scholarly journals Rigorous numerics for ODEs using Chebyshev series and domain decomposition

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jan Bouwe van den Berg ◽  
Ray Sheombarsing
Keyword(s):  
Domain Decomposition ◽  
Numerical Approximation ◽  
Lorenz System ◽  
Small Neighborhood ◽  
Analytic Solutions ◽  
Heuristic Procedure ◽  
Chebyshev Series ◽  
Connecting Orbits ◽  
Nonlinear Odes ◽  
The Lorenz System

<p style='text-indent:20px;'>In this paper we present a rigorous numerical method for validating analytic solutions of nonlinear ODEs by using Chebyshev-series and domain decomposition. The idea is to define a Newton-like operator, whose fixed points correspond to solutions of the ODE, on the space of geometrically decaying Chebyshev coefficients, and to use the so-called radii-polynomial approach to prove that the operator has an isolated fixed point in a small neighborhood of a numerical approximation. The novelty of the proposed method is the use of Chebyshev series in combination with domain decomposition. In particular, a heuristic procedure based on the theory of Chebyshev approximations for analytic functions is presented to construct efficient grids for validating solutions of boundary value problems. The effectiveness of the proposed method is demonstrated by validating long periodic and connecting orbits in the Lorenz system for which validation without domain decomposition is not feasible.</p>

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

CONTROL OF THE DIFFERENTIALLY FLAT LORENZ SYSTEM

2001 ◽  
Vol 11 (07) ◽  
pp. 1989-1996 ◽  
Author(s):  
JIN MAN JOO ◽  
JIN BAE PARK
Keyword(s):  
Equilibrium State ◽  
Lorenz System ◽  
Degree Of Freedom ◽  
Design Approach ◽  
State Trajectory ◽  
System A ◽  
Full State ◽  
Tracking Controller ◽  
The Lorenz System ◽  
Two Degree Of Freedom

This paper presents an approach for the control of the Lorenz system. We first show that the controlled Lorenz system is differentially flat and then compute the flat output of the Lorenz system. A two degree of freedom design approach is proposed such that the generation of full state feasible trajectory incorporates with the design of a tracking controller via the flat output. The stabilization of an equilibrium state and the tracking of a feasible state trajectory are illustrated.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

Chaos in the Periodically Parametrically Excited Lorenz System

2021 ◽  
Vol 31 (08) ◽  
pp. 2130024
Author(s):  
Weisheng Huang ◽  
Xiao-Song Yang
Keyword(s):  
Chaotic Dynamics ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Parametrically Excited ◽  
The Lorenz System

We demonstrate in this paper a new chaotic behavior in the Lorenz system with periodically excited parameters. We focus on the parameters with which the Lorenz system has only two asymptotically stable equilibrium points, a saddle and no chaotic dynamics. A new mechanism of generating chaos in the periodically excited Lorenz system is demonstrated by showing that some trajectories can visit different attractor basins due to the periodic variations of the attractor basins of the time-varying stable equilibrium points when a parameter of the Lorenz system is varying periodically.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

ADAPTIVE CONTROL OF UNCERTAIN LORENZ SYSTEM USING DECOUPLED BACKSTEPPING

2004 ◽  
Vol 14 (04) ◽  
pp. 1439-1445 ◽  
Author(s):  
S. S. GE
Keyword(s):  
Adaptive Control ◽  
Closed Loop ◽  
Lorenz System ◽  
Physical Property ◽  
Asymptotic Convergence ◽  
Closed Loop System ◽  
Feedback Form ◽  
Controlling Chaos ◽  
Simulation Results ◽  
The Lorenz System

In this letter, we reconsider the problem of controlling chaos in the well-known Lorenz system. Firstly, the difficulty in controlling the Lorenz system is discussed in the general strict-feedback form. Then, singularity-free adaptive control is presented for the Lorenz system with three key parameters unknown by exploiting the physical property of the system using decoupled backstepping design. The proposed controller guarantees the asymptotic convergence of the output and the boundedness of all the signals in the closed-loop system. Simulation results are conducted to show the effectiveness of the approach.

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