Related papers: Fractal structure of iterative time profiles
The scope of the paper is a theoretical analysis of the dynamical system, the model of which was reduced to Weierstrasse function. A fractal structure of the trajectory was proved and the entropy of the system information designated.
In this manuscript, we investigate a fractional stochastic neutral differential equation with time delay, which includes both deterministic and stochastic components. Our primary objective is to rigorously prove the existence of a unique…
Fluctuations in the return time statistics of a dynamical system can be described by a new spectrum of dimensions. Comparison with the usual multifractal analysis of measures is presented, and difference between the two corresponding sets…
The phase ordering dynamics of coupled chaotic maps on fractal networks are investigated. The statistical properties of the systems are characterized by means of the persistence probability of equivalent spin variables that define the…
The paper discusses linear fractional representations of parameter-dependent nonlinear systems with dynamics defined by real rational nonlinearities and a finite set of point delays. The global asymptotic stability is investigated via…
Intractable phase dynamics often challenge our understanding of complex oscillatory systems, hindering the exploration of synchronisation, chaos, and emergent phenomena across diverse fields. We introduce a novel conceptual framework for…
Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the parameter of the map and the fractional order of the…
Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling…
We describe a method for analyzing the phase space structures of Hamiltonian systems. This method is based on a time-frequency decomposition of a trajectory using wavelets. The ridges of the time-frequency landscape of a trajectory, also…
In the first section we review recent results on the harmonic analysis of fractals generated by iterated function systems with emphasis on spectral duality. Classical harmonic analysis is typically based on groups whereas the fractals are…
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves…
We study a deterministic dynamics with two time scales in a continuous state attractor network. To the usual (fast) relaxation dynamics towards point attractors (``patterns'') we add a slow coupling dynamics that makes the visited patterns…
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
We develop an effective model to describe the dynamics of a system of particle moving in circular configurations around a central mass, by considering the continuum limit of the angular distribution, to obtain the stable configurations for…
This paper deals with stability of a certain class of fractional order linear and nonlinear systems. The stability is investigated in the time domain and the frequency domain. The general stability conditions and several illustrative…
This paper numerically investigates the instability characteristics of decelerating flows. The flow dynamics and temporal evolution of coherent structures in a diverging section with mild spatial pressure gradient are analyzed using…
We identify a new type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently…
We investigate the time-periodic solutions to the nonlinear wave and beam equations and uncover their intricate, fractal-like structure. In particular, we identify a new class of large-energy solutions with complex mode compositions and…
Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…
Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation \begin{equation*} D^\alpha x(t)=\delta…