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Volume 12, Number 5

Volume 12, Number 5, 2007
On the 150th anniversary of A.M.Lyapunov

Moeckel R.
Symbolic dynamics in the planar three-body problem
Abstract
A chaotic invariant set is constructed for the planar three-body problem. The orbits in the invariant set exhibit many close approaches to triple collision and also excursions near infinity. The existence proof is based on finding appropriate "windows" in the phase space which are stretched across one another by flow-defined Poincaré maps.
Keywords: Celestial Mechanics, three-body problem
Citation: Moeckel R., Symbolic dynamics in the planar three-body problem, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 449-475
DOI:0.1134/S1560354707050012
Pesin Y.
Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents
Abstract
This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.
Keywords: Lyapunov exponents, hyperbolicity, genericity, ergodicity
Citation: Pesin Y., Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 476-489
DOI:10.1134/S1560354707050024
Sinai Y. G.
On a separating solution of a recurrent equation
Abstract
We consider the recurrent equation
$\Lambda_p = \frac{1}{p-1} \sum\limits_{p_1=1}^{p-1} f (\frac{p1}{p}) \Lambda_{p_1} · \Lambda_{p-p_1}$
which depends on the initial condition $\Lambda_1=x$. Under some conditions on $f$ we show that there exists the value of x for which $\Lambda_p$ tends to a constant as p tends to infinity.
Keywords: separating solution, recurrent equation
Citation: Sinai Y. G., On a separating solution of a recurrent equation, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 490-501
DOI:10.1134/S1560354707050036
Seipel J.,  Holmes P. J.
A simple model for clock-actuated legged locomotion
Abstract
The spring-loaded inverted pendulum (SLIP) model describes well the steady-state center-of-mass motions of a diverse range of walking and running animals and robots. Here we ask whether the SLIP model can also explain the dynamic stability of these gaits, and we find that it cannot do so in many physically-relevant parameter ranges. We develop an actuated, lossy, clock-torqued SLIP, or CT-SLIP, with more realistic hip-motor torque inputs, that can capture the robust stability properties observed in most animals and some legged robots. Variations of CT-SLIP at a similar level of detail and complexity may also be appropriate for capturing the whole-system center-of-mass dynamics of locomotion of legged animals and robots varying widely in size and morphology. This paper contributes to a broader program to develop mathematical models, at varied levels of detail, that capture the dynamics of integrated organismal systems exhibiting integrated whole-body motion.
Keywords: robotics, animal locomotion, biomechanics, stability
Citation: Seipel J.,  Holmes P. J., A simple model for clock-actuated legged locomotion, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 502-520
DOI:10.1134/S1560354707050048
Butta P.,  Negrini P.
Resonances and $O$-curves in Hamiltonian systems
Abstract
We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.
Keywords: Hamiltonian systems, resonances
Citation: Butta P.,  Negrini P., Resonances and $O$-curves in Hamiltonian systems, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 521-530
DOI:10.1134/S156035470705005X
Borisov A. V.,  Kozlov V. V.,  Mamaev I. S.
Asymptotic stability and associated problems of dynamics of falling rigid body
Abstract
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
Keywords: rigid body, ideal fluid, non-holonomic mechanics
Citation: Borisov A. V.,  Kozlov V. V.,  Mamaev I. S., Asymptotic stability and associated problems of dynamics of falling rigid body, Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 531-565
DOI:10.1134/S1560354707050061
Steklov V. A.
Aleksandr Mikhailovich Lyapunov (1857–1919)
Abstract
The speech presented by Academician V.A. Steklov of the Russian Academy of Sciences to the public session of the Academy on May 3rd 1919.
Keywords: A.M. Lyapunov, creative work
Citation: Steklov V. A., Aleksandr Mikhailovich Lyapunov (1857–1919), Regular and Chaotic Dynamics, 2007, vol. 12, no. 5, pp. 566-576
DOI:10.1134/S1560354707050073

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