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Lyapunov stability, the Glossary

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems.^[1]

49 relations: Absolute value, Aleksandr Lyapunov, American Mathematical Society, Asymptotic analysis, Attractor, Autonomous system (mathematics), BIBO stability, Chaos theory, Cold War (1953–1962), Continuous function, Control engineering, Control theory, Convergence proof techniques, Definite matrix, Differential equation, Discrete time and continuous time, Dynamical system, Eigenvalues and eigenvectors, Energy, Exponential stability, Guidance system, Hartman–Grobman theorem, Homoclinic orbit, Input-to-state stability, Jacobian matrix and determinant, Joint spectral radius, LaSalle's invariance principle, Lemma (mathematics), Libration point orbit, Linear system, Lyapunov exponent, Lyapunov function, Lyapunov–Malkin theorem, Markus–Yamabe conjecture, Metric space, Nikolay Gur'yevich Chetaev, Non-autonomous system (mathematics), Nonlinear system, Perturbation theory, Positive-definite function, Proceedings of the National Academy of Sciences of the United States of America, Providence, Rhode Island, Recurrence relation, Springer Science+Business Media, Stability theory, Stable manifold, State-space representation, Structural stability, Van der Pol oscillator.
Lagrangian mechanics
Stability theory
Three-body orbits

Absolute value

In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.

See Lyapunov stability and Absolute value

Aleksandr Lyapunov

Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в,; – 3 November 1918) was a Russian mathematician, mechanician and physicist.

See Lyapunov stability and Aleksandr Lyapunov

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

See Lyapunov stability and American Mathematical Society

Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

See Lyapunov stability and Asymptotic analysis

Attractor

In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system.

See Lyapunov stability and Attractor

Autonomous system (mathematics)

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. Lyapunov stability and autonomous system (mathematics) are dynamical systems.

See Lyapunov stability and Autonomous system (mathematics)

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. Lyapunov stability and BIBO stability are stability theory.

See Lyapunov stability and BIBO stability

Chaos theory

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics.

See Lyapunov stability and Chaos theory

Cold War (1953–1962)

The Cold War (1953–1962) discusses the period within the Cold War from the end of the Korean War in 1953 to the Cuban Missile Crisis in 1962.

See Lyapunov stability and Cold War (1953–1962)

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Lyapunov stability and Continuous function

Control engineering

Control engineering or control systems engineering or Automation engineering (In Some European Countries) is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments.

See Lyapunov stability and Control engineering

Control theory

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. Lyapunov stability and control theory are dynamical systems.

See Lyapunov stability and Control theory

Convergence proof techniques

Convergence proof techniques are canonical components of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.

See Lyapunov stability and Convergence proof techniques

Definite matrix

In mathematics, a symmetric matrix \ M\ with real entries is positive-definite if the real number \ \mathbf^\top M \mathbf\ is positive for every nonzero real column vector \ \mathbf\, where \ \mathbf^\top\ is the row vector transpose of \ \mathbf ~. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \ \mathbf^* M \mathbf\ is positive for every nonzero complex column vector \ \mathbf\, where \ \mathbf^*\ denotes the conjugate transpose of \ \mathbf ~.

See Lyapunov stability and Definite matrix

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives.

See Lyapunov stability and Differential equation

Discrete time and continuous time

In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Lyapunov stability and discrete time and continuous time are dynamical systems.

See Lyapunov stability and Discrete time and continuous time

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Lyapunov stability and dynamical system are dynamical systems.

See Lyapunov stability and Dynamical system

Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation.

See Lyapunov stability and Eigenvalues and eigenvectors

Energy

Energy is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat and light.

See Lyapunov stability and Energy

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 real parts (i.e., in the left half of the complex plane). Lyapunov stability and exponential stability are dynamical systems and stability theory.

See Lyapunov stability and Exponential stability

Guidance system

A guidance system is a virtual or physical device, or a group of devices implementing a controlling the movement of a ship, aircraft, missile, rocket, satellite, or any other moving object.

See Lyapunov stability and Guidance system

Hartman–Grobman theorem

In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point.

See Lyapunov stability and Hartman–Grobman theorem

Homoclinic orbit

In the study of dynamical systems, a homoclinic orbit is a path through phase space which joins a saddle equilibrium point to itself. Lyapunov stability and homoclinic orbit are dynamical systems.

See Lyapunov stability and Homoclinic orbit

Input-to-state stability

Input-to-state stability (ISS)Eduardo D. Sontag.

See Lyapunov stability and Input-to-state stability

Jacobian matrix and determinant

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.

See Lyapunov stability and Jacobian matrix and determinant

Joint spectral radius

In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices.

See Lyapunov stability and Joint spectral radius

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 autonomous (possibly nonlinear) dynamical system. Lyapunov stability and LaSalle's invariance principle are dynamical systems and stability theory.

See Lyapunov stability and LaSalle's invariance principle

Lemma (mathematics)

In mathematics, informal logic and argument mapping, a lemma (lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result.

See Lyapunov stability and Lemma (mathematics)

Libration point orbit

In orbital mechanics, a libration point orbit (LPO) is a quasiperiodic orbit around a Lagrange point.

See Lyapunov stability and Libration point orbit

Linear system

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Lyapunov stability and linear system are dynamical systems.

See Lyapunov stability and Linear system

Lyapunov exponent

In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Lyapunov stability and Lyapunov exponent are dynamical systems.

See Lyapunov stability and Lyapunov exponent

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. Lyapunov stability and Lyapunov function are stability theory.

See Lyapunov stability and Lyapunov function

Lyapunov–Malkin theorem

The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and) is a mathematical theorem detailing stability of nonlinear systems. Lyapunov stability and Lyapunov–Malkin theorem are stability theory.

See Lyapunov stability and Lyapunov–Malkin theorem

Markus–Yamabe conjecture

In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. Lyapunov stability and Markus–Yamabe conjecture are stability theory.

See Lyapunov stability and Markus–Yamabe conjecture

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Lyapunov stability and Metric space

Nikolay Gur'yevich Chetaev

Nikolay Gur'yevich Chetaev (23 November 1902 – 17 October 1959) was a Russian Soviet mechanician and mathematician.

See Lyapunov stability and Nikolay Gur'yevich Chetaev

Non-autonomous system (mathematics)

In mathematics, an autonomous system is a dynamic equation on a smooth manifold. Lyapunov stability and Non-autonomous system (mathematics) are dynamical systems.

See Lyapunov stability and Non-autonomous system (mathematics)

Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Lyapunov stability and nonlinear system are dynamical systems.

See Lyapunov stability and Nonlinear system

Perturbation theory

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.

See Lyapunov stability and Perturbation theory

Positive-definite function

In mathematics, a positive-definite function is, depending on the context, either of two types of function. Lyapunov stability and positive-definite function are dynamical systems.

See Lyapunov stability and Positive-definite function

Proceedings of the National Academy of Sciences of the United States of America

Proceedings of the National Academy of Sciences of the United States of America (often abbreviated PNAS or PNAS USA) is a peer-reviewed multidisciplinary scientific journal.

See Lyapunov stability and Proceedings of the National Academy of Sciences of the United States of America

Providence, Rhode Island

Providence is the capital and most populous city of the U.S. state of Rhode Island.

See Lyapunov stability and Providence, Rhode Island

Recurrence relation

In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms.

See Lyapunov stability and Recurrence relation

Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

See Lyapunov stability and Springer Science+Business Media

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. Lyapunov stability and stability theory are dynamical systems.

See Lyapunov stability and Stability theory

Stable manifold

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. Lyapunov stability and stable manifold are dynamical systems.

See Lyapunov stability and Stable manifold

State-space representation

In control engineering and system identification, a state-space representation is a mathematical model of a physical system specified as a set of input, output, and variables related by first-order differential equations or difference equations.

See Lyapunov stability and State-space representation

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 ''C''1-small perturbations). Lyapunov stability and structural stability are dynamical systems and stability theory.

See Lyapunov stability and Structural stability

Van der Pol oscillator

In the study of dynamical systems, the van der Pol oscillator (named for Dutch physicist Balthasar van der Pol) is a non-conservative, oscillating system with non-linear damping. Lyapunov stability and van der Pol oscillator are dynamical systems.

See Lyapunov stability and Van der Pol oscillator

References

[1] https://en.wikipedia.org/wiki/Lyapunov_stability

Also known as Asymptotic stability, Asymptotically stable, Barbalat's lemma, IsL, Lyapunov Theory, Lyapunov stability criterion, Lyapunov stable, Lyapunov's theory.

Lyapunov stability, the Glossary

Table of Contents

See also

Lagrangian mechanics

Stability theory

Three-body orbits

References