408 research outputs found
Normal form transforms separate slow and fast modes in stochastic dynamical systems
Modelling stochastic systems has many important applications. Normal form
coordinate transforms are a powerful way to untangle interesting long term
macroscale dynamics from detailed microscale dynamics. We explore such
coordinate transforms of stochastic differential systems when the dynamics has
both slow modes and quickly decaying modes. The thrust is to derive normal
forms useful for macroscopic modelling of complex stochastic microscopic
systems. Thus we not only must reduce the dimensionality of the dynamics, but
also endeavour to separate all slow processes from all fast time processes,
both deterministic and stochastic. Quadratic stochastic effects in the fast
modes contribute to the drift of the important slow modes. The results will
help us accurately model, interpret and simulate multiscale stochastic systems
Nonlinear structural vibrations by the linear acceleration method
Numerical integration method for calculating dynamic response of nonlinear elastic structure
Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators
In this set of lectures, we review briefly some of the recent developments in
the study of the chaotic dynamics of nonlinear oscillators, particularly of
damped and driven type. By taking a representative set of examples such as the
Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain
the various bifurcations and chaos phenomena associated with these systems. We
use numerical and analytical as well as analogue simulation methods to study
these systems. Then we point out how controlling of chaotic motions can be
effected by algorithmic procedures requiring minimal perturbations. Finally we
briefly discuss how synchronization of identically evolving chaotic systems can
be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in
Physics Please Lakshmanan for figures (e-mail: [email protected]
On the interaction of exponential non-viscous damping with symmetric nonlinearities
This paper studies the interaction between non-viscous damping and nonlinearities for nonlinear oscillators with reflection symmetry. The non-viscous damping function is an exponential damping model which adds a decaying memory property to the damping term of the oscillator. We consider the case of softening and hardening behaviour in the frequency response of the system. Numerical simulations of the Duffing oscillator show a significant enhancement of the resonance peaks for increasing levels of non-viscous damping parameter in the hardening case, but not in the softening case. This can be explained in the general context by an energy balance analysis of the undamped unforced oscillator, which shows that for hardening nonlinearities the growth of damping with the energy level is an order of magnitude smaller in the exponential case than in the viscous case
Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
In this paper we develop new techniques for revealing geometrical structures
in phase space that are valid for aperiodically time dependent dynamical
systems, which we refer to as Lagrangian descriptors. These quantities are
based on the integration, for a finite time, along trajectories of an intrinsic
bounded, positive geometrical and/or physical property of the trajectory
itself. We discuss a general methodology for constructing Lagrangian
descriptors, and we discuss a "heuristic argument" that explains why this
method is successful for revealing geometrical structures in the phase space of
a dynamical system. We support this argument by explicit calculations on a
benchmark problem having a hyperbolic fixed point with stable and unstable
manifolds that are known analytically. Several other benchmark examples are
considered that allow us the assess the performance of Lagrangian descriptors
in revealing invariant tori and regions of shear. Throughout the paper
"side-by-side" comparisons of the performance of Lagrangian descriptors with
both finite time Lyapunov exponents (FTLEs) and finite time averages of certain
components of the vector field ("time averages") are carried out and discussed.
In all cases Lagrangian descriptors are shown to be both more accurate and
computationally efficient than these methods. We also perform computations for
an explicitly three dimensional, aperiodically time-dependent vector field and
an aperiodically time dependent vector field defined as a data set. Comparisons
with FTLEs and time averages for these examples are also carried out, with
similar conclusions as for the benchmark examples.Comment: 52 pages, 25 figure
An algebraic criterion for the onset of chaos in nonlinear dynamic systems
The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one
ANALYSIS OF THE NONLINEAR VIBRATIONS OF ELECTROSTATICALLY ACTUATED MICRO-CANTILEVERS IN HARMONIC DETECTION OF RESONANCE (HDR)
Micro- and nano-cantilevers have the potential to revolutionize physical, chemical, and biological sensing; however, an accurate and scalable detection method is required. In this work, a fully electrical actuation and detection method is presented, known as the Harmonic Detection of Resonance (HDR). In HDR, harmonic components of the current are measured to determine the cantilever\u27s resonance frequency. These harmonics exist as a result of nonlinearities in the system, principally in the electrostatic actuation force. In order to better understand this rich harmonic structure, a theoretical investigation of the micro-cantilever is undertaken. Both a lumped parameter model and a more accurate continuum model are used to derive the governing nonlinear modal equations of motion (EOM) of the cantilever. Various approximate solution methods applicable to nonlinear equations are then discussed including numerical integration, perturbation, and averaging. An averaging method known as the method of harmonic balance is then used to obtain steady state solutions to the micro-cantilever EOM. Low-order closed-form harmonic balance solutions are derived which explain many of the important features of the HDR results, such as the presence of parasitic capacitance in the first harmonic and super-harmonic resonance peaks in higher harmonics. Finally, higher-order computer generated harmonic balance solutions are presented which show good agreement with the experimental HDR results, validating both the modeling and the solution methods used
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