Том 32, № 6 (2024)
Editorial
Scientific legacy of L. P. Shilnikov: to the 90th anniversary
Аннотация
This year we celebrate the 90th anniversary of Leonid Pavlovich Shilnikov (1934-2011), an outstanding Russian mathematician and one of the founders of the mathematical theory of dynamic chaos and the theory of global bifurcations of multidimensional systems.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):713-721
713-721
Bifurcation in dynamical systems. Deterministic chaos. Quantum chaos
Mixed dynamics: elements of theory and examples
Аннотация
The main goal of the paper is to present recent results obtained in the mathematical theory of dynamical chaos and related to the discovery of its new, third, form, the so-called mixed dynamics. This type of chaos is very different from its two classical forms, conservative and dissipative chaos, and its main difference is that attractors and repellers can intersect without coinciding. The main results of the paper are related to construction of theoretical schemes aimed to mathematical justification of this phenomenon using the most general methods of topological dynamics. The paper also provides a number of examples of systems from applications in which mixed dynamics is observed. It is shown that such dynamics can be of different types, from close to conservative to strongly dissipative, and also that it can arise as a result of various bifurcation mechanisms.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):722-765
722-765
On non-conservative perturbations of three-dimensional integrable systems
Аннотация
At present, non-conservative perturbations of two-dimensional nonlinear Hamiltonian systems have been studied quite fully. The purpose of the study is to generalize this theory to the three-dimensional case, when the unperturbed system is nonlinear, integrable and has a region filled with closed phase trajectories. In this paper, autonomous perturbations are considered and the main attention is paid to the problem of limit cycles. Methods. The study is based on the construction of special coordinates in which the variables are divided into two slow and one fast, and in the first approximation with respect to a small parameter the equations for the slow variables are separated. Results. It is shown that hyperbolic equilibrium states of a truncated system determine closed phase trajectories, in the vicinity of which cycles appear under the perturbation. Conclusion. Thus, the problem is reduced to the study of solutions of the “generating” system of two algebraic or transcendental equations, similar to the generating Poincare–Pontryagin equation for two-dimensional systems. As examples, we considere a three-imensional van der Pol type system and the Lorentz system in the case of large Rayleigh numbers.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):766-780
766-780
Quasinormal forms for systems of two equations with large delay
Аннотация
A system of two equations with delay is considered. The purpose of this work is to study the local dynamics of this system under the assumption that the delay parameter is sufficiently large. Critical cases in the problem of stability of an equilibrium state are identified and it is shown that they have infinite dimension. Methods. The research is based on the use of special methods of infinite-dimensional normalization. Classical methods based on the application of the theory of invariant integral manifolds and normal forms turn out to be directly inapplicable. Results. As the main results, special nonlinear boundary value problems are constructed, which play the role of normal forms. Their nonlocal dynamics determine the behavior of all solutions of the original system in the vicinity of the equilibrium state.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):782-795
782-795
On limit sets of simplest skew products defined on multidimensional cells
Аннотация
The purpose of this work is to describe two important types of limit sets of the most simple skew products of interval maps, the phase space of each of which is a compact n-dimensional cell (n ≥ 2): firstly, a non-wandering set and, secondly, ω-limit sets of trajectories. Methods. A method for investigating of a nonwandering set (new even for the two-dimensional case) is proposed, based on the use of the concept of C0-Ω-blow up in continuous closed interval maps, and the concept of C0-Ω-blow up introduced in the work in the family of continuous fibers maps. To describe the ω-limit sets, the technique of special series constructed for the trajectory and containing an information about its asymptotic behavior is used. Results. A complete description is given of the nonwandering set of the continuous simplest skew product of the interval maps, that is, a continuous skew product on a compact n-dimensional cell, the set of (least) periods of periodic points of which is bounded. The results obtained in the description of a nonwandering set are used in the study of ω-limit sets. The paper describes a topological structure of ω-limit sets of the maps under consideration. Sufficient conditions have been found under which the ω-limit set of the trajectory is a periodic orbit, as well as the necessary conditions for the existence of one-dimensional ω-limit sets. Conclusion. Further development of the C0-Ω-blow up technique in the family of maps in fibers will allow us to describe the structure of a nonwandering set of skew products of one-dimensional maps, in particular, with a closed set of periodic points defined on the simplest manifolds of arbitrary finite dimension. Further development of the theory of special divergent series constructed in the work will allow us to proceed to the description of ω-limit sets of arbitrary dimension d, where 2 ≤ d ≤ n - 1, n ≥ 3, in the simplest skew products.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):796-815
796-815
On Lorenz-type attractors in a six-dimensional generalization of the Lorenz model
Аннотация
The topic of the paper — Lorenz-type attractors in multidimensional systems. We consider a six-dimensional model that describes convection in a layer of liquid, taking into account impurities in the atmosphere and liquid, as well as the rotation of the Earth. The main purpose of the work is to study bifurcations in the corresponding system and describe scenarios for the emergence of chaotic attractors of various types. Results. It is shown that in the system under consideration, both a classical Lorenz attractor (the theory of which was developed in the works of Afraimovich–Bykov–Shilnikov) and an attractor of a new type, visually similar to the Lorenz attractor, but containing a symmetric pair of equilibrium states, can arise. It has been established that the Lorenz attractor in this system is born as a result of the classical scenario proposed by L. P. Shilnikov. We propose a new scenario for the emergence of an attractor of the second type via bifurcations inside the Lorenz attractor. In the paper we also discuss homoclinic and heteroclinic bifurcations that inevitably arise inside the found attractors, as well as their possible pseudohyperbolicity.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):816-831
816-831
On discrete Lorenz attractors of various types
Аннотация
The purpose of this work is to develop the theory of discrete attractors of Lorenz type in the case of three-dimensional maps. In this case, special attention will be paid to standard discrete Lorenz attractors, as well as discrete Lorenz attractors with axial symmetry (i.e. with symmetry x → -x, y → -y, z → -z characteristic of flows with the Lorenz attractors). The main results of the work are related to the construction of elements of classification of such attractors. For various types of discrete Lorenz attractors, we will describe their basic geometric and dynamical properties, and also present the main phenomenological bifurcation scenarios in which they arise. In the work we also consider specific examples of discrete Lorenz attractors of various types in three-dimensional quadratic maps such as three-dimensional Henon maps and quadratic maps with axial symmetry and constant Jacobian. For the latter, their normal forms will be constructed — universal maps, to which any map from a given class can be reduced by means of linear coordinate transformations.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):832-857
832-857
Trajectory attractors method for dissipative partial differential equations with small parameter
Аннотация
The purpose of this work is to study the limit behaviour of trajectory attractors for some equations and systems from mathematical physics depending on a small parameter when this small parameter approaches zero. The main attention is given to the cases when, for the limit equation, the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. The following problems are considered: approximation of the 3D Navier–Stokes system using the Leray α-model, homogenization of the complex Ginzburg–Landau equation in a domain with dense perforation, and zero viscosity limit of 2D Navier–Stokes system with Ekman friction. Methods. In this paper, the method of trajectory dynamical systems and trajectory attractors is used that is especially effective in the study of complicated partial differential equations for which the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. Results. For all problems under the consideration, we obtain the limit equations and prove the Hausdorff convergence for trajectory attractors of the initial equations to the trajectory attractors of the limit equations in the appropriate topology when the small parameter tends to zero. Conclusion. In the work, we demonstrate that the method of trajectory attractors is highly effective in the study of dissipative equations of mathematical physics with small parameter. We succeed to find the limit equations and to prove the convergence of trajectory attractors of the considered equations to the trajectory attractors of the limit (homogenized) equations in the corresponding topology as small parameter is vanishes.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):858-877
858-877
Spatial dynamics in the family of sixth-order differential equations from the theory of partial formation
Аннотация
Topic of the paper. Bounded stationary (i.e. independent in time) spatially one-dimensional solutions of a quasilinear parabolic PDE are studied on the whole real line. Its stationary solutions are described by a nonlinear ODE of the sixth order of the Euler–Lagrange–Poisson type and therefore can be transformed to the Hamiltonian system with three degrees of freedom being in addition reversible with respect two linear involutions. The system has three symmetric equilibria, two of them are hyperbolic in some region of the parameter plane. Goal of the paper. In this paper we, combining methods of dynamical systems theory and numerical simulations, investigate the orbit behavior near the symmetric heteroclinic connection based on these equilibria. It was found both simple (periodic) and complicated orbit behavior. To this end we use the theorem on a global center manifold near the heteroclinic connection. For the third symmetric equilibrium at the origin we found the region in the parameter plane where this equilibrium is of the saddle-focus-center type and found the existence of its homoclinic orbits, long-periodic orbits near homoclinic orbits and orbits with complicated structure.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):878-896
878-896
Groups of basic automorphisms of chaotic Cartan foliations with Eresmann connection
Аннотация
The purpose of the work is to study the groups of basic automorphisms of chaotic Cartan foliations with Ehresmann connection. Cartan foliations form a category where automorphisms preserve not only the foliation, but also its transverse Cartan geometry. The group of basic automorphisms of a foliation is the quotient group of the group of all automorphisms of this foliation by the normal subgroup of leaf automorphisms with respect to which each leaf is invariant. Cartan foliations include such wide classes of foliations as pseudo-Riemannian, Lorentzian, and foliations with transversal affine connection. No restrictions are imposed on the dimension of either the foliation or the foliated manifold. Compactness of the foliated manifold is not assumed. Methods. The proof of the structure theorem for chaotic Cartan foliations is based on the application of the foliated bundle construction, commonly used in the theory of foliations with transverse geometries. Results. The main result of this paper is the theorem stating that the group of basic automorphisms of any chaotic Cartan foliation with Ehresmann connection admits the structure of a Lie group and finding estimates for the dimension of this group. In particular, it is proved that if the set of closed leaves is countable, then the group of basic automorphisms of such a foliation is countable. Conclusion. In this paper, we prove a criterion according to which the chaoticity of a Cartan foliation of type (G, H) is equivalent to the chaoticity of a locally free action of the group H on the associated parallelizable manifold. Thus, the problem of the existence of chaos in Cartan foliations with Ehresmann connection reduces to the same problem for locally free actions of a Lie group on parallelizable manifolds.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):897-907
897-907
Modeling of global processes. Nonlinear dynamics and humanities
An asymptotic solution for the SIS epidemic model, taking into account migration and diffusion
Аннотация
The purpose of this work is to propose and investigate a simple and effective model of an epidemic in an animal population that takes into account migration along the plane of both diseased and healthy individuals. Within the framework of this model, the spatial migration of a population is described by introducing both diffusion and advective terms into its equations. Methods. In this paper, a method of many scales was used to find an asymptotic solution to the system of equations of the epidemic. Solutions of auxiliary linear equations of the parabolic type arising during this procedure were found using the Poisson integral. The simplification of the initial system of equations of the model is based on the assumption that the sum of densities of healthy and sick individuals on a single-connected region of large diameter on the plane is constant at the initial moment of time. Results. It is shown that in this case, designed for a slowly changing initial density of sick individuals concentrated inside this area at a considerable distance from its boundaries, the asymptotic solution of the model describes the effect of merging several spatially spaced small outbreaks of the disease into one large outbreak during migration of the entire population as a whole. In particular, for such an initial density obtained by the functional transformation of a Gaussian, a circular plateau is formed over long periods with an effective radius that grows linearly over time. Conclusion. The constructed asymptotic solution of the epidemic model proposed in this paper is simple in form and describes the transfer of the disease on a locally flat area of the earth’s surface without the use of numerical methods. This solution is convenient when describing the migration of a sick population under the influence of flooding, forest fire, man-made disaster with contamination of the area, etc.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2024;32(6):908-920
908-920