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2013
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Vol. 10, No. 4

Vol. 10, No. 4, 2014

Kuznetsov A. P.,  Migunova N. A.,  Sataev I. R.,  Sedova Y. V.,  Turukina L. V.
Abstract
Ensembles of several chaotic R¨ossler oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of invariant tori of different and sufficiently high dimension. The possibility of a quasi-periodic Hopf bifurcation and of the cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonant tori are revealed whose boundaries correspond to a saddle-node bifurcation. Within areas of resonant modes the torus-doubling bifurcations and tori destruction are observed.
Keywords: chaos, quasiperiodic oscillations, invariant tori, Lyapunov exponents, bifurcations
Citation: Kuznetsov A. P.,  Migunova N. A.,  Sataev I. R.,  Sedova Y. V.,  Turukina L. V., Dynamics of coupled chaotic oscillators: from chaos to quasiperiodicity, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 387-405
DOI:10.20537/nd1404001
Kulakov M. P.,  Neverova G. P.,  Frisman E. Y.
Abstract
This article researches model of two coupled an age structured populations. The model consists of two identical two-dimensional maps demonstrating the Neimark – Sacker and period-doubling bifurcations. The “bistability” of dynamic modes is found which is expressed in a co-existence the nontrivial fixed point and periodic points (stable 3-cycle). The mechanism of loss stability and formation of complex hierarchy for multistable states are investigated.
Keywords: metapopulation, multistability, maps, synchronization, basin of attraction
Citation: Kulakov M. P.,  Neverova G. P.,  Frisman E. Y., Multistability in dynamic models of migration coupled populations with an age structure, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 407-425
DOI:10.20537/nd1404002
Grines V. Z.,  Gurevich E. Y.,  Zhuzhoma E. V.,  Zinina S. K.
Abstract
We obtain properties of three-dimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in non-wandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma.
Keywords: Morse – Smale cascades, heteroclinic curves, mapping torus, locally trivial bundle, separators of magnetic field
Citation: Grines V. Z.,  Gurevich E. Y.,  Zhuzhoma E. V.,  Zinina S. K., Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 427-438
DOI:10.20537/nd1404003
Kozlov V. V.
Abstract
This paper is concerned with the problem of first integrals of the equations of geodesics on twodimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.
Keywords: conformal coordinates, rational integral, irreducible integrals, Cauchy–Kovalevskaya theorem
Citation: Kozlov V. V., On Rational Integrals of Geodesic Flows, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 439-445
DOI:10.20537/nd1404004
Markeev A. P.
Abstract
We consider the canonical differential equations describing the motion of a system with one degree of freedom. The origin of the phase space is assumed to be an equilibrium position of the system. It is supposed that in a sufficiently small neighborhood of the equilibrium Hamiltonian function can be represented by a convergent series. This series does not include terms of the second degree, and the terms of the third and fourth degrees are independent of time. Linear real canonical transformations leading the terms of the third and fourth degrees to the simplest forms are found. Classification of the systems in question being obtained on the basis of these forms is used in the discussion of the stability of the equilibrium position.
Keywords: Hamiltonian system, canonical transformation, stability
Citation: Markeev A. P., Simplifying the structure of the third and fourth degree forms in the expansion of the Hamiltonian with a linear transformation, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 447-464
DOI:10.20537/nd1404005
Polekhin I. Y.
Abstract
Two examples concerning application of topology in study of dynamics of inverted plain mathematical pendulum with pivot point moving along horizontal straight line are considered. The first example is an application of the Wazewski principle to the problem of existence of solution without falling. The second example is a proof of existence of periodic solution in the same system when law of motion is periodic as well. Moreover, in the second case it is also shown that along obtained periodic solution pendulum never becomes horizontal (falls).
Keywords: inverted pendulum, Lefschetz-Hopf theorem, Wazewski principle, periodic solution
Citation: Polekhin I. Y., Examples of topological approach to the problem of inverted pendulum with moving pivot point, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 465-472
DOI:10.20537/nd1404006
Kilin A. A.,  Bobykin A. D.
Abstract
The problem of motion of a vehicle in the form of a platform with an arbitrary number of Mecanum wheels fastened on it is considered. The controllability of this vehicle is discussed within the framework of the nonholonomic rolling model. An explicit algorithm is presented for calculating the control torques of the motors required to follow an arbitrary trajectory. Examples of controls for executing the simplest maneuvers are given.
Keywords: omniwheel, roller bearing wheel, nonholonomic constraint, dynamical system, integrability, controllability
Citation: Kilin A. A.,  Bobykin A. D., Control of a Vehicle with Omniwheels on a Plane, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 473-481
DOI:10.20537/nd1404007
Borisov A. V.,  Erdakova N. N.,  Ivanova T. B.,  Mamaev I. S.
Abstract
In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system’s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.
Keywords: dry friction, linear pressure distribution, two-dimensional motion, planar motion, Coulomb law
Citation: Borisov A. V.,  Erdakova N. N.,  Ivanova T. B.,  Mamaev I. S., On the dynamics of a body with an axisymmetric base sliding on a rough plane, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 483-495
DOI:10.20537/nd1404008
Kilin A. A.,  Karavaev Y. L.
Abstract
The kinematic control model for a spherical robot with an internal omniwheel platform is presented. We consider singularities of control of spherical robot with an unbalanced internal omniwheel platform. The general algorithm of control of spherical robot according to the kinematical quasi-static model and controls for simple trajectories (a straight line and in a circle) are presented. Experimental investigations have been carried out for all introduced control algorithms.
Keywords: spherical robot, kinematic model, nonholonomic constraint, omniwheel, displacement of center of mass
Citation: Kilin A. A.,  Karavaev Y. L., The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 497-511
DOI:10.20537/nd1404009
Rust  . C.,  Asada H. H.
Abstract
A Remotely Operated Vehicle (ROV) is developed for use in the inspection of underwater structures in hazardous environments. The vehicle presented can change orientation like an eyeball using a novel gimbal mechanism for moving an internal eccentric mass. Combined with a pair of thrusters, the Eyeball ROV can move in any direction with non-holonomic constraints. In this paper the design concept is presented first, followed by dynamic and hydrodynamic analysis. Due to poor open loop stability characteristics, stability a ugmentation is implemented using onboard sensors and was de signed and tested in simulation. A physical proof-of-concept prototype is also presented.
Citation: Rust  . C.,  Asada H. H., The eyeball ROV: Design and control of a spherical underwater vehicle steered by an internal eccentric mass, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 513-531
DOI:10.20537/nd1404010
Abstract
Citation: New books, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp. 533-535

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