The necessity of the Hopf bifurcation for periodically forced oscillators (original) (raw)
Resonance behaviour in two-parameter families of periodically forced oscillators
Physics Letters, 1988
We study resonance (periodic) behaviour in two-dimensional autonomous oscillators, periodically forced by discrete jumps in state space. Two different models are examined numerically using continuation methods. The results are qualitatively similar in both cases and show that regions ofvarious resonances arebounded in the forcing-amplitude-forcing-period parameter plane and have a rich internal bifurcation structure.
Recognition of resonance type in periodically forced oscillators
Physica D: Nonlinear Phenomena, 2010
This paper deals with families of periodically forced oscillators undergoing a Hopf-Neȋmarck-Sacker bifurcation. The interest is in the corresponding resonance sets, regions in parameter space for which subharmonics occur. It is a classical result that the local geometry of these sets in the non-degenerate case is given by an Arnol'd resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our main contribution is providing corresponding recognition conditions, that determine to which of these cases a given family of periodically forced oscillators corresponds. The conditions are constructed from known results for families of diffeomorphisms, which in the current context are given by Poincaré maps. Our approach also provides a skeleton for the local resonant Hopf-Neȋmarck-Sacker dynamics in the form of planar Poincaré-Takens vector fields. To illustrate our methods two case studies are included: A periodically forced generalized Duffing-Van der Pol oscillator and a parametrically forced generalized Volterra-Lotka system.
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
Nonlinearity, 2003
We perform a bifurcation analysis of normal-internal resonances in parametrised families of quasi-periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the 'backbone' system; forced, the system is a skew-product flow with a quasi-periodic driving with Ò basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The averaged system turns out to have the same structure as in the well-known case of periodic forcing (Ò ½); for a real analytic system, the non-integrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasi-periodic Ò-dimensional tori in the averaged system, filling normal-internal resonance 'gaps' that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of 'gaps within gaps' makes the quasi-periodic case more complicated than the periodic case.
Parametric Resonance of Hopf Bifurcation
Nonlinear Dynamics, 2005
We investigate the dynamics of a system consisting of a simple harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasi-periodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.
Bistability, period doubling bifurcations and chaos in a periodically forced oscillator
Physics Letters A, 1982
A two parameter mathematical model for a periodically forced nonlinear oscillator is analyzed using analytical and numerical techniques. The model displays phase locking, quasiperiodic dynamics, bistabifity, period-doubling bifurcations and chaotic dynamics. The regions in which the different dynamical behaviors occur as a function of the two parameters is considered.
Global parametrization and computation of resonance surfaces for periodically forced oscillators
Periodically forced planar oscillators are typically studied by varying the two parameters of forcing amplitude and forcing frequency. Such differential equations can be reduced via stroboscopic sampling to a two-parameter family of diffeomorphisms of the plane. A bifurcation analysis of this family almost always includes a study of the birth and death of periodic orbits. For low forcing amplitudes, this leads to a now-classic picture of Arnold resonance tongues. Studying these resonance tongues for higher forcing amplitudes requires numerical continuation. Previous work has revealed the usefulness of considering these tongues as projections of surfaces of periodic points from the cartesian product of the phase and parameter planes to the parameter plane. Many surfaces were displayed and described in [MP 1994], but their parametrization and computation was not discussed. In this paper, we do discuss their parametrization and computation. Especially useful are global parametrizations...
Small periodic perturbations of autonomous self-oscillating planar systems
IFAC Proceedings Volumes
In this paper a nonlinear planar autonomous system having a limit cycle of period T which is perturbed by a small parameter nonautonomous T-periodic nonlinear term is considered. By using the topological index of the equilibrium points of the unperturbed system and a geometric condition on the perturbation, similar to the condition of the guiding function method as introduced by M. A. Krasnoselskii-A. I. Perov, the existence of T-periodic solutions of the perturbed system close as we like to the limit cycle of the autonomous system is proved.
2010
Bifurcation theory is the mathematical investigation of changes in the qualitative or topological structure of a studied family. In this paper, we numerically investigate the qualitative behavior of nonlinear RLC circuit excited by sinusoidal voltage source based on the bifurcation analysis. Poincare mapping and bifurcation methods are applied to study both dynamics and qualitative properties of the periodic responses of such oscillator. As numerically illustrated here, a small variation of amplitude or frequency of the driver sinusoidal voltage may involve qualitative changes for witch the system exhibits fold, period doubling and pitchfork bifurcations. In fact, the presence of these kinds of bifurcation necessitates an examination of the role of these singularities in the dynamical behavior of circuit. Particularly, we numerically study the qualitative changes may affect number and stability of the periodic solutions and the shapes of its basins of attraction associated while app...