Category: Theorems in dynamical systems

Generalized Helmholtz theorem
The generalized Helmholtz theorem is the multi-dimensional generalization of the Helmholtz theorem which is valid only in one dimension. The generalized Helmholtz theorem reads as follows. Let be the
Oseledets theorem
In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by (al
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
No-wandering-domain theorem
In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985. The theorem states that a rational map f : Ĉ → Ĉ with deg(f) ≥ 2 does not have a wa
Lyapunov–Malkin theorem
The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and ) is a mathematical theorem detailing nonlinear stability of systems.
Artstein's theorem
Artstein's theorem states that a nonlinear dynamical system in the control-affine form has a differentiable control-Lyapunov function if and only if it admits a regular stabilizing feedback u(x), that
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.
Maximal ergodic theorem
The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that is a probability space, that is a (possibly noninvertible) measure-preserving transformation,
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
Krylov–Bogolyubov theorem
In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical sys
Stable manifold theorem
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given
Denjoy–Wolff theorem
In mathematics, the Denjoy–Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers i
Denjoy's theorem on rotation number
In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. D
Takens's theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamica
Liouville–Arnold theorem
In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent,Poisson commuting first integrals of
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
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
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
Ratner's theorems
In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner'
Liénard equation
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Mari
Sharkovskii's theorem
In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is
Classification of Fatou components
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
Bendixson–Dulac theorem
In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a function (called the Dulac function) such that the expression has the same sign almost everywhere in a si
Kharitonov's theorem
Kharitonov's theorem is a result used in control theory to assess the stability of a dynamical system when the physical parameters of the system are not known precisely. When the coefficients of the c
Kolmogorov–Arnold–Moser theorem
The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the that arises in the
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distributio
Krener's theorem
In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of of finite-dimensional control systems. It states that any at
Tikhonov's theorem (dynamical systems)
In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics. The theorem is nam
Poincaré recurrence theorem
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuou
Poincaré–Bendixson theorem
In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.