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Bifurcation Behaviour of Non-Linear Systems
Published in T. Thyagarajan, D. Kalpana, Linear and Non-Linear System Theory, 2020
Trajectory: The solution to a differential equation is called a trajectory. Flow: The collection of all such solutions of differential equation is called flow.Map: The collection of solutions of difference equation is called the map.Dissipative system: It is a system where energy loss takes place. (May be due to friction or damping, etc.)Hamiltonian system: It is a system where the total mechanical energy is preserved.
Soil shear–the physical model (part 2)
Published in Paul G. Joseph, Dynamical Systems-Based Soil Mechanics, 2017
As in Joseph (2009, 2010), the Poisson model derived in Chapter 3 continue to describe a dynamical system with a fixed, single-point attractor (the steady-state condition) defined by the steady-state void-ratio (and negligibly, the strain-rate). Given that the attractor is a fixed point, behavior is not chaotic; stress-paths for various initial conditions converge to this steady-state point attractor as described in Poulos (1981). A measure of convergence used in dynamical systems theory is the Lyapunov exponent. For soils the Lyapunov exponents are λq and λp and are both negative. They must be because negative Lyapunov exponents are characteristic of dissipative systems such as is the case for soil shear. Thermodynamically speaking, dissipative systems are non-conservative systems. A dissipative structure is a dissipative system that has a reproducible steady-state. Natural evolution of the system or artificial means, or a combination of the two can drive the system to this steady-state. Particulate materials such as soils are dissipative structures.
Chaos in Space
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
We know very well that any “dissipative” system abides by the Second Law of Thermodynamics and increases entropy because, in the course of its evolution, it degrades potential and/or kinetic energy into heat. We also know that any “conservative” non-chaotic system does not produce entropy because the sum of its kinetic and potential energies remains constant when it evolves. Non-chaotic Hamiltonian systems are theaters of reversible transformations. But, what about the entropy change in the evolution of a chaotic “conservative” system (Baranger 2000)? As we already know very well, the initial condition of the system can only be defined with a finite degree of accuracy. All the equally likely initial points are confined within a portion of the phase space whose volume is Vi=∑k=1Nink, where Ni is the total number of points contained in Vi. The initial uncertainty19 we have about the state of the system is [] Hi=−∑k=1Nipklog2pk=log2(Vi)
Three-dimensional soliton-like distortions in flexoelectric nematic liquid crystals: modelling and linear analysis
Published in Liquid Crystals, 2022
Ashley Earls, M. Carme Calderer
Here denotes the Lagrangian of the system, that is, the difference of the density of kinetic energy ), the density of potential energy , and represents the elastic force [28]. The variational statement of this equation is where is the total energy of the system and . That is, the system behaves in such a way that the rate of work is minimised with respect to the generalised velocities. Letting represent the Rayleigh dissipation function, the dissipative forces are given by . The dynamics of a dissipative system is then formulated as the balance of the conservative forces by the dissipative ones, that is, the statement
Output regulation control for switched stochastic delay systems with dissipative property under error-dependent switching
Published in International Journal of Systems Science, 2018
Dissipative property is raised in Willems (1972), which analyses systems based on a generalised notion of energy. A dissipative system means that the consumption of energy is no more than the supply of energy. Under some standard assumptions, the systems are stable once the dissipative property is achieved (Zhao & Hill, 2008). Therefore, the dissipative property has been extensively studied in many areas of control (Wu, Zheng, & Gao, 2013; Sakthivel, Rathika, Santra, & Muslim, 2017; Sakthivel et al., 2016; Shi et al., 2016). In Wu et al. (2013), dissipative property improves the transient performance of sliding mode control for linear switched stochastic systems. Employing the information of disturbance input, the dissipative property is conducive to reduce the conservatism of the results, which is used to control Markovian jump systems (Sakthivel et al., 2017), fuzzy switched systems (Shi et al., 2016) and switched stochastic systems (Sakthivel et al., 2016). However, there are still less result focused on the multiple dissipative parameters for switched systems, even if each subsystem meeting its own dissipative parameters is more reasonable and less conservative than all subsystems satisfying the common ones.
About robust hyperstability and dissipativity of linear time-invariant dynamic systems subject to hyperstable controllers and unstructured delayed state and output disturbances
Published in Cogent Engineering, 2018
Generally speaking, dissipative systems are open systems which interchange energy and matter with the environment. In the framework of Control Theory, those systems satisfy the dissipation inequality in terms that the time-derivative of the storage function is less than or equal to the supply rate given by the instantaneous input-output power. Dissipative is related to stability. Hyperstable closed-loop systems are systems which are stable not under just one stabilizing controller but for any member of a family satisfying a Popov´s- type inequality. This allows the achievement of the stability property under certain tolerance to controller parametrical uncertainties among the various control elements got from a production chain independently of each particular controller device. The hyperstability theory is relevant, for instance, in military and aeronautical targeting applications in which high precision is required in spite of potential dispersion in the fabrication procedure of the control devices.