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The Emergence of Chaos in Time
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
A concrete example of a chaotic system that can be easily found in the physics departments, but it can also be built at home, is the double pendulum (Shinbrot et al. 1992). A double pendulum consists of two simple double pendula, which are bound to one another as shown in Figure 10.2. The first point mass m1 is suspended from a fixed point (that is also the origin of the reference system) by a rigid weightless rod of length L1 and the second point mass m2 is suspended from m1 by another weightless rod of length L2 (see Figure 10.2). The top and center pivots are assumed frictionless, and the pendula rotate under the action of gravity in the absence of air. The double pendulum’s total energy, which is the sum of its kinetic and potential energies, is conserved. The double pendulum is a Hamiltonian system that exhibits chaotic behavior.
Lagrangian formulation in mechanics
Published in Bijan Kumar Bagchi, Advanced Classical Mechanics, 2017
The double pendulum is a combination of two simple pendulums positioned in the same vertical plane and connected in such a way that the lower one hangs from the bob of the upper one. The bob of the upper one of mass m1 is suspended from a ceiling by an inextensible string of negligible mass having a length l1 while the lower bob of mass m2 is tied to the bob of the upper one by an inextensible string of negligible mass having length l2. The setup is illustrated in Figure 3.4.
Chaos-enhanced multiple-choice strategy for particle swarm optimisation
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
Michal Pluhacek, Roman Senkerik, Adam Viktorin, Tomas Kadavy
In the past, the chaos has been observed in many of various systems including, for instance, weather, biological systems, many electronic circuits (Chua's circuit), mechanical systems, such as a double pendulum, magnetic pendulum, or so-called billiard problem. Since the area of ‘deterministic chaos’ presence is very wide, there exist several definitions of chaotic systems and their classification. The simplest chaotic systems are the discrete chaotic maps that usually have a form of iterated functions. These maps are the focus of our research. Chaotic flows, oscillators, and other time-continuous chaotic dynamical systems are often represented by the set of differential equations. Moreover, a special type of spatiotemporal chaotic behaviour is mostly defined in the form of a coupled map lattices system [31].