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Gravitational Search Algorithm
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
Parameter identification of a chaotic system is a multidimensional optimization problem. Different meta-heuristic algorithms have been applied to the identification of chaotic systems. Li et al. (2012b) applied the GSA and the chaotic GSA to identify the parameters of a Lorenz system described by {x˙=σ(y−x)y˙=ρx−xz−yz˙=xy−βz where 4 < σ < 14, 24 < ρ < 90, and 1.5 < β < 4.5 are the parameters of the Lorenz system, which defines the behavior of the chaotic system. The task of identification of the chaotic parameters [σ,ρ,β] of the Lorenz system will be to optimize the objective function defined in Equation 2.105. The position vector is then described as xi=[σ,ρ,β]
Chapter 20: Symbolic and Numerical Solutions of ODES with Mathematica
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
where σ, ρ, and β are the system parameters. The Lorenz system is an example of a dissipative chaotic system with a strange attractor. These features can be observed for certain values of the system parameters and initial conditions. This system models an unstable thermally convecting fluid (heated from below) and also arises in other simplified models.
Understanding the geometry of dynamics: the stable manifold of the Lorenz system
Published in Journal of the Royal Society of New Zealand, 2018
The Lorenz system is an exciting example of a dynamical system that exhibits chaotic dynamics, but it is somewhat removed from the original physical motivation of describing convection in the atmosphere. However, it is of fundamental importance to the theory of dynamical systems as a hallmark and prototypical example of a chaotic system. Moreover, it is an excellent test case to demonstrate, more generally, what can be achieved with advanced methods. We believe that our computational methods for the computation of invariant manifolds have reached such a maturity that detailed mathematical statements can be made about the overall organisation of phase space. Indeed, other more realistic models from applications can be studied in the same spirit; for example, these methods have been applied recently to explain the spiking behaviour of neurononal cells (Farjami et al. 2018) and complicated oscillations in chemical reactions (Hasan et al. 2017).