Search results for " DYNAMICAL SYSTEM"
showing 10 items of 188 documents
On the Kneser property for reaction–diffusion systems on unbounded domains
2009
Abstract We prove the Kneser property (i.e. the connectedness and compactness of the attainability set at any time) for reaction–diffusion systems on unbounded domains in which we do not know whether the property of uniqueness of the Cauchy problem holds or not. Using this property we obtain that the global attractor of such systems is connected. Finally, these results are applied to the complex Ginzburg–Landau equation.
Fixed point iterative schemes for variational inequality problems
2018
In a wide class of evolutionary processes, the problem of computing the solutions of an initial value problem is encountered. Here, we consider projected dynamical systems in the sense of \cite{Daniele} and references therein. Precisely, a projected dynamical system is an operator which solves the initial value problem: \begin{equation}\label{PDS}\frac{dx(t)}{dt}= \Pi_{\mathbb{K}}\left(x(t),-F(x(t))\right), \quad x(0)=x_0 \in \mathbb{K}, \, t \in [0,+\infty[,\tag{P}\end{equation} where $\mathbb{K}$ is a convex polyhedral set in $\mathbb{R}^n$, $F: \mathbb{K} \to \mathbb{R}^n$ and $\Pi_{\mathbb{K}}: \mathbb{R} \times \mathbb{K} \to \mathbb{R}^n$ is given as follows $\Pi_{\mathbb{K}}(x,-F(x))…
Asymptotic Behaviour of a Logistic Lattice System
2014
In this paper we study the asymptotic behaviour of solutions of a lattice dynamical system of a logistic type. Namely, we study a system of in nite ordinary di erential equations which can be obtained after the spatial discretization of a logistic equation with di usion. We prove that a global attractor exists in suitable weighted spaces of sequences.
On Differential Equations with Delay in Banach Spaces and Attractors for Retarded Lattice Dynamical Systems
2014
In this paper we first prove a rather general theorem about existence of solutions for an abstract differential equation in a Banach space by assuming that the nonlinear term is in some sense weakly continuous. We then apply this result to a lattice dynamical system with delay, proving also the existence of a global compact attractor for such system.
The minimal model of Hahn for the Calvin cycle.
2018
There are many models of the Calvin cycle of photosynthesis in the literature. When investigating the dynamics of these models one strategy is to look at the simplest possible models in order to get the most detailed insights. We investigate a minimal model of the Calvin cycle introduced by Hahn while he was pursuing this strategy. In a variant of the model not including photorespiration it is shown that there exists exactly one positive steady state and that this steady state is unstable. For generic initial data either all concentrations tend to infinity at lates times or all concentrations tend to zero at late times. In a variant including photorespiration it is shown that for suitable v…
Linear dynamics induced by odometers
2022
Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.
Control of Power Converters with Hybrid Affine Models and Pulse-Width Modulated Inputs
2021
In this paper, hybrid dynamical systems theory is applied to the analysis and control of switched converters with Pulse-Width Modulated (PWM) inputs. The system is described by a state-space model with continuous flows and discrete jumps, without averaged equations. The modulation effects are captured in full without using time-dependent signals, by enlarging the state vector to include the PWM waveform generation process. Furthermore, the sample-and-hold mechanism associated with the sampling frequency is also taken into account with this approach. A control law is proposed based on a Lyapunov function candidate. Furthermore, convergence sets and the steady state jitter, inherent to PWM-ba…
The Lyapunov dimension formula for the global attractor of the Lorenz system
2015
The exact Lyapunov dimension formula for the Lorenz system has been analytically obtained first due to G.A.Leonov in 2002 under certain restrictions on parameters, permitting classical values. He used the construction technique of special Lyapunov-type functions developed by him in 1991 year. Later it was shown that the consideration of larger class of Lyapunov-type functions permits proving the validity of this formula for all parameters of the system such that all the equilibria of the system are hyperbolically unstable. In the present work it is proved the validity of the formula for Lyapunov dimension for a wider variety of parameters values, which include all parameters satisfying the …
Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents
2021
Cyclicity and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run. We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy. Using of a mid-size firm model as an example, we demonstrate the use of effective analytical and numerical procedures for calculating the quantitative characteristics of its irregular limiting dynamics based on Lyapunov exponents, such as dimension and entropy. We use an analytical approach for localization of a global at…
Perturbation of the Lyapunov spectra of periodic orbits
2012
We describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism. The description is expressed in terms of the finest dominated splitting and Lyapunov exponents that appear in the limit of a sequence of periodic orbits, and involves the majorization partial order. Among the applications, we give a simple criterion for the occurrence of universal dynamics.