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In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts (i.e., in the left half of the complex plane). [1]
The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations.
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. [20] The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of reaction–diffusion systems in spite of large discrepancies e.g. in the local reaction ...
In chemistry, chemical stability is the thermodynamic stability of a chemical system, in particular a chemical compound or a polymer. [ 1 ] Thermodynamic stability occurs when a system is in its lowest energy state , or in chemical equilibrium with its environment.
The simplest non-trivial examples are the exponential growth model/decay (one unstable/stable equilibrium) and the logistic growth model (two equilibria, one stable, one unstable). The phase space of a two-dimensional system is called a phase plane , which occurs in classical mechanics for a single particle moving in one dimension, and where ...
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form / =, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part.
A molecular model is a physical model of an atomistic system that represents molecules and their processes. They play an important role in understanding chemistry and generating and testing hypotheses .
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation , for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature ...