Plot of a dynamical system's trajectories in phase space
Differential equations
Scope
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Continuum mechanics
Chaos theory
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List of named differential equations
Classification
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
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Features
Order
Operator
Notation
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay
Solution
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters
People
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank
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Potential energy and phase portrait of a simple pendulum. Note that the x-axis, being angular, wraps onto itself after every 2π radians.Phase portrait of damped oscillator, with increasing damping strength. The equation of motion is
In mathematics, 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 a different point or curve.
Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called a "sink". The repeller is considered as an unstable point, which 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 dots) and unstable steady states (with circles) in a phase space. The axes are of state variables.
In mathematics, a phaseportrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions...
composition. In mathematics, a phaseportrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions...
non-hyperbolic at the bifurcation point. The topological changes in the phaseportrait of the system can be confined to arbitrarily small neighbourhoods of...
entire field is the phaseportrait, a particular path taken along a flow line (i.e. a path always tangent to the vectors) is a phase path. The flows in...
and most informative to take a geometric approach and draw a phaseportrait. A phaseportrait is a qualitative sketch of the differential equation's behavior...
continually loses energy to the environment, and its time-dependent phaseportrait (velocity vs position) corresponds to an inward spiral toward 0 velocity...
mathematics, in the phaseportrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins...
its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine...
Book Company. ISBN 0-07-100276-6. p. 299, Theorem 15.4 "Why are the phaseportrait of the simple plane pendulum and a domain coloring of sin(z) so similar...
origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic...
ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of...
transients and settle the system into its typical behavior. The subset of the phase space of the dynamical system corresponding to the typical behavior is the...
behavior of a system of ODEs can be visualized through the use of a phaseportrait. Given a differential equation F ( x , y , y ′ , … , y ( n ) ) = 0 {\displaystyle...
the phase of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag...
numerical schemes with the same stepsize. For instance, Figure 1 shows the phaseportrait of the ODEs d x 1 d t = − 2 x 1 + x 2 + 1 − μ f ( x 1 , λ ) ( 4.10 )...
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