2,798 research outputs found
On Nonlinear Dynamics of the Pendulum with Periodically Varying Length
Dynamic behavior of a weightless rod with a point mass sliding along the rod
axis according to periodic law is studied. This is the pendulum with
periodically varying length which is also treated as a simple model of child's
swing. Asymptotic expressions for boundaries of instability domains near
resonance frequencies are derived. Domains for oscillation, rotation, and
oscillation-rotation motions in parameter space are found analytically and
compared with numerical study. Two types of transitions to chaos of the
pendulum depending on problem parameters are investigated numerically.Comment: 8 pages, 8 figure
The dynamics of the pendulum suspended on the forced Duffing oscillator
We investigate the dynamics of the pendulum suspended on the forced Duffing
oscillator. The detailed bifurcation analysis in two parameter space (amplitude
and frequency of excitation) which presents both oscillating and rotating
periodic solutions of the pendulum has been performed. We identify the areas
with low number of coexisting attractors in the parameter space as the
coexistence of different attractors has a significant impact on the practical
usage of the proposed system as a tuned mass absorber.Comment: Accepte
Mode competition in a system of two parametrically driven pendulums: the role of symmetry
This paper is the final part in a series of four on the dynamics of two coupled, parametrically driven pendulums. In the previous three parts (Banning and van der Weele, Mode competition in a system of two parametrically driven pendulums; the Hamiltonian case, Physica A 220 (1995) 485¿533; Banning et al., Mode competition in a system of two parametrically driven pendulums; the dissipative case, Physica A 245 (1997) 11¿48; Banning et al., Mode competition in a system of two parametrically driven pendulums with nonlinear coupling, Physica A 245 (1997) 49¿98) we have given a detailed survey of the different oscillations in the system, with particular emphasis on mode interaction. In the present paper we use group theory to highlight the role of symmetry. It is shown how certain symmetries can obstruct period doubling and Hopf bifurcations; the associated routes to chaos cannot proceed until these symmetries have been broken. The symmetry approach also reveals the general mechanism of mode interaction and enables a useful comparison with other systems
Dynamical Systems, Stability, and Chaos
In this expository and resources chapter we review selected aspects of the
mathematics of dynamical systems, stability, and chaos, within a historical
framework that draws together two threads of its early development: celestial
mechanics and control theory, and focussing on qualitative theory. From this
perspective we show how concepts of stability enable us to classify dynamical
equations and their solutions and connect the key issues of nonlinearity,
bifurcation, control, and uncertainty that are common to time-dependent
problems in natural and engineered systems. We discuss stability and
bifurcations in three simple model problems, and conclude with a survey of
recent extensions of stability theory to complex networks.Comment: 28 pages, 10 figures. 26/04/2007: The book title was changed at the
last minute. No other changes have been made. Chapter 1 in: J.P. Denier and
J.S. Frederiksen (editors), Frontiers in Turbulence and Coherent Structures.
World Scientific Singapore 2007 (in press
Melnikov's method in String Theory
Melnikov's method is an analytical way to show the existence of classical
chaos generated by a Smale horseshoe. It is a powerful technique, though its
applicability is somewhat limited. In this paper, we present a solution of type
IIB supergravity to which Melnikov's method is applicable. This is a brane-wave
type deformation of the AdSS background. By employing two
reduction ans\"atze, we study two types of coupled pendulum-oscillator systems.
Then the Melnikov function is computed for each of the systems by following the
standard way of Holmes and Marsden and the existence of chaos is shown
analytically.Comment: 37 pages, 5 figure
The dynamics of co- and counter rotating coupled spherical pendulums
The dynamics of co- and counter-rotating coupled spherical pendulums (two
lower pendulums are mounted at the end of the upper pendulum) is considered.
Linear mode analysis shows the existence of three rotating modes. Starting from
linear modes allow we calculate the nonlinear normal modes, which are and
present them in frequency-energy plots. With the increase of energy in one mode
we observe a symmetry breaking pitchfork bifurcation. In the second part of the
paper we consider energy transfer between pendulums having different energies.
The results for co-rotating (all pendulums rotate in the same direction) and
counter-rotating motion (one of lower pendulums rotates in the opposite
direction) are presented. In general, the energy fluctuations in
counter-rotating pendulums are found to be higher than in the co-rotating case.Comment: The European Physical Journal Special Topics 201
Edge of chaos as critical local symmetry breaking in dissipative nonautonomous systems
The fully nonlinear notion of resonance -- in
the general context of dissipative systems subjected to
potentials is discussed. It is demonstrated that there is an exact local
invariant associated with each geometrical resonance solution which reduces to
the system's energy when the potential is steady. The geometrical resonance
solutions represent a \textit{local symmetry} whose critical breaking leads to
a new analytical criterion for the order-chaos threshold. This physical
criterion is deduced in the co-moving frame from the local energy conservation
over the shortest significant timescale. Remarkably, the new criterion for the
onset of chaos is shown to be valid over large regions of parameter space, thus
being useful beyond the perturbative regime and the scope of current
mathematical techniques
- …