Research (grants 14-07-00116-а).
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