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Plot of a dynamical system's trajectories in phase space
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
Phase_portrait
Space of all possible states that a system can take
composition. 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
Phase_space
Study of sudden qualitative behavior changes caused by small parameter changes
non-hyperbolic at the bifurcation point. The topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of
Bifurcation_theory
Visual representation used in non-linear control system analysis
entire field is the phase portrait, a particular path taken along a flow line (i.e. a path always tangent to the vectors) is a phase path. The flows in
Phase_plane
Rigid pendulum
of the pendulum length a = l / 2 {\displaystyle a=l/2} leads to the phase portrait shown in the figure.[clarification needed] Further increase of the amplitude
Kapitza's_pendulum
decomposition theorem states that every flow of a dynamical system with compact phase portrait admits a decomposition into a chain-recurrent part and a gradient-like
Conley's fundamental theorem of dynamical systems
Conley's_fundamental_theorem_of_dynamical_systems
and most informative to take a geometric approach and draw a phase portrait. A phase portrait is a qualitative sketch of the differential equation's behavior
Biological applications of bifurcation theory
Biological_applications_of_bifurcation_theory
Stochastic differential equation
continually loses energy to the environment, and its time-dependent phase portrait (velocity vs position) corresponds to an inward spiral toward 0 velocity
Langevin_equation
Path between equilibrium points in a phase space
mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins
Heteroclinic_orbit
Initial estimate or framework to the solution of a mathematical problem
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Ansatz
2021–22 group of superhero media
Phase Four of the Marvel Cinematic Universe (MCU) is a group of American superhero films and television series produced by Marvel Studios based on characters
Marvel Cinematic Universe: Phase Four
Marvel_Cinematic_Universe:_Phase_Four
System where changes of output are not proportional to changes of input
analysis of nonlinear ordinary differential equations include: Phase plane or phase portrait methods Examination of any conserved quantities, especially
Nonlinear_system
Branch of differential geometry and differential topology
origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic
Symplectic_geometry
2016–2019 group of superhero films
Phase Three of the Marvel Cinematic Universe (MCU) is a group of American superhero films produced by Marvel Studios based on characters that appear in
Marvel Cinematic Universe: Phase Three
Marvel_Cinematic_Universe:_Phase_Three
Type of constraint on solutions to differential equations
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Dirichlet_boundary_condition
Differential equation containing derivatives with respect to only one variable
behavior of a system of ODEs can be visualized through the use of a phase portrait. Given a differential equation F ( x , y , y ′ , … , y ( n ) ) = 0 {\displaystyle
Ordinary differential equation
Ordinary_differential_equation
Theorem in dynamical system mathematics
geometric picture. In effect, the nonlinear phase portrait near the equilibrium is a thumbnail of the phase portrait of the linearized system. This is the meaning
Hartman–Grobman_theorem
Fundamental trigonometric functions
105. ISBN 978-0-07-338317-0. Howie (2003), p. 23–24. "Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar
Sine_and_cosine
Local bifurcation in which two fixed points of a dynamical system collide and anni
{dy}{dt}}=-y.} As can be seen by the animation obtained by plotting phase portraits by varying the parameter α {\displaystyle \alpha } , When α {\displaystyle
Saddle-node_bifurcation
Repetitive variation of some measure about a central value
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
Oscillation
Influence on an oscillating physical system which reduces or prevents its oscillation
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
Damping
Limiting set in dynamical systems
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
Attractor
Methods of mathematical approximation
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Perturbation_theory
Determinant of the matrix of first derivatives of a set of functions
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Wronskian
Partial differential equations with random force terms and coefficients
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Stochastic partial differential equation
Stochastic_partial_differential_equation
Linearization technique for nonlinear differential systems
because it is centred directly on the trajectory's initial condition. The phase portrait compares second-order Type A approximants constructed around the two
Carleman_linearization
Physical system that responds to a restoring force proportional to displacement
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
Harmonic_oscillator
Portrait of an artist made by that artist
Self-portraits are portraits artists make of themselves. Although self-portraits have been made since the earliest times, it is not until the mid-15th
Self-portrait
Type of problem involving ODEs or PDEs
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Boundary_value_problem
Non-linear second order differential equation and its attractor
successive period doubling. Time traces and phase portraits Some typical examples of the time series and phase portraits of the Duffing equation, showing the
Duffing_equation
Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator.
Lists_of_mathematics_topics
Critical point where a periodic solution arises
{dx}{dt}}=-x+ay+x^{2}y,~~{\frac {dy}{dt}}=b-ay-x^{2}y.} The figure shows a phase portrait illustrating the Hopf bifurcation in the Selkov model. In railway vehicle
Hopf_bifurcation
Property of differential equations describing physical phenomena
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Well-posed_problem
Paradigmatic model
at any time by 2 π l {\textstyle 2\pi l} (this is why stroboscopic phase portraits of the kicked rotator are usually displayed in a single momentum cell
Kicked_rotator
Differential equations involving stochastic processes
where x ∈ X {\displaystyle x\in X} is the position in the system in its phase (or state) space, X {\displaystyle X} , assumed to be a differentiable manifold
Stochastic differential equation
Stochastic_differential_equation
Type of functional equation (mathematics)
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Differential_equation
Generalized function whose value is zero everywhere except at zero
Bibcode:2007JMP....48h4101Y, doi:10.1063/1.2771422 Zhao, Ji-Cheng (2011), Methods for Phase Diagram Determination, Elsevier, ISBN 978-0-08-054996-5 Zee, Anthony (2013)
Dirac_delta_function
Type of differential equation
common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin x + sin
Partial_differential_equation
Numerical method for differential equations
numerical schemes with the same stepsize. For instance, Figure 1 shows the phase portrait of the ODEs d x 1 d t = − 2 x 1 + x 2 + 1 − μ f ( x 1 , λ ) ( 4.10 )
Local_linearization_method
Numerical method for solving physical or engineering problems
Benzerga, A A (2021-06-01). "A variational fast Fourier transform method for phase-transforming materials". Modelling and Simulation in Materials Science and
Finite_element_method
Differential equation that is linear with respect to the unknown function
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Linear_differential_equation
Class of numerical techniques
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Finite_difference_method
Geometric space with six dimensions
exponentiation. Phase space is a space made up of the position and momentum of a particle, which can be plotted together in a phase diagram to highlight
Six-dimensional_space
1916 novel by James Joyce
A Portrait of the Artist as a Young Man is the second book and first novel of Irish writer James Joyce, published in 1916. A Künstlerroman written in a
A Portrait of the Artist as a Young Man
A_Portrait_of_the_Artist_as_a_Young_Man
Bifurcation (disambiguation) Bifurcation diagram Bifurcation theory Phase portrait Rulkov map FitzHugh-Nagumo model The theorem on the continuous dependence
Bifurcation_memory
Camera manufacturer
together. Phase One's product line includes: Phase One XC Phase One XT Phase One XF Phase One 645DF+ (discontinued) Phase One 645DF (discontinued) Phase One
Phase_One_(company)
model of traffic flow theory Allen–Cahn equation in phase separation Cahn–Hilliard equation in phase separation Chemical reaction model Brusselator Oregonator
List of named differential equations
List_of_named_differential_equations
System of equations in mathematics
Periodic Wronskian Abel's identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence
Differential-algebraic system of equations
Differential-algebraic_system_of_equations
Free swinging suspended body
initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis
Pendulum_(mechanics)
Attempt to find a consistent theory of quantum gravity
approximation, arXiv:2210.11356 (2022) Donkin, Ivan; Pawlowski, Jan M. (2012). "The phase diagram of quantum gravity from diffeomorphism-invariant RG-flows". arXiv:1203
Asymptotic_safety
1914–1916 painting by Gustav Klimt
belongs to a group of Klimt's late portraits that depart from the geometric, hard-edged shapes of his 'golden phase' and use East Asian motifs more prominently
Portrait_of_Elisabeth_Lederer
Technique for visualizing complex functions
Elias Wegert (2012). Visual Complex Functions: An Introduction with Phase Portraits. Springer Basel. p. 29. ISBN 9783034801799. Retrieved 6 January 2016
Domain_coloring
Solution method for linear differential equations
function, semiclassically expanded, and then either the amplitude or the phase is taken to be changing slowly. The name is an initialism for Wentzel–Kramers–Brillouin
WKB_approximation
Overview of mechanics based on the least action principle
phase paths, the general solution to the differential equations, is the phase portrait: { ( q ( t ) , p ( t ) ) ∈ R 2 N : t ≥ 0 , t ∈ R } ⊆ P , {\displaystyle
Analytical_mechanics
French portrait painter (1735–1802)
the later phase of the French Revolution. He was appointed premier peintre de la reine (First Painter to the Queen). He drew the last portrait of King Louis
Joseph_Ducreux
Dynamical system
equation: once a strategy becomes extinct there is no way to revive it. Phase portrait solutions for the continuous linear-fitness replicator equation have
Replicator_equation
super-persistent chaotic transients. Intermittency Bifurcation diagram Phase portrait Grebogi, Celso; Ott, Edward; Yorke, James A. (1983). "Crises, sudden
Crisis_(dynamical_systems)
American rock band
adventures. In one tweet, Jennings mentions that The Funeral Portrait is currently in its second phase, named TFP 2.0, revealing that up until 2019 the band
The_Funeral_Portrait
Heidelberg. pp. 503-524 Holmengen, N. And Seip, K.L. 2009. Cycle lengths and phase portrait characteristics as probes for predator-prey interactions: comparing
Knut_Seip
Austrian symbolist painter (1862–1918)
this period include Portrait of Adele Bloch-Bauer I (1907), The Kiss (1907–08), and the Stoclet Frieze (1905–1911). Klimt's golden phase was marked by positive
Gustav_Klimt
Self-supporting non-linear waves in active media
excitation pulses arise spontaneously. It can be clearly seen on the phase portrait of the basic system of equations describing the active medium (see Fig
Autowave
Uniformity of handedness
racemic amino acid derivative continuously stirred, results in a 100% crystal phase of one of the enantiomers because the enantiomeric pair is able to equilibrate
Homochirality
units. The phase lag is ϕ = − Arg ( Q ( i ω ) / P ( i ω ) ) {\displaystyle \phi =-\operatorname {Arg} (Q(i\omega )/P(i\omega ))} . The phase lag has units
Exponential_response_formula
1907 painting by Gustav Klimt
Österreichische Galerie Belvedere. The portrait is the final and most fully representative work of Klimt's golden phase. It was the first of two depictions
Portrait of Adele Bloch-Bauer I
Portrait_of_Adele_Bloch-Bauer_I
1910 painting by Pablo Picasso
Portrait of Ambroise Vollard displays an important period in the evolution of Picasso's artwork, known as Analytic Cubism. Picasso said of this phase
Portrait of Ambroise Vollard (Picasso)
Portrait_of_Ambroise_Vollard_(Picasso)
microscopy Phase distortion Phase factor Phase noise Phase offset modulation Phase plane Phase portrait Phase problem Phase retrieval Phase space Phase space
Index_of_physics_articles_(P)
Painting by Pablo Picasso
Portrait of Gertrude Stein (French: Portrait de Gertrude Stein) is an oil-on-canvas painting of the American writer and art collector Gertrude Stein by
Portrait_of_Gertrude_Stein
Method for numerical differential equations
comprise: the Stefan problem which is modelling a melting material, two-phase flows in porous media, the Richards equation of underground water flow,
Gradient discretisation method
Gradient_discretisation_method
2025 Israel–Hamas ceasefire agreement
Israel and Hamas had reached an agreement to begin the first phase of the deal. Under this phase, all living hostages were to be released in exchange for
Gaza_peace_plan
parameters. The detailed analysis shows that there are 23 different phase portraits for this system, including oscillations, multiplicity of steady states
Grigoriy_Yablonsky
Polish mathematician (1931–2015)
doi:10.1016/0022-0396(66)90064-7. Olech, Czeslaw (1964). "Global phase- portrait of a plane autonomous system". Annales de l'Institut Fourier. 14: 87–97
Czesław_Olech
Most populous city in the United States
replaced Wuhan, China as the global epicenter of the pandemic during the early phase, straining the city's healthcare infrastructure. Through March 2023, New
New_York_City
Song by Kansas
"Portrait (He Knew)" is a song by the American progressive rock band Kansas. It was written by Kerry Livgren and Steve Walsh and was recorded for the band's
Portrait_(He_Knew)
Dutch astronomer (1921–2021)
plotted both components in a diagram, thus creating an experimental phase portrait. The track of the two components went from low to high activity around
Kees_de_Jager
British singer (born 1992)
reported that a portrait of Charli XCX representing her Brat era, photographed by Harley Weir, would be added to the National Portrait Gallery collection
Charli_XCX
American mathematician
Vector Fields (with Philip Holmes), Springer-Verlag, 1983, 453 pp. Phase portraits of planar vector fields: computer proofs, Journal of Experimental Mathematics
John_Guckenheimer
American singer-songwriter (born 1941)
Portrait Press Release" (PDF). expectingrain.com. July 16, 2013. Retrieved September 12, 2013. Jurek, Thom (August 27, 2013). "Another Self Portrait (1969–1971):
Bob_Dylan
American actress and model (born 1998)
September 16, 2025. Beresford, Trilby (January 3, 2020). "'Parasite,' 'Portrait of a Lady on Fire,' 'Pain and Glory' Lead Dorian Award Nominations". The
Hunter_Schafer
First-century Jewish preacher and religious leader
scholar to construct a portrait of Jesus that can be considered historically valid beyond the basic elements of his life. The portraits of Jesus constructed
Jesus
Darwinian evolution of self-replicating entities within framework of physical chemistry
1098/rstb.2010.0076. PMC 2880120. PMID 20478889. S Tseng Z (2008). Phase Portraits of Linear Systems. Eigen M., McCaskill J., Schuster P. (1989). "The
Quasispecies_model
South African artist and visual activist (born 1972)
contemporary art in (Germany), for a series of portraits of lesbians and transgender participants titled: Faces and Phases. The photos were also exhibited at Stedelijk
Zanele_Muholi
2021 Marvel Studios film
18, 2021, and was released in the United States on November 5 as part of Phase Four of the MCU. It grossed $402.1 million worldwide and received several
Eternals_(film)
American political party
tolerance for violence and conspiracy theories, he says, present "a chilling portrait of how far down the extremist tracks rank-and-file Republicans have gone
Republican Party (United States)
Republican_Party_(United_States)
Prime Minister of Singapore from 1959 to 1990
Kuan Yew at Wikimedia Commons Appearances on C-SPAN Portraits of Lee Kuan Yew at the National Portrait Gallery, London Quotations related to Lee Kuan Yew
Lee_Kuan_Yew
Graffiti artist, political activist and painter
(theater) Ecofeminism Eco-socialism Environmentalism Food Not Bombs Fossil fuel phase-out Green anarchism Green left Green politics Hippie Libertarian socialism
Banksy
Country in North Africa
Retrieved 5 October 2025. Darwish, Mustafa (1998). Dream Makers on the Nile: A Portrait of Egyptian Cinema. Cairo: American University in Cairo Press. pp. 12–13
Egypt
features. The Age of Enlightenment heralded a new phase in the development of the literary portrait. It invaded literature and even contaminated music
Portrait_(literature)
Rapid transit system in India serving Delhi NCR
December 2004. The network was developed in phases. Phase I was completed by 2006, followed by Phase II in 2011. Phase III was mostly complete by 2021, except
Delhi_Metro
reimbursed the slaveowners. According to Herbert Aptheker, "there were few phases of ante-bellum Southern life and history that were not in some way influenced
Slavery_in_the_United_States
Dictator of Germany from 1933 to 1945
ISBN 978-0-393-00280-5. Hamann, Brigitte (2010) [1999]. Hitler's Vienna: A Portrait of the Tyrant as a Young Man. Trans. Thomas Thornton. London; New York:
Adolf_Hitler
Israeli far-right politician (born 1976)
live television in 1995 shortly before his assassination; having had a portrait in his living room of Baruch Goldstein, a Jewish terrorist and mass murderer;
Itamar_Ben-Gvir
German-born theoretical physicist (1879–1955)
laid the groundwork for Bose–Einstein statistics. For much of the last phase of his academic life, Einstein worked on two endeavors that ultimately proved
Albert_Einstein
American actor (born 1939)
one more in 2013). A 2009 guest appearance on Saving Grace began a new phase of Abraham's career, wherein he became gradually more prolific onscreen
F._Murray_Abraham
British prince (born 1984)
official website of the British royal family Portraits of Prince Harry, Duke of Sussex at the National Portrait Gallery, London Prince Harry at IMDb Appearances
Prince_Harry,_Duke_of_Sussex
Prime Minister of Israel (1996–1999; 2009–2021; 2022–present)
shortly before the Israeli cabinet voted 17 to 5 to approve the initial phase of withdrawal from Gaza. Following the withdrawal of Sharon from the Likud
Benjamin_Netanyahu
1955–1975 war in Southeast Asia
three-point plan to win: Phase 1. Commitment of US and allied forces necessary to halt the losing trend by the end of 1965. Phase 2. US and allied forces
Vietnam_War
American actor (1976–2020)
commencement address at Howard University on May 12, 2018. Chadwick Boseman: Portrait of an Artist List of black Academy Award winners and nominees List of posthumous
Chadwick_Boseman
British racing driver (born 1985)
September 2019. "NPG x131991; Lewis Hamilton – Portrait – National Portrait Gallery". National Portrait Gallery. Retrieved 27 November 2020. "Silverstone
Lewis_Hamilton
Religious site in Jerusalem
Issac, Moadei Hare'iya, pp. 413–15. Levine, Lee I. (2002). Jerusalem: portrait of the city in the Second Temple period (538 BCE – 70 CE) (1st ed.). Philadelphia
Temple_Mount
Highest court of jurisdiction in the U.S
Retrieved May 10, 2025. Buchanan, Larry; Stevens, Matt (October 7, 2022). "The Portrait of Justice". The New York Times. ISSN 0362-4331. Archived from the original
Supreme Court of the United States
Supreme_Court_of_the_United_States
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