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In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. [1] Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.
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 ...
The Chetaev instability theorem for dynamical systems states that if there exists, for the system ˙ = with an equilibrium point at the origin, a continuously differentiable function V(x) such that the origin is a boundary point of the set G = { x ∣ V ( x ) > 0 } {\displaystyle G=\{\mathbf {x} \mid V(\mathbf {x} )>0\}} ;
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does ...
Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium.
Nonlinear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen.
Structural stability of the system provides a justification for applying the qualitative theory of dynamical systems to analysis of concrete physical systems. The idea of such qualitative analysis goes back to the work of Henri Poincaré on the three-body problem in celestial mechanics.
In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable.Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the steady state), and is unstable if it goes further and further away from any state, without being bounded.