Shadowing in a neighborhood of a separatrix

We consider a model example of a diffeomorphism of a two-dimensional manifold with two hyperbolic fixed points joined by a separatrix. We show that, in a tubular neighborhood of the separatrix, every pseudotrajectory satisfying an additional error smallness condition is shadowed by an exact trajectory.     

A route to chaos in a conservative system of three interacting Langmuir waves

This paper presents the results of numerical calculations of a route to chaos in a conservative Hamiltonian system of three Langmuir waves interacting with each other through three-wave couplings. The route is investigated by studying time series, power spectra, phase space portraits and Lyapnov exponents of wave variables for several combinations of wave vectors. The results show that the system follows a route which is very similar to the Ruelle–Takens–Newhouse scenario observed in dissipative systems, and widths and shifts of peaks in power spectra appeared due to the three moderate strength wave interactions. The breaks of tori in the system are also numerically investigated by studying the dependency of Maximum Lyapnov exponents for wave-variables on a parameter which represents the nonlinearity of the system.     

Study of a discrete dynamic system in a neighborhood of a quasi-periodic trajectory

We study a discrete system in a neighborhood of a quasi-periodic trajectory. We obtain conditions for reducing a system in this neighborhood to a system with quasi-periodic coefficients. We determine the behavior of this system under the action of small perturbations.This work was prepared with the financial support of the Ukrainian State Committee on Science and Technology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1702–1711, December, 1992.     

On the stability of a linear nongyroscopic conservative system

Zusammenfassung Im Anschluss an die Erklärung der Begriffe des Pseudofreiheitsgrades und des quasidynamischen Systems wurden durch Anwendung des Hamiltonschen Prinzips die Bewegungsdifferentialgleichungen solcher Systeme hergeleitet. An einem Beispiel wird gezeigt, dass für die Stabilitätsuntersuchung bei Systemen von diesem Typus nur die Schwingungsmethode zulässig ist, auch wenn es sich um einen konservativen nichtgyroskopischen Fall handelt.     

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic,chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.     

Regularity of a dynamic neighborhood of a regular language

Birational maps give the main research tool for the theory of Fano varieties, as we know from the fundamental works of V.A. Iskovskikh and his school. Nowadays one can exploit them in the new approach of D. Auroux to Mirror Symmetry of Fano varieties, which is based on a certain generalization of the notion of special Lagrangian fibration suitable for Fano varieties. We present a very simple example of how a special Lagrangian fibration can be transferred by a birational map. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 209–211.     

Oscillations near a separatrix in the duffing equation

A small periodic perturbation results in a complicated dynamics near separatrices and saddle points. A two-parameter family of asymptotic solutions staying near separatrices for a long time is constructed. Solutions from this family depend nonsmoothly on the perturbation parameter. An example is given in which the values of the perturbation parameter for this family of solutions are determined by a set with structure of the type of the Cantor set.     

Nature of chaos in conservative and dissipative systems of the Duffing-Holmes oscillator

We show that the chaotic dynamics of the conservative Duffing-Holmes oscillator obeys the universal Feigenbaum-Sharkovskii-Magnitskii theory of passage to chaos in dynamical systems of ordinary differential equations. Moreover, the cascades of bifurcations of the conservative and dissipative oscillators are continuously related to each other. Our study uses the stable control method, which permits rapidly stabilizing nearly any periodic solution and dynamically changing system parameters without moving far away from that periodic solution.     

On separatrix curves of a family of linear systems with pulse influence

For a family of linear systems of differential equations with pulse influence, we establish conditions of existence of separatrix curves and present a method for finding these curves.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No, 12, pp. 1682–1687, December, 1993.This work was supported by the Ukrainian State Committee on Science and Technology.     

Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos

This study aims to analyze the dynamic behavior of bevel-geared rotor system supported on a thrust bearing and journal bearings under nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

On the existence of piecewise-continuous separatrix curves for a certain class of pulse systems

Sufficient conditions for the existence of piecewise-continuous separatrix curves are obtained for a certain class of systems of differential equations with pulse influence for small 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 580–585, May, 1994.     

On the dynamic reconstruction of the input of a control system

We consider the problem of reconstructing the right-hand side of a dynamical system subjected to external disturbances. Under the assumption that the states of the system are measured with some error, we indicate an algorithm for solving the problem on the basis of the method of auxiliary positional control models. The algorithm is stable under noises and numerical errors.     

On a generalization of absolute neighborhood retracts

In this paper we generalize the concept of absolute neighborhood retract by introducing the notion of absolute neighborhood multi-retract. Furthermore, the Lefschetz fixed point theorem for admissible maps defined on absolute neighborhood multi-retracts is proved.     

Investigation of a discrete dynamical system in a neighborhood of the invariant torus

We study the behavior of a discrete dynamical system in a neighborhood of the invariant torus for the case where the trajectories may have arbitrary structure on the torus and establish conditions under which the system can be reduced to the canonical form in the indicated neighborhood.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1676–1685, December, 1995.     

On the genericity of chaos

Let M be a compact manifold with dimM?2. We prove that some iteration of the generic homeomorphism on M is semiconjugated to the shift map and has infinite topological entropy (Theorem 1.1).