
Scientific paper ID 1682 : 2018/3
BIFURCATION BEHAVIOR OF A HAMILTONIAN SYSTEM WITH TWO DEGREES OF FREEDOM
Svetoslav Nikolov, Daniela Zaharieva The investigation of the qualitative changes in the dynamics of a system is object of bifurcation theory. In this paper we analyze the bifurcation behavior of a Hamiltonian system with two degrees of freedom describing the rider and the swing (pumped from the seated position) as a compound pendulum. Our analytical calculations predict that a Hamiltonian Hopf bifurcation (1:1 resonance) takes place.
бифуркационно поведение хамилтонова система две степени на свободаbifurcation behavior Hamiltonian system two degrees of freedomSvetoslav Nikolov Daniela Zaharieva BIBLIOGRAPHY [1] Lyapunov, A., The general problem of the stability of motion, Int. J. of Control, vol. 55, 3, pp. 531773, 1992. [2] Abraham, R., Marsden, J., Foundations of mechanics, AMS Chelsea Publishing, vol. 364H, 2008. [3] Hansmann, H., Local and semilocal bifurcations in Hamiltonian dynamical systems results and examples, LNM 1893, Springer, 2007. [4] Case W., M. Swanson, The pumping of a swing from the seated position, American J. of Physics, 58, pp. 463467, 1990. [5] Nikolov, S., Zaharieva, D., Dynamics of swing oscillatory motion in Hamiltonian formalism, Mechanics, Transport, Communications (ISSN: 13123823), vol. 15, No 3, pp. VII7VII12, art. ID 1495, 2017. [6] Bridges, T., Bifurcation of periodic solutions near a collision of a eigenvalues of opposite signature, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 108, No 3, pp. 575601, 1990. [7] Lahiri, A., Roy, M., The Hamiltonian Hopf bifurcation: an elementary perturbative approach, Int. J. of Nonlinear Mechanics, vol. 36, pp. 787802, 2001. [8] Krein, M., A generalization of several investigations of A.M. Lyapunov on linear differential equations with periodic coefficients, Dokl. Acad. Nauk SSSR, vol. 73, pp. 445448, 1950. 