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On the region of attraction of dynamical systems: Application to Lorenz equations

Hammami M. A., Rettab N. H.

Archives of Control Sciences

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2020

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Vol. 30, No. 3

389--409

EN

Many nonlinear dynamical systems can present a challenge for the stability analysis in particular the estimation of the region of attraction of an equilibrium point. The usual methodis based on Lyapunov techniques. For the validity of the analysis it should be supposed that the initial conditions lie in the domain of attraction. In this paper, we investigate such problem for a class of dynamical systems where the origin is not necessarily an equilibrium point. In this case, a small compact neighborhood of the origin can be estimated as an attractor for the system. We give a method to estimate the basin of attraction based on the construction of a suitable Lyapunov function. Furthermore, an application to Lorenz system is given to verify the effectiveness of the proposed method.

2

Recognizing Sets in Evolutionary Multiobjective Optimization

Gajda-Zagórska E.

Journal of Telecommunications and Information Technology

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2012

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nr 1

74-82

EN

Among Evolutionary Multiobjective Optimization Algorithms (EMOA) there are many which find only Paretooptimal solutions. These may not be enough in case of multimodal problems and non-connected Pareto fronts, where more information about the shape of the landscape is required. We propose a Multiobjective Clustered Evolutionary Strategy (MCES) which combines a hierarchic genetic algorithm consisting of multiple populations with EMOA rank selection. In the next stage, the genetic sample is clustered to recognize regions with high density of individuals. These regions are occupied by solutions from the neighborhood of the Pareto set. We discuss genetic algorithms with heuristic and the concept of well-tuning which allows for theoretical verification of the presented strategy. Numerical results begin with one example of clustering in a single-objective benchmark problem. Afterwards, we give an illustration of the EMOA rank selection in a simple two-criteria minimization problem and provide results of the simulation of MCES for multimodal, multi-connected example. The strategy copes with multimodal problems without losing local solutions and gives better insight into the shape of the evolutionary landscape. What is more, the stability of solutions in MCES may be analyzed analytically.

3

Improving Population-Based Algorithms with Fitness Deterioration

Wolny A., Schaefer R.

Journal of Telecommunications and Information Technology

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2011

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nr 4

31-44

EN

This work presents a new hybrid approach for supporting sequential niching strategies called Cluster Supported Fitness Deterioration (CSFD). Sequential niching is one of the most promising evolutionary strategies for analyzing multimodal global optimization problems in the continuous domains embedded in the vector metric spaces. In each iteration CSFD performs the clustering of the random sample by OPTICS algorithm and then deteriorates the fitness on the area occupied by clusters. The selection pressure pushes away the next-step sample (population) from the basins of attraction of minimizers already recognized, speeding up finding the new ones. The main advantages of CSFD are low memory an computational complexity even in case of large dimensional problems and high accuracy of deterioration obtained by the flexible cluster definition delivered by OPTICS. The paper contains the broad discussion of niching strategies, detailed definition of CSFD and the series of the simple comparative tests.

4

Non-linear optimization methods for small earthquake locations

Pszczoła G., Leśniak A.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2004

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Vol. 8, No 4

583-590

EN

The problem of locating mine tremors using P-wave arrival times is revisited in the paper. A multidimensional, global, non-linear, constrained optimization method is used as a minimization algorithm for tremor location.In order to see the general properties of the minimized function a few images showing its basins of attractions have been constructed. These pictures enable us to choose efficient algorithms needed to solve location problems. The classical genetic algorithm, pure random search and the most efficient multistart algorithm have been tested. Local minimization methods should be introduced to the location procedure to increase the efficiency of tremor location.