1124:
1303:
1109:
1314:
1044:
929:, and so can be taken as constant. Within the context of a model system in classical mechanics, the phase-space coordinates of the system at any given time are composed of all of the system's dynamic variables. Because of this, it is possible to calculate the state of the system at any given time in the future or the past, through integration of Hamilton's or Lagrange's equations of motion.
29:
844:
positions and momenta (6 dimensions for an idealized monatomic gas), and for more complex molecular systems additional dimensions are required to describe vibrational modes of the molecular bonds, as well as spin around 3 axes. Phase spaces are easier to use when analyzing the behavior of mechanical
1285:
for each degree of freedom), one may integrate over continuous phase space. Such integration essentially consists of two parts: integration of the momentum component of all degrees of freedom (momentum space) and integration of the position component of all degrees of freedom (configuration space).
831:
initial condition. As a whole, the phase diagram represents all that the system can be, and its shape can easily elucidate qualities of the system that might not be obvious otherwise. A phase space may contain a great number of dimensions. For instance, a gas containing many molecules may require a
1587:
as describing part of this phase space. A point in this phase space is correspondingly called a macrostate. There may easily be more than one microstate with the same macrostate. For example, for a fixed temperature, the system could have many dynamic configurations at the microscopic level. When
1257:
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".
1603:
of much larger dimensions than in the second sense. Clearly, many more parameters are required to register every detail of the system down to the molecular or atomic scale than to simply specify, say, the temperature or the pressure of the system.
1520:-dependent quantum corrections, as the conventional commutative multiplication applying in classical mechanics is generalized to the noncommutative star-multiplication characterizing quantum mechanics and underlying its uncertainty principle.
1974:
Klabukov, I.; Tenchurin, T.; Shepelev, A.; Baranovskii, D.; Mamagulashvili, V.; Dyuzheva, T.; Krasilnikova, O.; Balyasin, M.; Lyundup, A.; Krasheninnikov, M.; Sulina, Y.; Gomzyak, V.; Krasheninnikov, S.; Buzin, A.; Zayratyants, G. (2023).
1453:
Expectation values in phase-space quantization are obtained isomorphically to tracing operator observables with the density matrix in
Hilbert space: they are obtained by phase-space integrals of observables, with the
1544:-dimensional phase space describes the dynamic state of every particle in that system, as each particle is associated with 3 position variables and 3 momentum variables. In this sense, as long as the particles are
819:. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a
1197:
860:
of classical systems in phase space (top). The systems are a massive particle in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time.
853:
901:
of configuration space, and in this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local
1880:
1040:
Here the horizontal axis gives the position, and vertical axis the velocity. As the system evolves, its state follows one of the lines (trajectories) on the phase diagram.
1025:, which occurs in classical mechanics for a single particle moving in one dimension, and where the two variables are position and velocity. In this case, a sketch of the
1004:
845:
systems restricted to motion around and along various axes of rotation or translation – e.g. in robotics, like analyzing the range of motion of a
2181:
1286:
Once the phase integral is known, it may be related to the classical partition function by multiplication of a normalization constant representing the number of
1281:(sum over states) known as the phase integral. Instead of summing the Boltzmann factor over discretely spaced energy states (defined by appropriate integer
1536:
contexts, the term "phase space" has two meanings: for one, it is used in the same sense as in classical mechanics. If a thermodynamic system consists of
823:
for the system) through the high-dimensional space. The phase-space trajectory represents the set of states compatible with starting from one particular
1977:"Biomechanical Behaviors and Degradation Properties of Multilayered Polymer Scaffolds: The Phase Space Method for Bile Duct Design and Bioengineering"
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1010:, and the qualitative behaviour of the system being immediately visible from the phase line. The simplest non-trivial examples are the
804:
202:
1399:
But they may alternatively retain their classical interpretation, provided functions of them compose in novel algebraic ways (through
1572:, thus describing the system at a microscopic level is often impractical. This leads to the use of phase space in a different sense.
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for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of
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36:. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)
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585:
239:
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1958:
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In classical statistical mechanics (continuous energies) the concept of phase space provides a classical analog to the
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For simple systems, there may be as few as one or two degrees of freedom. One degree of freedom occurs when one has an
2277:
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of the relevant process. (Other familiar deformations in physics involve the deformation of classical
Newtonian into
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897:, which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is the
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used in this sense, a phase is a region of phase space where the system in question is in, for example, the
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2216:
1442:, a complete and logically autonomous reformulation of quantum mechanics. (Its modern abstractions include
1360:
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22:
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of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a
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2147:
1580:
1080:
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Since there are many more microstates than macrostates, the phase space in the first sense is usually a
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A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable
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Thus, by expressing quantum mechanics in phase space (the same ambit as for classical mechanics), the
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827:, located in the full phase space that represents the set of states compatible with starting from
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Phase portrait of damped oscillator, with increasing damping strength. The equation of motion is
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states of the system, such as pressure, temperature, etc. For instance, one may view the
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317:
156:
123:
114:
2011:
1976:
1824:
1264:(with dots) and unstable steady states (with circles) in a phase space. The axes are of
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of typical trajectories in the phase space. This reveals information such as whether an
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Phase portraits are an invaluable tool in studying dynamical systems. They consist of a
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1617:
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A plot of position and momentum variables as a function of time is sometimes called a
2121:
2102:
2075:
2067:
2062:
2016:
1998:
1954:
1929:
1859:
1613:
1377:
1306:
Illustration of how a phase portrait would be constructed for the motion of a simple
824:
768:
619:
398:
133:
1840:
788:
674:
2652:
2632:
2057:
2047:
2006:
1988:
1917:
1828:
1734:
1616:, the branch of optics devoted to illumination. It is also an important concept in
1513:
1486:
1420:
1347:
1227:
1219:
1215:
1112:
898:
784:
780:
752:
684:
669:
1711:
for information about state space (similar to phase state) in control engineering.
849:
or determining the optimal path to achieve a particular position/momentum result.
2612:
2541:
2052:
2035:
1993:
1944:
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1569:
1291:
1282:
1238:
1116:
624:
540:
67:
2617:
1119:. Note that the x-axis, being angular, wraps onto itself after every 2Ď€ radians.
679:
2607:
2414:
1739:
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1657:
1529:
1364:
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per unit phase space. This normalization constant is simply the inverse of the
1265:
1207:
1101:
1047:
1029:
may give qualitative information about the dynamics of the system, such as the
1026:
776:
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736:
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423:
142:
28:
1921:
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2002:
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1424:
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1393:
1076:
852:
1898:
Curtright, T. L.; Zachos, C. K. (2012). "Quantum
Mechanics in Phase Space".
1317:
Time-series flow in phase space specified by the differential equation of a
1294:
raised to a power equal to the number of degrees of freedom for the system.
2662:
2597:
2511:
2079:
2020:
1717:
for information about state space with discrete states in computer science.
1416:
1353:
1339:
1261:
664:
614:
500:
128:
1108:
2159:
2036:"A phase space model of hemopoiesis and the concept of stem cell renewal"
1714:
1668:
1624:
1575:
The phase space can also refer to the space that is parameterized by the
1313:
1250:
1223:
1203:
1088:
1030:
1022:
942:
846:
816:
72:
2647:
1408:
783:. The concept of phase space was developed in the late 19th century by
689:
1832:
759:
are represented, with each possible state corresponding to one unique
2409:
1950:
1242:
1211:
808:
430:
151:
94:
84:
1973:
1471:(generalization) of classical mechanics, with deformation parameter
2449:
1600:
1318:
1307:
1246:
1084:
772:
1912:
105:
100:
89:
2118:
Galileo
Unbound: A Path Across Life, the Universe and Everything
2099:
Introduction to Modern
Dynamics: Chaos, Networks, Space and Time
1589:
1226:. Each set of initial conditions is represented by a different
1192:{\displaystyle {\ddot {x}}+2\gamma {\dot {x}}+\omega ^{2}x=0.}
32:
Diagram showing the periodic orbit of a mass-spring system in
1593:
1231:
1043:
767:, the phase space usually consists of all possible values of
1512:
Classical expressions, observables, and operations (such as
1523:
1253:
is present for the chosen parameter value. The concept of
1856:
1560:! points, corresponding to all possible exchanges of the
1006:
with the resulting one-dimensional system being called a
1811:
Nolte, D. D. (2010). "The tangled tale of phase space".
1021:
The phase space of a two-dimensional system is called a
912:
1325:
axis corresponds to the pendulum's position, and the
1133:
960:
925:. The local density of points in such systems obeys
1791:
Characteristics in phase space of quantum mechanics
18:
Space of all possible states that a system can take
1465:facilitates recognition of quantum mechanics as a
1191:
998:
2577:List of nonlinear ordinary differential equations
1946:Introduction to Nonimaging Optics, Second Edition
1014:/decay (one unstable/stable equilibrium) and the
921:of systems in this space is studied by classical
2680:
2582:List of nonlinear partial differential equations
1415:on phase space, and conversely, as specified by
1407:of quantum mechanics. Every quantum mechanical
1897:
1501:; or the deformation of Newtonian gravity into
1435:(1949), these completed the foundations of the
2572:List of linear ordinary differential equations
1638:, the phase space method is used to visualize
2175:
815:, while a two-dimensional system is called a
712:
1781:for information about state space in physics
1018:(two equilibria, one stable, one unstable).
2189:
2182:
2168:
1853:
719:
705:
2061:
2051:
2010:
1992:
1936:
1911:
1548:, a point in phase space is said to be a
2033:
1858:. New York: Cambridge University Press.
1524:Thermodynamics and statistical mechanics
1338:Classic examples of phase diagrams from
1312:
1301:
1122:
1107:
1042:
851:
27:
1878:
832:separate dimension for each particle's
2681:
1942:
1427:(1932); and, in a grand synthesis, by
881:for the position (i.e. coordinates on
869:In classical mechanics, any choice of
2163:
2115:
2096:
1810:
1456:Wigner quasi-probability distribution
2567:List of named differential equations
1411:corresponds to a unique function or
1371:
932:
913:Statistical ensembles in phase space
864:
751:in which all possible "states" of a
167:List of named differential equations
2492:Method of undetermined coefficients
2273:Dependent and independent variables
1612:Phase space is extensively used in
1079:", is more usually reserved in the
1057:
240:Dependent and independent variables
13:
2090:
1556:a microstate consists of a set of
1458:effectively serving as a measure.
1075:. However the latter expression, "
14:
2715:
2135:
1653:Configuration space (mathematics)
1568:is typically on the order of the
1272:
1094:
2389:Carathéodory's existence theorem
1540:particles, then a point in the 6
1100:This section is an excerpt from
375:Carathéodory's existence theorem
1900:Asia Pacific Physics Newsletter
1854:Laurendeau, Normand M. (2005).
1403:). This is consistent with the
1388:of phase space normally become
1333:
1297:
2027:
1967:
1891:
1872:
1847:
1804:
1401:Groenewold's 1946 star product
990:
984:
952:ordinary differential equation
462: / Integral solutions
1:
1797:
1505:, with deformation parameter
1493:, with deformation parameter
1361:complex quadratic polynomials
1062:
887:conjugate generalized momenta
798:
2217:Notation for differentiation
2053:10.1016/j.exphem.2004.02.013
1994:10.3390/biomedicines11030745
1509:/characteristic dimension.)
506:Exponential response formula
252:Coupled / Decoupled
23:Phase space (disambiguation)
7:
2313:Exact differential equation
2148:Encyclopedia of Mathematics
2120:. Oxford University Press.
2101:. Oxford University Press.
1645:
1629:
1585:temperature–entropy diagram
1554:indistinguishable particles
1419:(1927) and supplemented by
999:{\displaystyle dy/dt=f(y),}
10:
2720:
1622:
1429:H. J. Groenewold
1099:
936:
20:
2623:Józef Maria Hoene-Wroński
2603:Gottfried Wilhelm Leibniz
2590:
2559:
2469:
2402:
2394:Cauchy–Kowalevski theorem
2371:
2364:
2326:
2265:
2204:
2197:
1922:10.1142/S2251158X12000069
1642:physiological responses.
1607:
640:Józef Maria Hoene-Wroński
586:Undetermined coefficients
495:Method of characteristics
380:Cauchy–Kowalevski theorem
2517:Finite difference method
1881:"Configuration integral"
1444:deformation quantization
1352:population growth (i.e.
1115:and phase portrait of a
1012:exponential growth model
939:Phase line (mathematics)
803:In a phase space, every
763:in the phase space. For
733:dynamical systems theory
365:Picard–Lindelöf theorem
359:Existence and uniqueness
2497:Variation of parameters
2487:Separation of variables
2384:Peano existence theorem
2379:Picard–Lindelöf theorem
2266:Attributes of variables
2040:Experimental Hematology
2034:Kirkland, M.A. (2004).
1785:Phase-space formulation
1581:pressure–volume diagram
1438:phase-space formulation
1255:topological equivalence
871:generalized coordinates
591:Variation of parameters
581:Separation of variables
370:Peano existence theorem
2658:Carl David Tolmé Runge
2232:Differential-algebraic
2191:Differential equations
1943:Chaves, Julio (2015).
1709:State space (controls)
1491:relativistic mechanics
1448:geometric quantization
1330:
1310:
1214:representation of the
1199:
1193:
1120:
1054:
1052:Van der Pol oscillator
1037:shown in the diagram.
1035:Van der Pol oscillator
1000:
954:in a single variable,
909:on a cotangent space.
861:
821:phase-space trajectory
660:Carl David Tolmé Runge
203:Differential-algebraic
44:Differential equations
37:
34:simple harmonic motion
2704:Hamiltonian mechanics
2643:Augustin-Louis Cauchy
2638:Joseph-Louis Lagrange
2532:Finite element method
2522:Crank–Nicolson method
2456:Numerical integration
2435:Exponential stability
2327:Relation to processes
2212:Differential operator
2116:Nolte, D. D. (2018).
2097:Nolte, D. D. (2015).
2063:10536/DRO/DU:30101092
1779:State space (physics)
1769:Hamiltonian mechanics
1750:Wigner–Weyl transform
1623:Further information:
1534:statistical mechanics
1433:J. E. Moyal
1405:uncertainty principle
1316:
1305:
1288:quantum energy states
1194:
1126:
1111:
1046:
1016:logistic growth model
1001:
923:statistical mechanics
855:
775:variables. It is the
650:Augustin-Louis Cauchy
635:Joseph-Louis Lagrange
467:Numerical integration
449:Exponential stability
312:Relation to processes
31:
2699:Dimensional analysis
2537:Finite volume method
2461:Dirac delta function
2430:Asymptotic stability
2372:Existence/uniqueness
2237:Integro-differential
1879:Vu-Quoc, L. (2008).
1787:of quantum mechanics
1774:Lagrangian mechanics
1671:, 2-dimensional case
1665:, 1-dimensional case
1552:of the system. (For
1507:Schwarzschild radius
1440:of quantum mechanics
1131:
958:
907:symplectic structure
793:Josiah Willard Gibbs
779:of direct space and
472:Dirac delta function
208:Integro-differential
21:For other uses, see
2689:Concepts in physics
2547:Perturbation theory
2527:Runge–Kutta methods
2507:Integral transforms
2440:Rate of convergence
2336:(discrete analogue)
1825:2010PhT....63d..33N
1764:Classical mechanics
1745:Symplectic manifold
1704:Optical phase space
1390:Hermitian operators
1359:parameter plane of
1091:, and composition.
927:Liouville's theorem
903:Darboux coordinates
883:configuration space
568:Perturbation theory
563:Integral transforms
454:Rate of convergence
320:(discrete analogue)
157:Population dynamics
124:Continuum mechanics
115:Applied mathematics
2668:Sofya Kovalevskaya
2502:Integrating factor
2425:Lyapunov stability
2345:Stochastic partial
1887:on April 28, 2012.
1721:Molecular dynamics
1680:Phase space method
1618:Hamiltonian optics
1516:) are modified by
1503:general relativity
1380:, the coordinates
1331:
1311:
1279:partition function
1200:
1189:
1121:
1055:
996:
862:
765:mechanical systems
558:Integrating factor
399:Initial conditions
334:Stochastic partial
38:
2694:Dynamical systems
2676:
2675:
2555:
2554:
2360:
2359:
2127:978-0-19-880584-7
2108:978-0-19-965703-2
1833:10.1063/1.3397041
1614:nonimaging optics
1378:quantum mechanics
1372:Quantum mechanics
1164:
1143:
1081:physical sciences
933:In low dimensions
917:The motion of an
905:for the standard
865:Conjugate momenta
825:initial condition
805:degree of freedom
729:
728:
620:Gottfried Leibniz
511:Finite difference
303:
302:
164:
163:
134:Dynamical systems
2711:
2653:Phyllis Nicolson
2633:Rudolf Lipschitz
2470:Solution methods
2445:Series solutions
2369:
2368:
2202:
2201:
2184:
2177:
2170:
2161:
2160:
2156:
2131:
2112:
2084:
2083:
2065:
2055:
2031:
2025:
2024:
2014:
1996:
1971:
1965:
1964:
1940:
1934:
1933:
1915:
1895:
1889:
1888:
1883:. Archived from
1876:
1870:
1869:
1851:
1845:
1844:
1808:
1735:Cotangent bundle
1640:multidimensional
1634:In medicine and
1514:Poisson brackets
1421:John von Neumann
1348:Lorenz attractor
1220:dynamical system
1198:
1196:
1195:
1190:
1179:
1178:
1166:
1165:
1157:
1145:
1144:
1136:
1113:Potential energy
1058:Related concepts
1005:
1003:
1002:
997:
971:
899:cotangent bundle
856:Evolution of an
785:Ludwig Boltzmann
781:reciprocal space
753:dynamical system
721:
714:
707:
685:Phyllis Nicolson
670:Rudolf Lipschitz
553:Green's function
529:Infinite element
520:
485:Solution methods
463:
321:
232:By variable type
186:
185:
68:Natural sciences
61:
60:
40:
39:
2719:
2718:
2714:
2713:
2712:
2710:
2709:
2708:
2679:
2678:
2677:
2672:
2613:Jacob Bernoulli
2586:
2551:
2542:Galerkin method
2465:
2403:Solution topics
2398:
2356:
2322:
2261:
2193:
2188:
2141:
2138:
2128:
2109:
2093:
2091:Further reading
2088:
2087:
2032:
2028:
1972:
1968:
1961:
1941:
1937:
1896:
1892:
1877:
1873:
1866:
1852:
1848:
1809:
1805:
1800:
1795:
1685:Parameter space
1648:
1632:
1627:
1610:
1570:Avogadro number
1546:distinguishable
1526:
1374:
1336:
1329:axis its speed.
1300:
1292:Planck constant
1283:quantum numbers
1275:
1270:
1269:
1266:state variables
1174:
1170:
1156:
1155:
1135:
1134:
1132:
1129:
1128:
1117:simple pendulum
1105:
1097:
1065:
1060:
967:
959:
956:
955:
945:
937:Main articles:
935:
915:
896:
880:
867:
801:
725:
696:
695:
694:
625:Jacob Bernoulli
609:
596:
595:
577:
546:Petrov–Galerkin
514:
499:
486:
478:
477:
476:
458:
404:Boundary values
393:
385:
384:
360:
347:
346:
345:
319:
313:
305:
304:
292:
269:
227:
183:
170:
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165:
143:Social sciences
99:
77:
58:
26:
19:
12:
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1885:the original
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1819:(4): 33–38.
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1696:Applications
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1596:phase, etc.
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1413:distribution
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1340:chaos theory
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1334:Chaos theory
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680:Émile Picard
665:Martin Kutta
655:George Green
615:Isaac Newton
447: /
443: /
263: /
129:Chaos theory
15:
2420:Phase space
2278:Homogeneous
1727:Mathematics
1715:State space
1669:Phase plane
1625:Vague torus
1577:macroscopic
1468:deformation
1251:limit cycle
1224:phase plane
1204:mathematics
1089:temperature
1031:limit cycle
1023:phase plane
943:Phase plane
847:robotic arm
817:phase plane
745:state space
741:phase space
573:Runge–Kutta
318:Difference
261:Homogeneous
73:Engineering
2683:Categories
2648:John Crank
2477:Inspection
2340:Stochastic
2334:Difference
2308:Autonomous
2252:Non-linear
2242:Fractional
2205:Operations
1987:(3): 745.
1798:References
1690:Separatrix
1663:Phase line
1592:phase, or
1550:microstate
1409:observable
1069:phase plot
1063:Phase plot
1008:phase line
949:autonomous
885:) defines
813:phase line
799:Principles
690:John Crank
491:Inspection
445:Asymptotic
329:Stochastic
248:Autonomous
223:Non-linear
213:Fractional
2452:solutions
2410:Wronskian
2365:Solutions
2293:Decoupled
2257:Holonomic
2153:EMS Press
2072:0301-472X
2003:2227-9059
1951:CRC Press
1930:119230734
1913:1104.5269
1906:: 37–46.
1243:attractor
1212:geometric
1172:ω
1162:˙
1153:γ
1141:¨
809:parameter
431:Wronskian
409:Dirichlet
152:Economics
95:Chemistry
85:Astronomy
2560:Examples
2450:Integral
2222:Ordinary
2080:15183891
2021:36979723
2012:10044742
1841:17205307
1646:See also
1630:Medicine
1601:manifold
1481:, where
1463:Weyl map
1423:(1931);
1319:pendulum
1308:pendulum
1247:repellor
1085:pressure
919:ensemble
858:ensemble
773:momentum
769:position
541:Galerkin
441:Lyapunov
352:Solution
296:Notation
288:Operator
274:Features
193:Ordinary
2288:Coupled
2227:Partial
2155:, 2001
1821:Bibcode
1756:Physics
1485:is the
1222:in the
1050:of the
1033:of the
414:Neumann
198:Partial
106:Geology
101:Biology
90:Physics
2303:Degree
2247:Linear
2124:
2105:
2078:
2070:
2019:
2009:
2001:
1957:
1928:
1862:
1839:
1608:Optics
1590:liquid
1487:action
1321:. The
1216:orbits
791:, and
601:People
513:
460:Series
218:Linear
57:Fields
2352:Delay
2298:Order
1926:S2CID
1908:arXiv
1837:S2CID
1594:solid
1392:in a
1363:with
1342:are:
1232:curve
1228:point
1218:of a
1210:is a
1071:or a
761:point
755:or a
749:space
747:is a
501:Euler
419:Robin
341:Delay
283:Order
256:Exact
182:Types
50:Scope
2122:ISBN
2103:ISBN
2076:PMID
2068:ISSN
2017:PMID
1999:ISSN
1955:ISBN
1860:ISBN
1532:and
1446:and
1384:and
1346:the
1245:, a
1239:plot
1206:, a
941:and
840:and
771:and
739:, a
735:and
608:List
2058:hdl
2048:doi
2007:PMC
1989:doi
1918:doi
1829:doi
1583:or
1528:In
1450:.)
1376:In
1249:or
1230:or
1202:In
829:any
807:or
743:or
731:In
2685::
2151:,
2145:,
2074:.
2066:.
2056:.
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2015:.
2005:.
1997:.
1985:11
1983:.
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1949:.
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1902:.
1835:.
1827:.
1817:63
1815:.
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1234:.
1187:0.
1087:,
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2176:t
2169:v
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2111:.
2082:.
2060::
2050::
2023:.
1991::
1963:.
1932:.
1920::
1910::
1868:.
1843:.
1831::
1823::
1566:N
1562:N
1558:N
1542:N
1538:N
1518:ħ
1499:c
1497:/
1495:v
1483:S
1479:S
1477:/
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1382:p
1367:.
1356:)
1327:Y
1323:X
1268:.
1184:=
1181:x
1176:2
1168:+
1159:x
1150:2
1147:+
1138:x
1104:.
994:,
991:)
988:y
985:(
982:f
979:=
976:t
973:d
969:/
965:y
962:d
894:i
890:p
878:i
874:q
842:z
838:y
834:x
720:e
713:t
706:v
519:)
515:(
25:.
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