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# 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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