It is well known that chaotic dynamics is inherent in almost all natural and social processes and phenomena described by nonlinear systems of ordinary and partial differential equations. However, many years there was no clear understanding how there are formed irregular attractors which are differ from stable fixed points, limit cycles and tori. It was considered that there are differences between attractors of non-autonomous and autonomous nonlinear systems, ordinary and partial differential equations and the equations with delay arguments. Chaos in Hamiltonian and conservative systems was considered to be essentially different from chaos in dissipative systems. There was also an opinion that irregular attractors of complex nonlinear systems could not be described by systems of differential equations and that a new mathematical apparatus had to be developed to describe them. And only recently it has been proved and confirmed by numerous examples that irregular (chaotic) attractors can be understood and described within the framework of the theory of bifurcations in nonlinear systems of differential equations. It was proved that there is a universal bifurcation scenario of the transition to chaos in nonlinear systems of differential equations of all kinds and types: dissipative and conservative, autonomous and non-autonomous, ordinary, partial and with a delayed argument (see, for example, [1,2,3,4]). This is the Feigenbaum-Sharkovsky-Magnitskii (FShM) bifurcation scenario. It begins with a cascade of Feigenbaum bifurcations of doubling the period of some stable cycle or torus of arbitrary dimension, and then continues with a subharmonic cascade of Sharkovsky bifurcations of stable cycles or tori of any period up to a cycle or torus of period three. Then, if possible, the bifurcation scenario continues with a homoclinic or heteroclinic cascade of Magnitskii bifurcations of stable cycles or tori converging to a homoclinic or heteroclinic separatrix contour or toroidal manifold. All irregular (chaotic) attractors that are born during the implementation of such a scenario are exclusively singular attractors, that is, non-periodic bounded trajectories in a finite-dimensional or infinite-dimensional phase space, which are the limits of the cycles of the Feigenbaum cascade and contain an infinite number of unstable periodic trajectories in any of their neighborhoods. The birth of cycles (tori) of the universal bifurcation scenario occurs in accordance with the order The left part of the order is the Feigenbaum cascade of bifurcations of cycle (torus) period doubling and ends with the first simplest singular attractor - the Feigenbaum attractor. The middle part of the order is a subharmonic cascade of Sharkovsky bifurcations and ends with the birth of a cycle (torus) of period three. The right-hand side of the order is a homoclinic cascade of Magnitskii bifurcations and ends in the limit, as a rule, with a homoclinic separatrix loop of the saddle-focus. All cycles (tori) previously born as a result of saddle-node bifurcations become unstable, but remain in the system. Therefore, if, for example, a stable cycle of period three is found in the system, which completes the subharmonic cascade of bifurcations, then in the system, together with a stable cycle of period three, there are infinitely many unstable cycles of all periods.

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. For conservative and Hamiltonian systems, only an incomplete cascade of Feigenbaum bifurcations is usually realized with the birth of elliptic tori around the cycles of the cascade.

It is proved in papers of author and in other papers, that the FShM bifurcation scenario of transition to chaos takes place in such classical two-dimensional dissipative systems with periodic coefficients, as Mathieu, Duffing-Holmes, Croquette, Krasnoschekov systems; in three-dimensional autonomous dissipative systems, as systems of Lorenz, Chua, Sprott, Ressler, Chen, Rabinovich-Fabricant, Vallis, Magnitskii, Anishchenko-Astakhov, Volterra-Gause, Pikovskii-Rabinovich-Trakhtengertz, Sviregev, Rucklidge, Genezio-Tesi, Wiedlich-Trubetskov and many others. This scenario transition to chaos takes place also in many- and infinitely dimensional systems of nonlinear ordinary differential equations, such as Rikitaki system, Lorenz complex five-dimensional system, Mackey-Glass equation with delay argument and many others. This scenario transition to chaos takes place also in many partial differential equations and systems, such as Brusselator and Kuramoto-Tsuzuki (Time Dependent Ginzburg- Landau) equations, reaction-diffusion and FitzHugh-Nagumo type systems of equations, nonlinear Schrodinger and Kuramoto-Sivashinskii equations. Moreover, this scenario describes also the laminar-turbulent transitions in any tasks for Navier-Stokes equations and transition to chaos in Hamiltonian and conservative systems, such as conservative Croquette and Duffing-Holmes equations, Mathieu-Magnitskii and Yang-Mills-Higgs Hamiltonian systems. The listed systems of equations describe a variety of complex natural, social, scientific and technical processes and phenomena in physics, chemistry, biology, economics, medicine and sociology, which emphasizes the universal applicability of the considered bifurcation approach.

However, works continue to appear in the scientific literature, in which the authors, not understanding the essence of the ongoing processes, write about new attractors allegedly discovered by them in nonlinear systems of differential equations. “Hidden” attractors for systems with stable singular points or no singular points at all are explained by the authors by the presence of the Smale horseshoe or by the found numerically positive Lyapunov exponent, or by the so-called "intermittency". Hyperchaos in the system is explained by the numerically found two positive Lyapunov exponents. Diffusion chaos in nonlinear system of equations with partial derivatives is explained by the RT (Ruelle-Takens) 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 КАМ (Kolmogorov-Arnold-Mozer) theory and is connected with consecutive destruction in the system of rational and mostly irrational tori of nonperturbed system.

But, in the works of the author (see, for example, [1,2,3,4] and others) it is theoretically proved and demonstrated on numerous examples that the intermittency and positiveness of the Lyapunov exponent are only calculation errors and cannot serve as criteria for the existence of chaotic dynamics in a system. It is proved that the leading characteristic Lyapunov exponent is equal to zero on any singular attractor. The effect of the positiveness of the Lyapunov exponent is exclusively a consequence of computational errors, since due to the presence of an everywhere dense set of non-periodic trajectories, numerical motion is possible only over the entire region in which the trajectory of the singular attractor is located, and not along its trajectory itself. In addition, the calculated Lyapunov exponent will also be positive when moving along a stable periodic trajectory of large period, located in the neighborhood of some singular attractor. The same errors lead to the determination of the alleged presence of “intermittency” in the system. Тhe presence of the Smale horseshoe in the system indicates the complex dynamics of solutions, but even in the neighborhood of the saddle-focus separatrix loop, where, according to Shilnikov’s theorem, there are an infinite number of Smale horseshoes, the dynamics of solutions is determined not by horseshoes, but by a much more complex infinite set of unstable periodic solutions generated on all stages of all three cascades of bifurcations of the FShM scenario. A birth in the system of three-dimensional and even multi-dimensional stable torus leads not to its destruction with birth of mythical strange attractor, as it postulated by RT theorem, but 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, as it postulated by KAM theorem.

The purpose of this article is to present five new systems of ordinary and partial differential equations in which transition to chaos occurs in accordance with the universal bifurcation FShM theory. The two systems describe autocatalytic chemical processes. The two next systems describe the numbers of interacting populations, one of them is the system of partial differential equations. The last system has no singular points and so it has so-called "hidden" attractor. It is well known that models of chemical autocatalytic reactions and models of the dynamics of the numbers of interacting populations have complex dynamics of the behavior of their solutions up to chaotic dynamics, called chemical or biological turbulence. However, the theoretical explanation of the development of chaotic dynamics in models of autocatalytic chemical processes and in models of the numbers of interacting populations has so far been reduced at best to the numerical determination of the Feigenbaum cascade of cycle period doubling bifurcations and further postulation of the transition to chaos either through the so-called "intermittency" or through the existence of a mythical "positive" Lyapunov exponent in the system. This explains the appearance of chaotic dynamics in the models proposed in [5,6,7] of an autocatalytic chemical process with feedback and an autocatalytic process in which the autocatalyst undergoes a mutation process in a fully stirred tank reactor. The appearance of chaotic dynamics in the model of the dynamics of the numbers of interacting populations of a predator and two prey proposed by A.D. Bazykin et al. in [8] has the similar explanation.

The obtained results ones again indicate the wide applicability of the universal bifurcation FShM theory for describing laminar-turbulent transitions to chaotic dynamics in complex nonlinear systems of differential equations.

Let us consider the model of an autocatalytic chemical process with feedback proposed in [5], which is a development of the well-known Gray-Scott model [6] in the case of homogeneity of its solutions. The model is a system of three nonlinear ordinary differential equations with positive fixed parameters $k=10, \sigma =0.005, \delta =0.02$ and changing bifurcation parameter μ. It was numerically shown in [5] that as the values of the parameter μ ≥ 0.1 increase, the system of equations (1) has a cascade of Feigenbaum bifurcations of doubling the period of stable cycles and then chaotic dynamics at μ = 0.153 with one window of periodicity of the cycle of period five at μ = 0.155 and cycle with double period ten. Then it is shown numerically that in system (1) the inverse Feigenbaum tree is realized up to the birth of a stable cycle of period one. But, since the stable cycles of the Feigenbaum bifurcation cascade are regular attractors, the numerical analysis carried out in [5] does not allow us to answer the question about the nature of the chaotic dynamics found in system (1) by the authors of [5].

In this section, an analytical and numerical analysis of the transition to chaos in the system of equations (1) is carried out. It is proved that for certain values of the bifurcation parameters, the transition to chaos in the system occurs in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation scenario [1,2,3,4] through the Feigenbaum bifurcation cascade of doubling the period of stable cycles, then through the subharmonic cascade of Sharkovsry bifurcations of the birth of stable cycles up to a cycle of period three, and then through the initial stage of the homoclinic cascade of Magnitskii bifurcations.

Let us consider the model of autocatalytic chemical reactions with mutation proposed in [7], in which the autocatalyst undergoes a mutation process in a fully stirred tank reactor. The model of autocatalytic reactions with mutation proposed in [7] is described by a system of nonlinear differential equations System (2) has six parameters: $\mathsf{\alpha },\mathsf{\beta },\mathsf{\theta },{\gamma }_1,{\gamma }_2,\delta .$ In [7], the last three parameters are fixed: ${\gamma }_1=450,{\gamma }_2=11.25,\delta =0.067$ and several cases with different values of mutation coefficients α and its efficiency β are considered. In this case, the coefficient θ is a bifurcation parameter. Chaos in system (2) was discovered in [7] at $\mathsf{\alpha }$ = 0.29,$\mathsf{\beta }$ = 0.68 and with a change in the bifurcation parameter in the interval $0.10902\le \mathsf{\theta }\le 0.10984$. For $\mathsf{\theta }\approx 0.10902$ in system (2) a hard loss of stability of the singular point occurs. The cycle becomes stable, and, as the values of the bifurcation parameter increase, it undergoes the Feigenbaum cascade of bifurcations of doubling period. A stable cycle of period two is born at $\mathsf{\theta }\approx 0.10937,$ a cycle of period four is born at $\mathsf{\theta }\approx 0.10642$, a cycle of period eight is born at $\mathsf{\theta }\approx 0.10973$, etc. Then, at $\mathsf{\theta }=0.10984$ chaotic dynamics was discovered in the system, which, according to the authors of [7], is confirmed by the found numerically positive value of the Lyapunov exponent, and the transition to chaos occurs allegedly through “intermittency”. The question of the nature of the chaotic dynamics found in system (2) by the authors of the paper [7] also remains open.

In [8], A.D. Bazykin et al. proposed a model for the dynamics of the interacting populations of a predator and two prey. The model is a system of three nonlinear ordinary differential equations with positive fixed parameters $\alpha$ = 11,$\beta$ = 6 and a changing bifurcation parameter δ, which is the growth rate of the second prey. It was numerically shown in [8] that, as the values of the parameter δ decrease in system (3), a cascade of period-doubling bifurcations of stable limit cycles is observed. In [9], a hypothesis was put forward that such a cascade is a transition to a complex inexplicable quasi-stochastic dynamics. The purpose of this work is to show that the dynamics of the numbers of interacting populations in model (3) is completely subject to the universal law of transition to dynamical chaos in nonlinear systems of differential equations in accordance with the bifurcation scenario of Feigenbaum-Sharkovsky-Magnitskii.

Consider a spatio-temporal model of the dynamics of numbers of interacting populations, which is a system of equations in partial derivatives of the reaction-diffusion type

By linearization in the neighborhood of the thermodynamic branch $\left(U,V\right)=(\sqrt{A},\mu /\sqrt{A}$) system (4) can be reduced in the first approximation to the system where $u=\sqrt{A}+z, v=\mu /\sqrt{A}+y$. Therefore, for $0<\mu <2$ the dynamics of system of equations (4) in the neighborhood of its thermodynamic branch is the dynamics of the interaction between the numbers of the predator described by the variable y, and the numbers of the prey described by the variable z.

Let us turn to the analysis of the dynamics of solutions and the transition to space-time chaos in the system of partial differential equations (4). Let us prove that, for certain values of the parameters, system (4) has, in full accordance with the universal bifurcation theory of Feigenbaum-Sharkovsky-Magnitskii, an infinite number of different stable wave solutions travelling along the spatial axis with arbitrary velocities, as well as an infinite number of different regimes of space-time chaos. Moreover, the bifurcation parameter is the velocity of propagation of traveling waves along the spatial axis, which is clearly not included in the original system of equations.

In the modern literature on nonlinear and chaotic dynamics, the concept of a “hidden” attractor of an autonomous system of ordinary differential equations is widely used in the case when the system, together with an irregular attractor, either has no singular points at all (see [10]) or has stable singular points (see [11]). The system from [11] with a stable singular point was considered by the author in [4], where it was shown that its chaotic attractor is also a singular attractor generated by a subharmonic cascade of FShM bifurcations theory. The system of equations with “hidden” attractor and without singular points proposed in [10] has the following form

The “hidden” attractor was found by the authors of [10] in system (7) at $a=4, b=40$, and the allegedly positive Lyapunov exponent was calculated in MATLAB by integrating the equations of system (7) by the fourth-order Runge-Kutta method, that, as noted above, is an absolutely meaningless procedure and is not a characteristic of the attractor.

The fact that system (7) must have some attractor for all $a>0$ follows from its dissipativity, since for the divergence of its right-hand side $F(x,y,z,w)$ we have

Usually a complex singular attractor is born as a result of FShM cascade of stable cycle or torus bifurcations, which themselves are born as a result of one or two Andronov-Hopf bifurcations from an initially stable singular point. It is clear that this scenario does not work for system (7), since it has no singular points at all at $b\ne 0$, as in this case its right-hand sides cannot be equal to zero at the same time. Nevertheless, let us show numerically that the “hidden” attractor found in system (7) is also a complex singular attractor of FShM cascades of bifurcations with respect to the parameter $a$ of some initially stable limit cycles of system (7) not born from singular points as a result of Andronov-Hopf bifurcations, but those born as a result of saddle-node bifurcations.

In this work, an analytical and numerical analysis of laminar-turbulent transitions in five nonlinear systems of ordinary and partial differential equations, which are models of autocatalytic chemical processes, the numbers of interacting populations and “hidden” attractor is carried out. It is numerically shown that in all considered systems of equations, the transition to chemical and biological turbulence is carried out in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory through subharmonic and homoclinic cascades of bifurcations of stable limit cycles. It is proved that irregular attractors in all cases are exclusively singular attractors in the sense of the FShM theory.

Let us make some general remarks about the causes and scenarios of the emergence of chaos in nonlinear systems of differential equations, as the publication of papers [7,10,11] and many others about “hyperchaotic”, “hidden”, “strange”, “stochastic”, “hyperbolic” and other attractors even in prestigious refereed journals, indicates to a complete lack of understanding of the mechanism of chaotic dynamics in nonlinear systems of differential equations. In this paper, author's papers [1,2,3,4] and others, on numerous examples, it is convincingly demonstrated that there exists universal FShM bifurcation scenario of transition to chaos in all systems of nonlinear differential equations without exception: autonomous and non-autonomous, 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 may not be included explicitly 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 homoclinic or heteroclinic separatrix contours of saddle-nodes or saddle-focuses, is not a criterion for the appearance of chaotic dynamics in the system. Also, neither the positivity of the numerical senior Lyapunov exponent, nor the proof of existence of Smale's horseshoe, nor the KAM theory, nor the theory of RT, 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 non-periodic 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 neighborhood 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 the system is determined not by horseshoes, but 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 FShM-cascades of bifurcations.