Bifurcations and Chaos in an Experimental Based Quasi-Continuum Nonlinear Dynamical System for the ‘Clapper’ Nanoresonator

2007 ◽  
Author(s):  
O. Gottlieb ◽  
A. Gemintern ◽  
R. H. Blick
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Field Equations ◽  
Bifurcation Structure ◽  
Nonlinear Dynamical ◽  
Spatio Temporal ◽  
Initial Boundary Value ◽  
Temporal Field

In this paper we formulate and numerically investigate an experimentally based quasi-continuum nonlinear initial-boundary-value problem for the three-field ‘Clapper’ nanoresonator that consistently incorporates the system geometric nonlinearity with nonlinear contributions of both magnetomotive and electrodynamic excitation. The spatio-temporal field equations are then reduced via symmetry and a modal projection to an equivalent quasiperiodically excited, low order, nonlinear dynamical system. The governing parameters of the resulting system are matched with the experimentally measured resonance conditions for small amplitude response. Numerical analysis reveals a complex bifurcation structure of torus doubling culminating with a chaotic strange attractor that exhibits similar features to that previously measured in the ‘Clapper’ experiment.

Long-time behaviour for a model of phase-field type

1996 ◽  
Vol 126 (1) ◽  
pp. 167-185 ◽  
Author(s):  
Ph. Laurençot
Keyword(s):  
Phase Field ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Behaviour ◽  
Field Equations ◽  
Field Type ◽  
Long Time Behaviour ◽  
Long Time ◽  
Phase Field Equations ◽  
Initial Boundary Value

In this paper, we study a model of phase-field type for the kinetics of phase transitions which was considered by Halperin, Hohenberg and Ma and which includes the phase-field equations. We study the well-posedness of the corresponding initial boundary value problem in an open bounded subset in space dimension lower than or equal to 3 and prove that, under suitable conditions, the long-time behaviour of the solutions to this problem is described by a maximal attractor.

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

scholarly journals Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoqiang Dai ◽  
Wenke Li
Keyword(s):  
Dynamical System ◽  
Control System ◽  
Source Term ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Source ◽  
Sharp Condition ◽  
Global Nonexistence ◽  
Existence And Nonexistence ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.</p>

Computational Methods for Solving Dynamic Ice Structure Interaction Problems

2011 ◽  
Author(s):  
Ibrahim Konuk
Keyword(s):  
Dynamical System ◽  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Element Model ◽  
Cohesive Element ◽  
Structure Interaction ◽  
Initial Boundary Value ◽  
Well Posed ◽  
Cohesive Element Model

A framework based on a complex dynamical system viewpoint for formulating and solving dynamic ice-structure interaction problems is introduced. Important constituents required for formulating a well posed initial-boundary value problem are discussed. Significance of these constituents is illustrated using a Cohesive Element model of several example problems.

Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions

2019 ◽  
Vol 29 (03) ◽  
pp. 373-418 ◽  
Author(s):  
Michael Winkler
Keyword(s):  
Nutrient Concentration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solutions ◽  
Cross Diffusion ◽  
Temporal Decay ◽  
Mild Assumption ◽  
Neumann Type ◽  
Spatio Temporal ◽  
Initial Boundary Value

This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model [Formula: see text] for the population densities [Formula: see text] and [Formula: see text] of foragers and exploiters, as well as the nutrient concentration [Formula: see text], is considered in smoothly bounded domains [Formula: see text], [Formula: see text]. It is first shown that under an explicit condition linking the sizes of the resource production rate [Formula: see text] and of the initial nutrient concentration, an associated Neumann-type initial-boundary value problem admits a global solution within an appropriate generalized concept. The second of the main results asserts stabilization of these solutions toward spatially homogeneous equilibria in the large time limit, provided that [Formula: see text] satisfies a mild assumption on temporal decay. To the best of our knowledge, these are the first rigorous analytical results addressing taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆).

Study of rigidity of a first-order algebro-differential system with perturbation in the right-hand side

2021 ◽  
pp. 172-181
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Dynamical System ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Bifurcation Equation ◽  
First Order ◽  
Initial Boundary Value ◽  
The Right

The rigidity of a dynamical system described by a first-order differential equationwith an irreversible operator at the highest derivative is investigated. The system is perturbed by an operator addition of the order of the second power of a small parameter. Conditions under which the system is robust with respect to these disturbances are determined as well as conditions under which the influence of disturbances is significant. For this, the bifurcation equation is derived. It is used to set the type of boundary layer functions. As an example, we investigate the initial boundary value problem for a system of partial differential equations with a mixed second partial derivative which occurs in the study of the processes of sorption anddesorption of gases, drying processes, etc.

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