1,074 research outputs found
Computed Chaos or Numerical Errors
Discrete numerical methods with finite time-steps represent a practical
technique to solve initial-value problems involving nonlinear differential
equations. These methods seem particularly useful to the study of chaos since
no analytical chaotic solution is currently available. Using the well-known
Lorenz equations as an example, it is demonstrated that numerically computed
results and their associated statistical properties are time-step dependent.
There are two reasons for this behavior. First, chaotic differential equations
are unstable so that any small error is amplified exponentially near an
unstable manifold. The more serious and lesser-known reason is that stable and
unstable manifolds of singular points associated with differential equations
can form virtual separatrices. The existence of a virtual separatrix presents
the possibility of a computed trajectory actually jumping through it due to the
finite time-steps of discrete numerical methods. Such behavior violates the
uniqueness theory of differential equations and amplifies the numerical errors
explosively. These reasons imply that, even if computed results are bounded,
their independence on time-step should be established before accepting them as
useful numerical approximations to the true solution of the differential
equations. However, due to these exponential and explosive amplifications of
numerical errors, no computed chaotic solutions of differential equations
independent of integration-time step have been found. Thus, reports of computed
non-periodic solutions of chaotic differential equations are simply
consequences of unstably amplified truncation errors, and are not approximate
solutions of the associated differential equations.Comment: pages 24, Figures
Why Patterns Appear Spontaneously in Dissipative Systems?
It is proposed that the spatial (and temporal) patterns spontaneously
appearing in dissipative systems maximize the energy flow through the pattern
forming interface. In other words - the patterns maximize the entropy growth
rate in an extended conservative system (consisting of the pattern forming
interface and the energy bathes). The proposal is supported by examples of the
pattern formation in different systems. No example contradicting the proposal
is known.Comment: 7 pages, 1 figur
Compound orbits break-up in constituents: an algorithm
In this paper decomposition of periodic orbits in bifurcation diagrams are
derived in unidimensional dynamics system , being an
unimodal function. We proof a theorem which states the necessary and sufficient
conditions for the break-up of compound orbits in their simpler constituents. A
corollary to this theorem provides an algorithm for the computation of those
orbits. This process closes the theoretical framework initiated in (Physica D,
239:1135--1146, 2010)
Existence of the solution to a nonlocal-in-time evolutional problem
This work is devoted to the study of a nonlocal-in-time evolutional problem
for the first order differential equation in Banach space. Our primary
approach, although stems from the convenient technique based on the reduction
of a nonlocal problem to its classical initial value analogue, uses more
advanced analysis. That is a validation of the correctness in definition of the
general solution representation via the Dunford-Cauchy formula. Such approach
allows us to reduce the given existence problem to the problem of locating
zeros of a certain entire function. It results in the necessary and sufficient
conditions for the existence of a generalized (mild) solution to the given
nonlocal problem. Aside of that we also present new sufficient conditions which
in the majority of cases generalize existing results.Comment: This article is an extended translation of the part of Dmytro
Sytnyk's PhD Thesi
Notion of a virtual derivative
Diagrams as a graphic expresion of derivatives is proposed for calculation of
derivatives for composed function. The concret diagram is understood as a
virtual derivative in contrast of concret derivative. In polynomial expression
of functions derivative the concret derivative will be every monomic member,
and the virtual derivative represent the sum of similar monomic members. The
word virtual denotes that we dont need to know every virtual derivative, we
don't write all the sequence of these virtual derivatives, and simply pick the
needed one. This is in contrast of tradition to write the whole algebraic
expresion as a denotion of whole function's derivative. Such graphic expresion
can be helpful in the problems of differential geometry, in the various
asymptotic expantions, also in the solution of some differential equations.Comment: The diagrams are drown with the help of xy-pic and can be automaticly
generated for the derivative of large degry of more general composed functio
On comparison of the estimators of the Hurst index and the diffusion coefficient of the fractional Gompertz diffusion process
We study some estimators of the Hurst index and the diffusion coefficient of
the fractional Gompertz diffusion process and prove that they are strongly
consistent and most of them are asymptotically normal. Moreover, we compare the
asymptotic behavior of these estimators with the aid of computer simulations.Comment: 17 pages, 4 figure
Fixed point theorems for --contractive mappings of Meir--Keeler type and applications
In this paper, we introduce the notion of --contractive mapping of
Meir--Keeler type in complete metric spaces and prove new theorems which assure
the existence, uniqueness and iterative approximation of the fixed point for
this type of contraction. The presented theorems extend, generalize and improve
several existing results in literature. To validate our results, we establish
the existence and uniqueness of solution to a class of third order two point
boundary value problems
Discrete Multistage Optimization and Hierarchical Market
New simple form of mixed solutions is described by bilinear continuous optimization processes. It enables investigate an analytic solutions and the connection between discrete and continuous optimization processes. Connection between discrete and continuous processes is stochastic. Discrete optimization processes are used for the control works in levels and groups of the hierarchical market. Equilibrium between local and global levels of works is investigated in hierarchical market
On the backward bifurcation of a vaccination model with nonlinear incidence
A compartmental epidemic model, introduced by Gumel and Moghadas [1], is considered. The model incorporates a nonlinear incidence rate and an imperfect preventive vaccine given to susceptible individuals. A bifurcation analysis is performed by applying the bifurcation method introduced in [2], which is based on the use of the center manifold theory. Conditions ensuring the occurrence of backward bifurcation are derived. The obtained results are numerically validated and then discussed from both the mathematical and the epidemiological perspective
Numerical algorithms for solving the optimal control problem of simple bioreactors
The modified nonlocal feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion–convection–reaction equations. The analysis of the influence of the convection transport is one the aims of this paper. The control loop is defined using the relation, which shows how the amount of the drug produced in the bioreactor and delivered into a human body depends on the substrate concentration specified on the external boundary of the bioreactor. The system of PDEs is solved by using the finite volume and finite difference methods, the control loop parameters are defined from the analysis of stationary linearized equations. The second aim of this paper is to solve the inverse problem and to determine optimal boundary conditions. These results enable us to estimate the potential accuracy of the proposed devices.
 
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