# Category: Stability theory

Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-
Control-Lyapunov function
In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical
Equilibrium point
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability o
Exponential stability
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
Linear stability
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 linearizatio
Popov criterion
In nonlinear control and stability theory, the Popov criterion is a stability criterion by Vasile M. Popov for the absolute stability of a class of nonlinear systems whose nonlinearity must satisfy an
Chetaev instability theorem
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 1. * t
Lyapunov–Malkin theorem
The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and ) is a mathematical theorem detailing nonlinear stability of systems.
Marginal stability
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
Multidimensional system
In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables. Importan
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the S
Autonomous convergence theorem
In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system.
Orbital stability
In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form is said to be orbitally stable if any solution with the initial data sufficiently close
Resistive ballooning mode
The resistive ballooning mode (RBM) is an instability occurring in magnetized plasmas, particularly in magnetic confinement devices such as tokamaks, when the pressure gradient is opposite to the effe
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. L
Olech theorem
In dynamical systems theory, the Olech theorem establishes sufficient conditions for global asymptotic stability of a two-equation system of non-linear differential equations. The result was establish
Jury stability criterion
In signal processing and control theory, the Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic po
Butterfly effect
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a la
Kalman–Yakubovich–Popov lemma
The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controlla
Hyperstability
In stability theory, hyperstability is a property of a system that requires the state vector to remain bounded if the inputs are restricted to belonging to a subset of the set of all possible inputs.
Ballooning instability
The ballooning instability (a.k.a. ballooning mode instability) is a type of internal pressure-driven plasma instability usually seen in tokamak fusion power reactors or in space plasmas. It is import
BIBO stability
In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for signals and systems that take inputs. If a system is BIBO stable, then the
Stable polynomial
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or *
Peixoto's theorem
In the theory of dynamical systems, Peixoto theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may
Stability criterion
In control theory, and especially stability theory, a stability criterion establishes when a system is stable. A number of stability criteria are in common use: * Circle criterion * Jury stability c
Markus–Yamabe conjecture
In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptoti
Hyperbolic equilibrium point
In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimens
Circle criterion
In nonlinear control and stability theory, the circle criterion is a stability criterion for nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion for
Briggs–Bers criterion
In stability theory, the Briggs–Bers criterion is a criterion for determining whether the trivial solution to a linear partial differential equation with constant coefficients is stable, or . This is
Firehose instability
The firehose instability (or hose-pipe instability) is a dynamical instability of thin or elongated galaxies. The instability causes the galaxy to buckle or bend in a direction perpendicular to its lo
Routh–Hurwitz stability criterion
In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system
Vakhitov–Kolokolov stability criterion
The Vakhitov–Kolokolov stability criterion is a condition for linear stability (sometimes called spectral stability) of solitary wave solutions to a wide class of Hamiltonian systems, named after Sovi
Lagrange stability
Lagrange stability is a concept in the stability theory of dynamical systems, named after Joseph-Louis Lagrange. For any point in the state space, in a real continuous dynamical system , where is , th
Instability
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stab
Liénard–Chipart criterion
In control system theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed by A. Liénard and M. H. Chipart. This criterion has a com
Comparison function
In applied mathematics, comparison functions are several classes of continuous functions, which are used in stability theory to characterize the stability properties of control systems as Lyapunov sta
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which
Stability theory
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 equat
LaSalle's invariance principle
LaSalle's invariance principle (also known as the invariance principle, Barbashin-Krasovskii-LaSalle principle, or Krasovskii-LaSalle principle) is a criterion for the asymptotic stability of an auton
Massera's lemma
In stability theory and nonlinear control, Massera's lemma, named after José Luis Massera, deals with the construction of the Lyapunov function to prove the stability of a dynamical system. The lemma
Derrick's theorem
Derrick's theorem is an argument by physicist G.H. Derrickwhich shows that stationary localized solutions to a or nonlinear Klein–Gordon equationin spatial dimensions three and higher are unstable.
Bistritz stability criterion
In signal processing and control theory, the Bistritz criterion is a simple method to determine whether a discrete linear time invariant (LTI) system is stable proposed by . Stability of a discrete LT
Plasma stability
The stability of a plasma is an important consideration in the study of plasma physics. When a system containing a plasma is at equilibrium, it is possible for certain parts of the plasma to be distur