An Extension of Discrete Lagrangian Descriptors for Unbounded Maps

In this paper, we provide an extension for the method of Discrete Lagrangian Descriptors with the purpose of exploring the phase space of unbounded maps. The key idea is to construct a working definition, that is built on the original approach introduced in [ Lopesino et al., 2015a ], and which relies on stopping the iteration of initial conditions when their orbits leave a certain region in the plane. This criterion is partly inspired by the classical analysis used in Dynamical Systems Theory to study the dynamics of maps by means of escape time plots. We illustrate the capability of this technique to reveal the geometrical template of stable and unstable invariant manifolds in phase space, and also the intricate structure of chaotic sets and strange attractors, by applying it to unveil the phase space of a well-known discrete-time system, the Hénon map.

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Market Share Delegation in a Bertrand Duopoly: Synchronisation and Multistability

Discrete Dynamics in Nature and Society ◽

10.1155/2015/394810 ◽

2015 ◽

Vol 2015 ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

Luciano Fanti ◽

Luca Gori ◽

Cristiana Mammana ◽

Elisabetta Michetti

Keyword(s):

Market Share ◽

Discrete Time ◽

Price Competition ◽

Initial Conditions ◽

Two Dimensional ◽

Discrete Time System ◽

Multiple Attractors ◽

Time System ◽

Market Share Delegation

This paper tackles the issue of local and global analyses of a duopoly game with price competition and market share delegation. The dynamics of the economy is characterised by a differentiable two-dimensional discrete time system. The paper stresses the importance of complementarity between products as a source of synchronisation in the long term, in contrast to the case of their substitutability. This means that when products are complements, players may coordinate their behaviour even if initial conditions are different. In addition, there exist multiple attractors so that even starting with similar conditions may end up generating very different dynamic patterns.

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A COMPUTER-ASSISTED STUDY OF GLOBAL DYNAMIC TRANSITIONS FOR A NONINVERTIBLE SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740701780x ◽

2007 ◽

Vol 17 (04) ◽

pp. 1305-1321 ◽

Cited By ~ 8

Author(s):

RAYMOND A. ADOMAITIS ◽

IOANNIS G. KEVREKIDIS ◽

RAFAEL DE LA LLAVE

Keyword(s):

Phase Space ◽

Basin Of Attraction ◽

Basins Of Attraction ◽

Computer Assisted ◽

Discrete Time System ◽

Time System ◽

Invariant Circle ◽

Global Dynamic ◽

Assisted Analysis ◽

The Basin Of Attraction

We present a computer-assisted analysis of the phase space features and bifurcations of a noninvertible, discrete-time system. Our focus is on the role played by noninvertibility in generating disconnected basins of attraction and the breakup of invariant circle solutions. Transitions between basin of attraction structures are identified and organized according to "levels of complexity," a term we define in this paper. In particular, we present an algorithm that provides a computational approximation to the boundary (in phase space) separating points with different preimage behavior. The interplay between this boundary and other phase space features is shown to be crucial in understanding global bifurcations and transitions in the structure of the basin of attraction.

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Discrete-Time System Conditional Optimization in the Parameter Space with Nonzero Initial Conditions

Tehnicki vjesnik - Technical Gazette ◽

10.17559/tv-20191031145233 ◽

2022 ◽

Vol 29 (1) ◽

Keyword(s):

Parameter Space ◽

Discrete Time ◽

Initial Conditions ◽

Discrete Time System ◽

Time System ◽

Conditional Optimization

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ALGORITHMS FOR SYNTHESIS OF DISCRETE CONTROLLERS IN NONLINEAR CONTROL SYSTEMS WITH DELAY

Technical science and innovation ◽

10.51346/tstu-02.21.1-77-0012 ◽

2020 ◽

Vol 2021 (1) ◽

pp. 80-86

Author(s):

Alisher Mallayev ◽

◽

Suban Xusanov

Keyword(s):

Phase Space ◽

Nonlinear Control ◽

Control Systems ◽

Invariant Manifolds ◽

Initial Conditions ◽

Continuous System ◽

Nonlinear Control Systems ◽

Image Point ◽

Process Control Systems ◽

Discrete Controller

Algorithms for the synthesis of discrete controllers in nonlinear control systems, taking into account the delay, have been proposed. The synthesized vector controller provides a solution to the control problem, for example, the translation of the image point of a closed system from arbitrary initial conditions to the origin of the phase space coordinates. Built on the basis of a series-parallel set of invariant manifolds, the dynamic discrete controller ensures the fulfillment of the specified technological requirements, the asymptotic stability of a closed discrete-continuous system, and has the property of predicting the behavior of the system after sampling. These algorithms have made it possible to effectively solve the problems of synthesis of discrete controllers, taking into account the delay in process control systems.

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