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Type of mathematical system
Linear dynamical systems are dynamical systems whose evolution functions are linear. While dynamical systems, in general, do not have closed-form solutions
Linear_dynamical_system
Mathematical model of the time dependence of a point in space
qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers
Dynamical_system
Graphical method of determining the stability of a dynamical system
determining the stability of a linear dynamical system. Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly
Nyquist_stability_criterion
In control theory, visible state of a system
Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems. A dynamical system designed to estimate the state of a system from measurements of the outputs
Observability
Area of mathematics
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations
Dynamical_systems_theory
Time behavior of a system controlled by Heaviside step functions
the overall system response. Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its
Step_response
System where changes of output are not proportional to changes of input
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well
Nonlinear_system
Finding linear approximation of function at given point
In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential
Linearization
Test for whether a polynomial's roots lie in the left-half complex plane
important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable, linear system has roots
Routh–Hurwitz_theorem
Field of mathematics and science based on non-linear systems and initial conditions
study. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought
Chaos_theory
linear dynamical system under the addition of nonlinearities. SSM theory provides conditions for when invariant properties of eigenspaces of a linear
Spectral_submanifold
Square matrix whose off-diagonal entries are nonnegative
stability analysis of time delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of
Metzler_matrix
dynamical system and differential equation topics. Deterministic system (mathematics) Linear system Partial differential equation Dynamical systems and
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Equation from stability analysis
Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. In particular, the discrete-time Lyapunov equation (also known
Lyapunov_equation
systems and in a data-driven robust linear matrix inequality-based model predictive control scheme. Finsler's lemma can be used to give novel linear matrix
Finsler's_lemma
Physical system satisfying the superposition principle
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features
Linear_system
Dynamic system property
also be specified, the dynamical system is often referred to as being strongly controllable. Consider the continuous linear system 𝐱˙(t)=A(t)𝐱(t)+B(t
Controllability
Pattern in control theory
A weighting pattern for a linear dynamical system describes the relationship between an input u {\displaystyle u} and output y {\displaystyle y} . Given
Weighting_pattern
Theorem in dynamical system mathematics
study of dynamical systems, the Hartman–Grobman theorem or linearization theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood
Hartman–Grobman_theorem
Mathematical operation
have a number of properties that make them useful for analysing linear dynamical systems. There is an integral formula for the inverse Laplace transform
Inverse_Laplace_transform
Statistical Markov model
those where the Markov process over hidden variables is a linear dynamical system, with a linear relationship among related variables and where all hidden
Hidden_Markov_model
Thermodynamically open system which is not in equilibrium
dynamical system, the role of Lyapunov functions. Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear
Dissipative_system
Branch of mathematical biology
neuroscience that dynamical systems encompasses. In 2007, a canonical text book was written by Eugene Izhikivech called Dynamical Systems in Neuroscience
Dynamical_neuroscience
Topics referred to by the same term
through which electromagnetic waves propagate Excitable medium, a non-linear dynamic system which has the capacity to propagate a wave Data storage medium,
Medium
Linear optimal control technique
concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and
Linear–quadratic_regulator
Linear optimal control technique
control applies to both linear time-invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that
Linear–quadratic–Gaussian control
Linear–quadratic–Gaussian_control
Part of mathematics that addresses the stability of solutions
stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation
Stability_theory
Family of second-order differential equations
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a type of second-order ordinary
Liénard_equation
Formulas to determine the energy balance of a nonlinear wave
since been found to describe systems in non-linear optics, fluid mechanics and the theory of non-linear dynamical systems, as they provide a pair of invariants
Manley–Rowe_relations
Computer science concept
can happen statically (at compile time), dynamically (at runtime), or as a combination of both. Type systems have other purposes as well, such as expressing
Type_system
Dimensionality reduction algorithm
These snapshots are assumed to be related via a linear mapping that defines a linear dynamical system v i + 1 = A v i , {\displaystyle v_{i+1}=Av_{i}
Dynamic_mode_decomposition
Family of type systems based on substructural logic
it was introduced. Linear types correspond to linear logic and ensure that objects are used exactly once. This allows the system to safely deallocate
Substructural_type_system
conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state. It therefore provides the
Kushner_equation
Visual representation used in non-linear control system analysis
paths, providing a pictorial interpretation of the solution to the dynamical system, as shown next. The phase plane is then first set-up by drawing straight
Phase_plane
Space of all possible states that a system can take
is also known as a "source". A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with
Phase_space
trajectory. A "linear time-invariant differential system" is a dynamical system Σ=(ℝ,ℝq,ℬ) whose behavior ℬ is the solution set of a system of constant coefficient
Behavioral_modeling
Flatness in systems theory is a system property that extends the notion of controllability from linear systems to nonlinear dynamical systems. A system that
Flatness_(systems_theory)
Differential equation that is linear with respect to the unknown function
appear in the equation are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations
Linear_differential_equation
Linear subspace generated from a vector acted on by a power series of a matrix
finding approximate solutions to high-dimensional linear algebra problems. Many linear dynamical system tests in control theory, especially those related
Krylov_subspace
Block diagonal matrix of Jordan blocks
a dynamical system may substantially change as the versal deformation of the Jordan normal form of A(c). The simplest example of a dynamical system is
Jordan_matrix
Technique in mathematical modeling
such problems, for example, in simulations of large-scale dynamical systems and control systems. By a reduction of the model's associated state space dimension
Model_order_reduction
Fixed point that does not have any center manifolds
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.
Hyperbolic_equilibrium_point
Hungarian-American mathematician (1930–2016)
Mexicana. Kalman, R. E. (1963). "Mathematical description of linear dynamical systems". Journal of the Society for Industrial and Applied Mathematics
Rudolf_E._Kálmán
Regulation of nonlinear systems
modelled as parametrized linear systems whose parameters change with their state. In designing feedback controllers for dynamical systems a variety of modern
Linear parameter-varying control
Linear_parameter-varying_control
Type of motion that is approximately periodic
general theory of non-linear dynamic systems. Quasiperiodic motion does not exhibit the butterfly effect characteristic of chaotic systems. In other words,
Quasiperiodic_motion
Limiting set in dynamical systems
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. System
Attractor
Identification of nonlinear systems
S2CID 11396163. M. Abdalmoaty, ‘Learning Stochastic Nonlinear Dynamical Systems Using Non-stationary Linear Predictors’, Licentiate dissertation, Stockholm, Sweden
Nonlinear system identification
Nonlinear_system_identification
Field of mathematics
maps, in particular, piecewise linear self-maps of the interval and the circle. Unlike the theory of smooth dynamical systems, where the main object of study
Topological_dynamics
System that manages the behavior of other systems
feedback to control states or outputs of a dynamical system. Its name comes from the information path in the system: process inputs (e.g., voltage applied
Control_system
Control theory for nonlinear or time-variant systems
of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to
Nonlinear_control
Linear scaling of Barycentric Coordinate Time
basis was Terrestrial Dynamical Time (TDT). The new time scale to supersede ET for planetary ephemerides was to be Barycentric Dynamical Time (TDB). TDB was
Barycentric_Dynamical_Time
Actuator that creates motion in a straight line
A linear actuator is an actuator that creates linear motion (i.e., in a straight line), in contrast to the circular motion of a conventional electric motor
Linear_actuator
Property of certain dynamical systems
integrability is a property of certain dynamical systems, that means very roughly that the solutions of the system are "simple" enough that they can be
Integrable_system
Algorithm that estimates unknowns from a series of measurements over time
exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Regardless
Kalman_filter
American control theorist
Kalman, R.E.; Ho, Y.C.; Narendra, K.S. (1963). "Controllability of linear dynamical systems". Contributions to Differential Equations. 1 (2): 189–213. Ho,
Yu-Chi_Ho
Plot of a dynamical system's trajectories in phase space
a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by
Phase_portrait
Advanced method of process control
Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification. The main advantage
Model_predictive_control
Dynamical system that exhibits continuous and discrete dynamic behavior
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both flow (described by a differential
Hybrid_system
Chinese-American mechanical engineer
C.J. (2026). Single Linear-Bivariate Cubic Systems. Luo, Albert C.J. (2025). Introduction to Infinite-Equilibriums in Dynamical Systemss. Xu, Yeyin;
Albert_C.J._Luo
Method of spatial referencing
Linear referencing, also called linear reference system or linear referencing system (LRS), is a method of spatial referencing over linear or curvilinear
Linear_referencing
Differential equation containing derivatives with respect to only one variable
equations Laplace transform applied to differential equations List of dynamical systems and differential equations topics Matrix differential equation Method
Ordinary differential equation
Ordinary_differential_equation
Dutch mathematician
(1976). "Moduli and canonical forms for linear dynamical systems II: The topological case" (PDF). Mathematical Systems Theory. 10 (1): 363–385. doi:10.1007/BF01683285
Michiel_Hazewinkel
the behavior of biological networks modeled as dynamical systems. In the context of a biological system, bifurcation theory describes how small changes
Biological applications of bifurcation theory
Biological_applications_of_bifurcation_theory
Parameter in differential equations and dynamical systems
In mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time t=0) of a differential equation, difference
Initial_condition
Method for solving continuous operator problems (such as differential equations)
physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination
Galerkin_method
Mathematical relation defining a sequence
combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Output of a dynamic system when given a brief input
response of a linear time-invariant system for all frequencies. Mathematically, how the impulse is described depends on whether the system is modeled in
Impulse_response
Pattern of oscillating motion in a system
the dynamic system in a linear fashion, in which linear superposition of states can be performed. Typical examples include: In a mechanical dynamical system
Normal_mode
Type of functional equation (mathematics)
determine solutions with a given degree of accuracy. The theory of dynamical systems analyzes the qualitative aspects of solutions, such as their average
Differential_equation
Concept in physics
In statistical physics, frenesy is a measure of the dynamical activity of a system's microscopic trajectories under non-equilibrium conditions. Frenesy
Frenesy_(physics)
Polish psychologist
and linear versus non-linear societal transitions.[citation needed] At FAU, he conducts both simulation and experimental research in the Dynamical Social
Andrzej_Nowak_(psychologist)
Dynamic data structure
Linear hashing (LH) is a dynamic data structure which implements a hash table and grows or shrinks one bucket at a time. It was invented by Witold Litwin
Linear_hashing
Determinant of the matrix of first derivatives of a set of functions
and is used in the study of differential equations, where it can show the linear independence of certain sets of solutions. The Wronskian of two differentiable
Wronskian
Algorithm for measuring similarity between temporal sequences
is the tightest known lower bound that can be computed in linear time. Averaging for dynamic time warping is the problem of finding an average sequence
Dynamic_time_warping
especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus𝕋n=S1×S1×⋯×S1⏟n
Linear_flow_on_the_torus
Rate of separation of infinitesimally close trajectories
mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the exponential rate of separation
Lyapunov_exponent
Process of removing noise from a signal
surface non-linearities. Single-ended dynamic range expanders like the Phase Linear Autocorrelator Noise Reduction and Dynamic Range Recovery System (Models
Noise_reduction
Mathematical model of a system in control engineering
dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector. For linear
State-space_representation
Neural networks
addition to this, the energy function is bounded from below the non-linear dynamical equations are guaranteed to converge to a fixed point attractor state
Modern_Hopfield_network
Electric motor that produces a linear force
A linear motor is an electric motor that has had its stator and rotor "unrolled", thus it produces a linear force along its length, rather than a rotational
Linear_motor
Class of numerical techniques
nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations
Finite_difference_method
Linear operator in mathematics
operator has been used in data-driven techniques for dynamical systems in the context of dynamic mode decomposition algorithms, which approximate the
Composition_operator
Continuous-time linear system with only negative real parts
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
Exponential_stability
Branch of engineering and mathematics
the control of dynamical systems. The aim is to develop a model or algorithm governing the application of system inputs to drive the system to a desired
Control_theory
Awarded every year by the American Mathematical Society
ISSN 0021-9223. Kálmán, R. E. (1963). "Mathematical description of linear dynamical systems". J. SIAM Control. 1 (2): 152–192. doi:10.1137/0301010. ISSN 0887-4603
Leroy_P._Steele_Prize
Mathematical map
Energy Systems. 29 (1) e2660. arXiv:1809.03616. doi:10.1002/etep.2660. Brock, William A.; Dechert, W. Davis (1991-01-01), Chapter 40 Non-linear dynamical systems:
Tent_map
Method to solve optimization problems
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical
Linear_programming
Method in numerical analysis
to parameterized autonomous nonlinear dynamical systems of the form: 𝐮′=F(𝐮,λ). However, in an algebraic system there is no distinction between unknowns
Numerical_continuation
State of linear equations
and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the
Linear_stability
Relationship of a signal transducer
Clozeaux,Linear Response Theory, in: E. Antončik et al., Theory of condensed matter, IAEA Vienna, 1968 Christian Maes, "Frenesy: Time-symmetric dynamical activity
Linear_response_function
Mathematician
Clàudia Valls Anglés is a mathematician and an expert in dynamical systems. She is an associate professor in the Instituto Superior Técnico of the University
Clàudia_Valls
of Liège. J. C. Willems. Dissipative dynamical systems – Part I: General theory, Part II: Part II: Linear systems with quadratic supply rates , Volume
Jan_Camiel_Willems
Statistical methods to build mathematical models of dynamical systems from measured data
The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification
System_identification
X-ray computed tomography. Kalman filter: estimate the state of a linear dynamic system from a series of noisy measurements Odds algorithm (Bruss algorithm)
List_of_algorithms
Concept in the analysis of dynamical systems
second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general
Lyapunov_function
Frequencies in coupled oscillators
almost stationary. The frequency response function (FRF) of any linear dynamic system composed of many coupled components will in general display distinctive
Antiresonance
set of linear differential equations that reformulates continuous-time algebraic Riccati equation (CARE). Consider a linear dynamical system x ˙ ( t
Chandrasekhar_algorithm
Branch of mathematics that studies dynamical systems
mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties"
Ergodic_theory
Topics referred to by the same term
conditional probability density of the state of a stochastic non-linear dynamical system Search for "kushner" on Wikipedia. Cedric Kushner Promotions, Ltd
Kushner_(disambiguation)
Branch of ordinary differential equations
time-periodic systems. Formally, Floquet theory is a branch of ordinary differential equations relating to the class of solutions to periodic linear differential
Floquet_theory
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