On a Computational Method for Finding the Fixed Points of a Map

1997 ◽  
Author(s):  
Ramesh S. Guttalu
Keyword(s):  
Differential Equation ◽  
Fixed Points ◽  
Vector Function ◽  
Multiple Equilibria ◽  
Nonlinear Dynamical Systems ◽  
Computational Method ◽  
Continuation Methods ◽  
Multiple Zeros ◽  
Highly Nonlinear ◽  
Nonlinear Vector

Abstract One of the problems arising in the analysis of periodic nonlinear dynamical systems is to determine multiple equilibria and periodic solutions. These may be formulated in terms of finding zeros of a nonlinear vector function. The iterative methods commonly employed for this purpose concentrate on locating a single zero from a given initial guess. The homotopic and continuation methods are also used for locating multiple zeros but they primarily have been used to follow a single solution branch with a parameter variation. The purpose of this paper is to explore a method based on a differential equation for finding multiple zeros. The equilibria of this differential equation correspond with the zeros of a given vector function. Since the equilibria is asymptotically stable, they can be obtained by examining a large number of trajectories. For this purpose, the cell-to-cell mapping is ideally suited for finding the equilibria, and hence the zeros of a nonlinear vector function. Preliminary results for finding multiple fixed points of a highly nonlinear map are provided in this paper.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

scholarly journals On Unique and Nonunique Fixed Points and Fixed Circles in M v b -Metric Space and Application to Cantilever Beam Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Meena Joshi ◽  
Anita Tomar ◽  
Hossam A. Nabwey ◽  
Reny George
Keyword(s):  
Differential Equation ◽  
Fixed Points ◽  
Metric Space ◽  
Order Differential Equation ◽  
Unique Fixed Point ◽  
Elastic Beam ◽  
Closed Ball ◽  
Open Ball ◽  
Ball Convergence ◽  
Fourth Order Differential Equation

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

2013 ◽  
Vol 5 (04) ◽  
pp. 407-422 ◽  
Author(s):  
Matthew A. Beauregard ◽  
Qin Sheng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Difference ◽  
Nonlinear Partial Differential Equation ◽  
Partial Differential ◽  
Spatial Adaptation ◽  
Monitor Function ◽  
Monitoring Strategies ◽  
Highly Nonlinear ◽  
Equidistribution Principle

AbstractFinite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-calledprimitiveregime, the rest belong to a later category of themodifiedtype, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

Non-Linear Systems Under Parametric Alpha-Stable Le´vy White Noises

2005 ◽  
Author(s):  
Mario Di Paola ◽  
Antonina Pirrotta ◽  
Massimiliano Zingales
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Characteristic Function ◽  
Stochastic Analysis ◽  
Nonlinear Dynamical Systems ◽  
Parametrically Excited ◽  
Nonlinear Dynamical ◽  
Non Linear Systems ◽  
The Fourier Transform ◽  
White Noises

In this study stochastic analysis of nonlinear dynamical systems under a-stable, multiplicative white noise has been performed. Analysis has been conducted by means of the Itoˆ rule extended to the case of α-stable noises. In this context the order of increments of Levy process has been evaluated and differential equations ruling the evolutions of statistical moments of either parametrically and external dynamical systems have been obtained. The extended Itoˆ rule has also been used to yield the differential equation ruling the evolution of the characteristic function for parametrically excited dynamical systems. The Fourier transform of the characteristic function, namely the probability density function is ruled by the extended Einstein-Smoluchowsky differential equation to case of parametrically excited dynamical systems. Some numerical applications have been reported to assess the reliability of the proposed formulation.

scholarly journals On diagrammatic technique for nonlinear dynamical systems

2014 ◽  
Vol 29 (35) ◽  
pp. 1430039
Author(s):  
Mykola Semenyakin
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Radius Of Convergence ◽  
Evolution Operators ◽  
Finite Degree ◽  
Nonlinear Dynamical ◽  
Diagrammatic Technique

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev {V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

scholarly journals Observer Design for a Core Circadian Rhythm Network

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuhuan Zhang
Keyword(s):  
Differential Equation ◽  
Circadian Rhythm ◽  
Biological Networks ◽  
Original System ◽  
Equation Model ◽  
Observer Design ◽  
Differential Equation Model ◽  
Highly Nonlinear ◽  
Input Signals ◽  
Potential Applications

The paper investigates the observer design for a core circadian rhythm network inDrosophilaandNeurospora. Based on the constructed highly nonlinear differential equation model and the recently proposed graphical approach, we design a rather simple observer for the circadian rhythm oscillator, which can well track the state of the original system for various input signals. Numerical simulations show the effectiveness of the designed observer. Potential applications of the related investigations include the real-world control and experimental design of the related biological networks.

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