AN OPTIMIZATION APPROACH TO LOCATING AND STABILIZING UNSTABLE PERIODIC ORBITS OF CHAOTIC SYSTEMS

2002 ◽  
Vol 12 (05) ◽  
pp. 1163-1172 ◽  
Author(s):  
YU-PING TIAN
Keyword(s):  
Periodic Orbits ◽  
Chaotic Systems ◽  
Main Idea ◽  
Optimal Solution ◽  
Optimization Process ◽  
Optimization Approach ◽  
Unstable Periodic Orbits ◽  
Global Optimal Solution ◽  
Local Optimal Solution ◽  
Global Optimal

In this paper, a novel method for locating and stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems is proposed. The main idea of the method is to formulate the UPO locating problem as an optimization issue by using some inherent properties of UPOs of chaotic systems. The global optimal solution of this problem yields the desired UPO. To avoid a local optimal solution, the state of the controlled chaotic system is absorbed into the initial condition of the optimization problem. The ergodicity of chaotic dynamics guarantees that the optimization process does not stay forever at any local optimal solution. When the chaotic orbit approaches the global optimal solution, which is the desired UPO, the controller will stabilize it at the UPO, and the optimization process will cease simultaneously. The method has been developed for both discrete-time and continuous-time systems, and validated for some typical chaotic systems such as the Hénon map and the Duffing oscillator, among others.

AO-BBO: A Novel Optimization Algorithm and Its Application in Plant Drug Extraction

2019 ◽  
Vol 19 (2) ◽  
pp. 139-145 ◽  
Author(s):  
Bote Lv ◽  
Juan Chen ◽  
Boyan Liu ◽  
Cuiying Dong
Keyword(s):  
Optimization Algorithm ◽  
Learning Strategy ◽  
Optimal Solution ◽  
Population Diversity ◽  
Global Optimal Solution ◽  
Plant Medicine ◽  
Suitability Index ◽  
Sensing Model ◽  
Local Optimal Solution ◽  
Global Optimal

<P>Introduction: It is well-known that the biogeography-based optimization (BBO) algorithm lacks searching power in some circumstances. </P><P> Material & Methods: In order to address this issue, an adaptive opposition-based biogeography-based optimization algorithm (AO-BBO) is proposed. Based on the BBO algorithm and opposite learning strategy, this algorithm chooses different opposite learning probabilities for each individual according to the habitat suitability index (HSI), so as to avoid elite individuals from returning to local optimal solution. Meanwhile, the proposed method is tested in 9 benchmark functions respectively. </P><P> Result: The results show that the improved AO-BBO algorithm can improve the population diversity better and enhance the search ability of the global optimal solution. The global exploration capability, convergence rate and convergence accuracy have been significantly improved. Eventually, the algorithm is applied to the parameter optimization of soft-sensing model in plant medicine extraction rate. Conclusion: The simulation results show that the model obtained by this method has higher prediction accuracy and generalization ability.</P>

scholarly journals Hybrid GA-SOCP Approach for Placement and Sizing of Distributed Generators in DC Networks

2020 ◽  
Vol 10 (23) ◽  
pp. 8616 ◽  
Author(s):  
Oscar Danilo Montoya ◽  
Walter Gil-González ◽  
Luis Fernando Grisales-Noreña
Keyword(s):  
Power Flow ◽  
Optimal Power Flow ◽  
Mathematical Optimization ◽  
Optimal Solution ◽  
Distribution Networks ◽  
Optimization Approach ◽  
Global Optimal Solution ◽  
Distributed Generators ◽  
Second Order Cone ◽  
Global Optimal

This research addresses the problem of the optimal location and sizing distributed generators (DGs) in direct current (DC) distribution networks from the combinatorial optimization. It is proposed a master–slave optimization approach in order to solve the problems of placement and location of DGs, respectively. The master stage applies to the classical Chu & Beasley genetic algorithm (GA), while the slave stage resolves a second-order cone programming reformulation of the optimal power flow problem for DC grids. This master–slave approach generates a hybrid optimization approach, named GA-SOCP. The main advantage of optimal dimensioning of DGs via SOCP is that this method makes part of the exact mathematical optimization that guarantees the possibility of finding the global optimal solution due to the solution space’s convex structure, which is a clear improvement regarding classical metaheuristic optimization methodologies. Numerical comparisons with hybrid and exact optimization approaches reported in the literature demonstrate the proposed hybrid GA-SOCP approach’s effectiveness and robustness to achieve the global optimal solution. Two test feeders compose of 21 and 69 nodes that can locate three distributed generators are considered. All of the computational validations have been carried out in the MATLAB software and the CVX tool for convex optimization.

The Research on Modified K-Means Algorithm Based on GA&SA

2013 ◽  
Vol 347-350 ◽  
pp. 3242-3246
Author(s):  
Zhe Feng Zhu ◽  
Xiao Bin Hui ◽  
Yi Qian Cao ◽  
Wan Xiang Lian
Keyword(s):  
Local Search ◽  
Clustering Algorithm ◽  
Optimal Solution ◽  
Global Search ◽  
Local Optimization ◽  
Effective Algorithm ◽  
Global Optimal Solution ◽  
Local Optimal Solution ◽  
Global Optimal ◽  
Clustering Center

The traditional K-means clustering algorithm has the disadvantage of weakness in overall search, easily falling into local optimization, highly reliance on initial clustering center. Aiming at the drawback of falling into partial optimization, putting forward a modified K-means algorithm mixing GA and SA, which combined the advantages of global search ability of GA and local search, to avoid K-means algorithm to lost into local optimal solution. The results of simulation show that the performance of above-mentioned algorithm is better in the optimization capacity than before, and easier to get the global optimal solution. It is an effective algorithm.

scholarly journals On Locally and Globally Optimal Solutions in Scalar Variational Control Problems

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 829 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Control Problem ◽  
Optimal Solution ◽  
Control Problems ◽  
Optimal Solutions ◽  
Global Optimal Solution ◽  
New Class ◽  
Minimal Criterion ◽  
Local Optimal Solution ◽  
Global Optimal ◽  
Curvilinear Integral

In this paper, optimality conditions are studied for a new class of PDE and PDI-constrained scalar variational control problems governed by path-independent curvilinear integral functionals. More precisely, we formulate and prove a minimal criterion for a local optimal solution of the considered PDE and PDI-constrained variational control problem to be its global optimal solution. The effectiveness of the main result is validated by a two-dimensional non-convex scalar variational control problem.

A Self-Adaptive and Variable Step Length Alopex Algorithm

2012 ◽  
Vol 433-440 ◽  
pp. 4302-4307
Author(s):  
Dong Li
Keyword(s):  
Optimal Solution ◽  
Step Length ◽  
Global Optimal Solution ◽  
Optimum Algorithm ◽  
Variable Step ◽  
Local Optimal Solution ◽  
Global Optimal ◽  
Simulation Results ◽  
Self Adaptive

Alopex is a heuristic and optimum algorithm. A self-adaptive and variable step length Alopex algorithm was raised to exceed local optimal solution and to approximate global optimal solution based on modified Alopex algorithm. To improve modified Alopex approximation precision further and eliminate the follow-on shocks appearance, the reasonable alter of δin was implemented. The simulation results show algorithm optimized is practicable and effective.

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Entangled States ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Qualitative Information ◽  
Chaotic Dynamical System ◽  
Naturally Occurring ◽  
External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

scholarly journals Best Proximity Point Results in Complex Valued Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Binayak S. Choudhury ◽  
Nikhilesh Metiya ◽  
Pranati Maity
Keyword(s):  
Minimum Distance ◽  
Metric Spaces ◽  
Optimal Solution ◽  
Global Optimal Solution ◽  
Complex Valued Metric Space ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Global Optimal ◽  
Metric Function ◽  
Complex Valued

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples.

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.

Sign in / Sign up

Export Citation Format

Share Document