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Visual representation used in non-linear control system analysis
system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being
Phase_plane
Space of all possible states that a system can take
space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or
Phase_space
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
Shape of the Moon's sunlit portion as viewed from Earth
A lunar phase or Moon phase is the apparent shape of the Moon's day and night phases of the lunar day as viewed from afar. Because the Moon is tidally
Lunar_phase
Curves on which differential equations are zero
they are arbitrary curves. Nullclines are useful for visualization in phase plane plot analysis. Nullclines split the plot into regions of potentially
Nullcline
Elapsed fraction of a cycle of a periodic function
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F {\displaystyle F} of some real variable t {\displaystyle
Phase_(waves)
Convention in engineering geometry
output of E-Plane Tee is 180° out of phase where the output of H-plane Tee is in phase. For a linearly-polarized antenna, this is the plane containing
E-plane_and_H-plane
Array of antennas creating a steerable beam
are conveniently small. Phased arrays were originally invented for use in military radar systems, to detect fast moving planes and missiles, but are now
Phased_array
Algorithm for phase retrieval
iterative phase retrieval algorithm for retrieving the phase of a complex-valued wavefront from two intensity measurements acquired in two different planes. Typically
Gerchberg–Saxton_algorithm
Type of avalanche
Interaction between velocity and effective density in turbidity flow; phase-plane analysis, with criteria for autosuspension. Marine Geology 31:59--99
Powder_snow_avalanche
Type of complex number
{\displaystyle z^{*}z=1.} Multiplying the equation of a plane wave Aei(k·r − ωt) by a phase factor eiθ shifts the phase of the wave by θ: e i θ A e i ( k ⋅ r − ω t
Phase_factor
Biological neuron model
bifurcation parameter and φ, gCa, V3, V4 as secondary parameters for phase plane analysis. Possible Bifurcations Computational neuroscience Biological
Morris–Lecar_model
Diagram used to analyze autonomous ordinary differential equations
differential calculus Phase plane, 2-dimensional form Phase space, n {\displaystyle n} -dimensional form Equilibria and the Phase Line Archived 2024-02-02
Phase_line_(mathematics)
In control theory, when an LTI system and its inverse are causal and stable
side (s-plane imaginary line) or outside (z-plane unit circle) of the complex plane lead to unstable systems, only the class of minimum-phase systems
Minimum_phase
Type of plane wave
sinusoidal plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. It
Sinusoidal_plane_wave
System where changes of output are not proportional to changes of input
qualitative analysis of nonlinear ordinary differential equations include: Phase plane or phase portrait methods Examination of any conserved quantities, especially
Nonlinear_system
Method for processing electrocardiograms
the scalar ECG-signal z(t) in any of the leads to its mapping on the phase plane with the coordinates z(t), dz/dt, where dz/dt is the rate of change in
Fasegraphy
Chart used to show conditions at which physical phases of a substance occur
equilibria a three-dimensional phase diagram is required. Often such a diagram is drawn with the composition as a horizontal plane and the temperature on an
Phase_diagram
Wave shaped like the sine function
some phase is chosen as a zero reference, a sine wave of arbitrary phase can be written as the linear combination of two sine waves with phases of zero
Sine_wave
Property of waves that can oscillate with more than one orientation
just consider coherent plane waves; these are sinusoidal waves of one particular direction (or wavevector), frequency, phase, and polarization state
Polarization_(waves)
Apparent path of the Sun on the celestial sphere
The ecliptic or ecliptic plane is the apparent path of the Sun on the celestial sphere, resulting from Earth's orbit around the Sun. It was a central
Ecliptic
, the Riemann sphere), and we call S {\displaystyle S} the phase plane or dynamical plane. One possible asymptotic behavior for a point z 0 {\displaystyle
Siegel_disc
Holography of electron waves
and it interferes with the unscattered wave (reference wave) the detector plane. The spatial coherence in the in-line scheme is defined by the size of the
Electron_holography
SI derived unit of angle
ISBN 978-92-822-2272-0: "The radian is the coherent unit for plane angle. ... The radian is also the coherent unit for phase angle." Ocean Optics Protocols for Satellite
Radian
Geometric representation of the complex numbers
plane represent real numbers: the radian measure of the physical angle (i.e. the number of radians in the angle). In particular, the argument (phase)
Complex_plane
Type of wave propagating in 3 dimensions
\varphi } is its "phase shift". A true plane wave cannot physically exist, because it would have to fill all space. Nevertheless, the plane wave model is
Plane_wave
Oscillating dynamical system with nonlinear damping
Van der Pol oscillator. Such a Hamiltonian also connects the geometric phase of the limit cycle system having time dependent parameters with the Hannay
Van_der_Pol_oscillator
Mathematical model of the interactions between neuronal populations
neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal
Wilson–Cowan_model
Control theory for nonlinear or time-variant systems
nonlinear feedback systems (see, e.g., and ): Describing function method Phase plane method Lyapunov stability analysis Singular perturbation method The Popov
Nonlinear_control
Phase of a cycle
mechanics, the geometric phase (also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase) is a phase difference acquired over the
Geometric_phase
Lunar phase when the Moon is fully illuminated
The full moon is the lunar phase when the Moon appears fully illuminated from Earth's perspective. This occurs when Earth is located between the Sun and
Full_moon
Concept in geometry and physics
plane of a revolving body is the geometric plane in which its orbit lies. Three non-collinear points in space suffice to determine an orbital plane.
Orbital_plane
Complex number representing a particular sine wave
complex plane. It is sometimes convenient to refer to the entire function as a phasor, as we do in the next section. Multiplication of the phasor A e i
Phasor
Locus of points at equal phase in a wave
time-varying wave field is the set (locus) of all points having the same phase. The term is generally meaningful only for fields that, at each point, vary
Wavefront
Optical microscopy technique
the image plane. First, the background light is phase-shifted by −90° by passing it through a phase-shift ring, which eliminates the phase difference
Phase-contrast_microscopy
States of matter for water as a solid
rise to different phases of ice, which have varying properties and molecular geometries. Currently, twenty-two crystalline phases have been observed
Phases_of_ice
Type of foot movement
during foot landing while running or walking. Composed of three cardinal plane components: subtalar eversion, ankle dorsiflexion, and forefoot abduction
Pronation_of_the_foot
Toy aircraft made of folded paper
A paper plane (also known as a paper airplane or paper dart in American English, or paper aeroplane in British English) is a toy aircraft, usually a glider
Paper_plane
Concept in mathematics
systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional
Autonomous system (mathematics)
Autonomous_system_(mathematics)
Optical prism
each reflection introduces a phase difference of 45° between the components polarized parallel and perpendicular to the plane of reflection. For a given
Fresnel_rhomb
Overview article
of the plane waves used to create it, it can be shown that one of the diffracted waves is a re-construction of the other plane wave. When a plane wave is
Physics_of_optical_holography
Light microscope accessory
it allows observation of the back focal plane of the objective lens and its conjugated focal planes. The phase telescope/Bertrand lens is inserted into
Phase_telescope
1988 attempted aircraft hijacking
Goiânia and Belo Horizonte. In the final phase of the flight, between Belo Horizonte and Rio de Janeiro, the plane was hijacked and diverted to Brasília
VASP_Flight_375
Fundamental principle of physics
electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as
Superposition_principle
Beam focusing, typically horn-fed planar array of unit cells
backed by a ground plane, and the incident wave reflects off them towards the direction of the beam, but each cell adds a different phase delay to the reflected
Reflectarray_antenna
Ability of lipid molecules to move within a bilayer relative to temperature
constrained to the two dimensional plane of the membrane, but in liquid phase bilayers the molecules diffuse freely within this plane. Thus, in a liquid bilayer
Lipid_bilayer_phase_behavior
American experimental aircraft
The Vertical Take-Off and Landing Experimental Aircraft (VTOL X-Plane) program was an American research project sponsored by the Defense Advanced Research
VTOL_X-Plane
Single-stage solid-fueled booster rocket
algorithm to provide stability whilst the Orbus 21 was burning, while a phase-plane controller was used to manage its reaction-control system. Only two Transfer
Transfer_Orbit_Stage
Method of digital signal processing
calculate amplitude and phase. Together with a vector of one crystallographic plane depicted in the image, the amplitude and phase can be used to generate
Geometric_phase_analysis
Signal representation
amplitude modulation or phase-shift keying. It displays the signal as a two-dimensional xy-plane scatter diagram in the complex plane at symbol sampling instants
Constellation_diagram
Limiting set in dynamical systems
describes the evolution of a free particle in one dimension then the phase space is the plane R 2 {\displaystyle \mathbb {R} ^{2}} with coordinates ( x , v )
Attractor
Dynamic disturbance in a medium or field
finite domains. A plane wave is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific
Wave
permits the expression of the pupil-plane phase aberrations φ {\displaystyle \varphi } to the image plane Fourier phase as a linear transformation described
Kernel-phase
Terrorist attack planned for January 1995
plot (/boʊˈdʒɪŋkə/ boh-JING-kə; Arabic: بوجينكا) was a large-scale, three-phase terrorist attack planned – but never executed – by Ramzi Yousef and Khalid
Bojinka_plot
direction (i.e. in it its own plane), the phase of one beam increases, while that of the other decreases, so that the relative phase of the two beams changes
Electronic speckle pattern interferometry
Electronic_speckle_pattern_interferometry
Rate at which the phase of the wave propagates in space
The phase velocity of a wave is the speed of any wavefront, a surface of constant phase. This is the velocity at which the phase of any constant-frequency
Phase_velocity
Ordered arrangement of atoms, ions, or molecules in a crystalline material
Frank–Kasper phases Geometrical crystallography before X-rays Hermann–Mauguin notation – a notation to represent symmetry in point groups, plane groups and
Crystal_structure
Study of classical optics using Fourier transforms
wave is far from its sources, it is locally tangent to a planar phase front (a single plane wave out of the infinite spectrum), which is transverse to the
Fourier_optics
History of research by Augustin-Jean Fresnel
the prescribed angle, they would no longer have the same phase in a common perpendicular plane, and every vibration would be cancelled by a nearby vibration
Fresnel's_physical_optics
Proposed Orbital Sciences Corporation spaceplane
forward by Orbital Sciences Corporation in late 2010 as part of the second phase of NASA's Commercial Crew Development (CCDev) program. The Prometheus design
Prometheus_(spacecraft)
Analysis of multiaxial stresses and strains
Fatemi, A.; Socie, D. F. (1988). "A Critical Plane Approach to Multiaxial Fatigue Damage Including Out-Of-Phase Loading". Fatigue & Fracture of Engineering
Critical_plane_analysis
Mathematical function in general imaging
HRTEM comes from interference in the image plane between the phases of scattered electron waves with the phase of the transmitted electron wave. Complex
Contrast_transfer_function
Geological feature resulting from the movement of bed material by fluid flow
diagrammatically: "Lower plane bed" refers to the flat configuration the bed of a river that is produced in via low rates of sediment transport. "Upper plane bed" features
Bedform
Far-field diffraction
is effectively a plane wave so that the phase of the light at each point on the aperture is the same. At a sufficiently distant plane of observation from
Fraunhofer_diffraction
with a proportional-integral function and rotational commands with a phase plane function. The application then combined these inputs into a single command
Seeker_(spacecraft)
Brand of a system on a chip
compensation and frame rate conversion technology based on enhanced phase-plane correlation technology, which converts 24 or 60 Hz input video signals
Xilleon
Short "burst" or "envelope" of restricted wave action that travels as a unit
carrier waves, phase velocity, and group velocity – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first
Wave_packet
Measurement method using interference of waves
Measure Out-Of-Plane Displacement" (PDF). Engineering Mechanics. 14 (1/2): 37–44. Moustafa, N. A.; Hendawi, N. (2003). "Comparative Phase-Shifting Digital
Interferometry
Wayback Machine Description of strange attractors using invariants of phase-plane Skarya Archived 2015-12-22 at the Wayback Machine Van der Pol Oscillator
List_of_chaotic_maps
Resolution-improving photomask
Phase-shift masks are photomasks that take advantage of the interference generated by phase differences to improve image resolution in photolithography
Phase-shift_mask
Theory of traffic flow
when Kerner's F → S → J phase transitions occurring in real traffic (Figure 9 (a)) are presented in the speed-density plane (Figure 9 (b)) (or speed-flow
Three-phase_traffic_theory
1982 studio album by Crispy Ambulance
Plateau Phase" – 5:50 "Bardo Plane" – 3:32 "Chill" – 4:47 "Federation" –3:44 "Simon's Ghost" –3:42 The CD, first issued as Comprising : The Plateau Phase, Live
The_Plateau_Phase
Sweeping audio effect
Phaser on strings synth A short, unprocessed string sample, followed by a few different phasing effects on the same sample Problems playing this file
Phaser_(effect)
Search for a missing Boeing 777 in the southern Indian Ocean
Retrieved 6 May 2014. Donnison, Jon (28 April 2014). "Missing plane: Search enters 'new phase'" (text, images & video). BBC News. Archived from the original
Search for Malaysia Airlines Flight 370
Search_for_Malaysia_Airlines_Flight_370
Diagram showing the singularities of a given control system's transfer function
Region of convergence (ROC) Minimum phase / non minimum phase A pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer
Pole–zero_plot
Phenomenon resulting from the superposition of two waves
lower amplitude (destructive interference) if the two waves are in phase or out of phase, respectively. Interference effects can be observed with all types
Wave_interference
Crankshaft with throws extending in two planes
The crossplane or cross-plane is a crankshaft design for piston engines with a 90° angle (phase in crank rotation) between the crank throws. The crossplane
Crossplane
Flying for a long time over some small region
and aviation, loiter is the phase of flight consisting of flying over some small region. In general aviation, the loiter phase generally occurs at the end
Loiter_(aeronautics)
Type of data encoding
This shows the points in the complex plane where, in this context, the real and imaginary axes are termed the in-phase and quadrature axes respectively due
Phase-shift_keying
Period when eclipses can occur
orbital plane intersects with the Earth's orbital plane – align with the Sun and Earth, such that a solar eclipse is formed during the new moon phase and
Eclipse_season
mathematically characterized by biophysical models of calcium signaling. Phase plane and bifurcation analysis of these models, can reveal in fact how frequency
Calcium_encoding
Type of radio-frequency coupler
injected into the H-plane port will be divided equally between ports 1 and 2, and will be in phase. A signal injected into the E-plane port will also be
Magic_tee
Interference phenomenon of waves
the "half-plane problem", originally solved by Arnold Sommerfeld using a plane wave spectrum formulation. A generalization of the half-plane problem is
Diffraction
American scientist, scientific instrument-maker, and artist
Jovian Decametric Radio Emission Occurrence Probabilities in the CML-Io Phase Plane, American Geophysical Union/NASA conference poster Shing F. Fung, Todd
Thomas_Ashcraft
Type of antenna
passive phased array, is an antenna in which the beam of radio waves can be electronically steered to point in different directions (that is, a phased array
Passive electronically scanned array
Passive_electronically_scanned_array
Fourier transform of a real-space lattice, important in solid-state physics
R n {\displaystyle \mathbf {R} _{n}} . Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of 2 π {\displaystyle
Reciprocal_lattice
Complete reflection of a wave
accompanied by a non-trivial phase shift (not just zero or 180°) for each component of polarization (perpendicular or parallel to the plane of incidence), and the
Total_internal_reflection
2007 suborbital spaceplane concept
is shut down and the plane continues to climb up to a maximum altitude of 100 km. This is the weightlessness phase. Then the plane gets down to 15 km at
Airbus Defence and Space Spaceplane
Airbus_Defence_and_Space_Spaceplane
Airplane crash in Milan, Italy
found that the pilot was likely unable to "adequately manage the final phase of the flight in the presence of technical, operational, and environmental
2002 Pirelli Tower airplane crash
2002_Pirelli_Tower_airplane_crash
Statistical mechanics model for phase transitions
known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in large physical systems in the thermodynamic limit based on
Lee–Yang_theory
Method of specimen illumination used in optical microscopy
back focal plane of the objective (e.g. the phase ring for phase contrast microscopy) and at the condenser diaphragm (e.g. the annulus for phase contrast
Köhler_illumination
Equations of light transmission and reflection
quantity. The electric field for a uniform plane sine wave will then be represented by the location-dependent phasor For fields of that form, Faraday's law
Fresnel_equations
Imaging mode of electron microscopes
distribution in the image plane, similar to a "classic" light microscope. For disambiguation, the technique is also sometimes referred to as phase contrast transmission
High-resolution transmission electron microscopy
High-resolution_transmission_electron_microscopy
American musician
Andrzejewski (born McIlwee on March 11, 1989), often known by the stage name Wicca Phase Springs Eternal (WPSE), is an American musician from Scranton, Pennsylvania
Wicca_Phase_Springs_Eternal
Number with a real and an imaginary part
This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers
Complex_number
Polarization state
changing phase angle. Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is
Circular_polarization
Optical phenomenon
of plane wave components of a finite-sized beam undergo different phase shifts. A finite-width beam can be expressed as a superposition of plane waves
Goos–Hänchen_effect
Part of planet seen to reflect sunlight
A planetary phase is a certain portion of a planet's area that reflects sunlight as viewed from a given vantage point, as well as the period of time during
Planetary_phase
undertaken to launch paper planes from the stratosphere or higher. The Guinness World Record for the highest altitude paper plane launch is 35,043 metres
Paper planes launched from space
Paper_planes_launched_from_space
Device to determine relative phase shift
Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light
Mach–Zehnder_interferometer
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