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

Compound orbits break-up in constituents: an algorithm

In this paper decomposition of periodic orbits in bifurcation diagrams are derived in unidimensional dynamics system $x_{n+1}=f(x_{n};r)$, being $f$ 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)

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

Linearized analysis versus optimization-based nonlinear analysis for nonlinear systems

For autonomous nonlinear systems stability and input-output properties in small enough (infinitesimally small) neighborhoods of (linearly) asymptotically stable equilibrium points can be inferred from the properties of the linearized dynamics. On the other hand, generalizations of the S-procedure and sum-of-squares programming promise a framework potentially capable of generating certificates valid over quantifiable, finite size neighborhoods of the equilibrium points. However, this procedure involves multiple relaxations (unidirectional implications). Therefore, it is not obvious if the sum-of-squares programming based nonlinear analysis can return a feasible answer whenever linearization based analysis does. Here, we prove that, for a restricted but practically useful class of systems, conditions in sum-of-squares programming based region-of-attraction, reachability, and input-output gain analyses are feasible whenever linearization based analysis is conclusive. Besides the theoretical interest, such results may lead to computationally less demanding, potentially more conservative nonlinear (compared to direct use of sum-of-squares formulations) analysis tools

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

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

Simplicial Nonlinear Principal Component Analysis

We present a new manifold learning algorithm that takes a set of data points lying on or near a lower dimensional manifold as input, possibly with noise, and outputs a simplicial complex that fits the data and the manifold. We have implemented the algorithm in the case where the input data can be triangulated. We provide triangulations of data sets that fall on the surface of a torus, sphere, swiss roll, and creased sheet embedded in a fifty dimensional space. We also discuss the theoretical justification of our algorithm.Comment: 21 pages, 6 figure

Piecewise Volterra modelling of the Duffing oscillator in the frequency domain

When analysing the nonlinear Duffing oscillator, the weak nonlinearity is basically dependent on the amplitude range of the input excitation. The nonlinear differential equation models of such nonlinear oscillators, which can be transformed into the frequency domain, can generally only provide Volterra modelling and analysis in the frequency-domain over a fraction of the entire framework of weak nonlinearity. This paper discusses the problem of using a new non-parametric routine to extend the capability of Volterra analysis, in the frequency domain, to weakly nonlinear Duffing systems at a wider range of excitation amplitude range which the current underlying nonlinear differential equation models fail to address

Four conjectures in Nonlinear Analysis

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.Comment: arXiv admin note: text overlap with arXiv:1504.01010, arXiv:1409.5919, arXiv:1612.0819