STABILITY OF EQUILIBRIUM STATES OF NONLINEAR STRUCTURES AND CHAOS PHENOMENON.

1992 ◽  
Vol 02 (02) ◽  
pp. 271-283 ◽  
Author(s):  
D. SHILKRUT
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Geometrically Nonlinear ◽  
Geometrical Approach ◽  
Initial Value ◽  
Nonlinear Structure ◽  
Deterministic Systems ◽  
Classical Chaos ◽  
Problem Characteristic

The “classical” chaos of deterministic systems is characteristic for the motion of dynamical systems. Recently, some attempts were made to find static analogies of chaos [Thompson & Virgin, 1988; Naschie & Athel, 1989; Naschie, 1989]. However, this was considered for structures in specific artificial conditions (for example, infinitely long bars with sinusoidal geometric imperfections) transferring de facto the boundary value problem (which always describes static deformation of structures) into an initial value problem characteristic for problems of motion. In this article, chaotic (unpredictable) behavior is described for a usual (not special) nonlinear structure in statics, which is governed, naturally, by a boundary value problem in a finite interval of the argument. The behavior of this structure (geometrically nonlinear plate), which is an example of the class of static chaotic structures, is investigated by a new geometrical approach called the “deformation map.” The presented results are one of the first steps in the chapter of chaos in statics, and therefore the link between “classical” and static chaos needs further investigations.

scholarly journals Stabilization of higher order Schrödinger equations on a finite interval: Part II

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Türker Özsarı ◽  
Kemal Cem Yılmaz
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Initial Boundary ◽  
Higher Order ◽  
Observer Design ◽  
Target System ◽  
Schrodinger Equations ◽  
Left Endpoint ◽  
Schrödinger Equations

<p style='text-indent:20px;'>Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [<xref ref-type="bibr" rid="b3">3</xref>] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.</p>

SHOOTING METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

2013 ◽  
Vol 10 (06) ◽  
pp. 1350041 ◽  
Author(s):  
J. CHRISTY ROJA ◽  
A. TAMILSELVAN
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Outer Region ◽  
Coupled System ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
The Given

A class of singularly perturbed boundary value problems (SPBVPs) for fourth-order ordinary differential equations (ODEs) is considered. The SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. In order to solve them numerically, a method is suggested in which the given interval is divided into two inner regions (boundary layer regions) and one outer region. Two initial-value problems associated with inner regions and one boundary value problem corresponding to the outer region are derived from the given SPBVP. In each of the two inner regions, an initial value problem is solved by using fitted mesh finite difference (FMFD) scheme on Shishkin mesh and the boundary value problem corresponding to the outer region is solved by using classical finite difference (CFD) scheme on Shishkin mesh. A combination of the solution so obtained yields a numerical solution of the boundary value problem on the whole interval. First, in this method, we find the zeroth-order asymptotic expansion approximation of the solution of the weakly coupled system. Error estimates are derived. Examples are presented to illustrate the numerical method. This method is suitable for parallel computing.

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