In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
Stable limit cycle (shown in bold) and two other trajectories spiraling into itStable limit cycle (shown in bold) for the Van der Pol oscillator
We consider a two-dimensional dynamical system of the form
where
is a smooth function. A trajectory of this system is some smooth function with values in which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some such that for all . An orbit is the image of a trajectory, a subset of . A closed orbit, or cycle, is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.
Propertiesedit
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching , then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching . The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching , and also for trajectories in the exterior approaching the limit cycle.
Stable, unstable and semi-stable limit cyclesedit
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a stable or attractive limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an unstable limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a semi-stable limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
Stable limit cycles are examples of attractors. They imply self-sustained oscillations: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.
Finding limit cyclesedit
Every closed trajectory contains within its interior a stationary point of the system, i.e. a point where . The Bendixson–Dulac theorem and the Poincaré–Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
Open problemsedit
Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert's sixteenth problem. It is unknown, for instance, whether there is any system in the plane where both components of are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
Applicationsedit
Examples of limit cycles branching from fixed points near Hopf bifurcation. Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.
Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled. Some examples include:
The daily oscillations in gene expression, hormone levels and body temperature of animals, which are part of the circadian rhythm,[3][4] although this is contradicted by more recent evidence.[5]
The migration of cancer cells in confining micro-environments follows limit cycle oscillations.[6]
^Thomas, Jeffrey P.; Dowell, Earl H.; Hall, Kenneth C. (2002), "Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations" (PDF), AIAA Journal, American Institute of Aeronautics and Astronautics, 40 (4): 638, Bibcode:2002AIAAJ..40..638T, doi:10.2514/2.1720, retrieved December 9, 2019
^Sel'kov, E. E. (1968). "Self-Oscillations in Glycolysis 1. A Simple Kinetic Model". European Journal of Biochemistry. 4 (1): 79–86. doi:10.1111/j.1432-1033.1968.tb00175.x. ISSN 1432-1033. PMID 4230812.
^Leloup, Jean-Christophe; Gonze, Didier; Goldbeter, Albert (1999-12-01). "Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora". Journal of Biological Rhythms. 14 (6): 433–448. doi:10.1177/074873099129000948. ISSN 0748-7304. PMID 10643740. S2CID 15074869.
^Meijer, JH; Michel, S; Vanderleest, HT; Rohling, JH (December 2010). "Daily and seasonal adaptation of the circadian clock requires plasticity of the SCN neuronal network". The European Journal of Neuroscience. 32 (12): 2143–51. doi:10.1111/j.1460-9568.2010.07522.x. PMID 21143668. S2CID 12754517.
^Brückner, David B.; Fink, Alexandra; Schreiber, Christoph; Röttgermann, Peter J. F.; Rädler, Joachim; Broedersz, Chase P. (2019). "Stochastic nonlinear dynamics of confined cell migration in two-state systems". Nature Physics. 15 (6): 595–601. Bibcode:2019NatPh..15..595B. doi:10.1038/s41567-019-0445-4. ISSN 1745-2481. S2CID 126819906.
^Ginoux, Jean-Marc; Letellier, Christophe (2012-04-30). "Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept". Chaos: An Interdisciplinary Journal of Nonlinear Science. 22 (2): 023120. arXiv:1408.4890. Bibcode:2012Chaos..22b3120G. doi:10.1063/1.3670008. ISSN 1054-1500. PMID 22757527. S2CID 293369.
^Mackey, M.; Glass, L (1977-07-15). "Oscillation and chaos in physiological control systems". Science. 197 (4300): 287–289. Bibcode:1977Sci...197..287M. doi:10.1126/science.267326. ISSN 0036-8075. PMID 267326.
Further readingedit
Steven H. Strogatz (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Avalon. ISBN9780813349114.
M. Vidyasagar (2002). Nonlinear Systems Analysis (Second ed.). SIAM. ISBN9780898715262.
Philip Hartman, "Ordinary Differential Equation", Society for Industrial and Applied Mathematics, 2002.
Witold Hurewicz, "Lectures on Ordinary Differential Equations", Dover, 2002.
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In mathematics in the study of dynamical systems with two dimensional phase space a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity Such behavior is exhibited in some nonlinear systems Limit cycles have been used to model the behavior of many real world oscillatory systems The study of limit cycles was initiated by Henri Poincare 1854 1912 Stable limit cycle shown in bold and two other trajectories spiraling into itStable limit cycle shown in bold for the Van der Pol oscillator Contents 1 Definition 2 Properties 3 Stable unstable and semi stable limit cycles 4 Finding limit cycles 5 Open problems 6 Applications 7 See also 8 References 9 Further reading 10 External linksDefinition editWe consider a two dimensional dynamical system of the formx t V x t displaystyle x t V x t nbsp where V R 2 R 2 displaystyle V mathbb R 2 to mathbb R 2 nbsp is a smooth function A trajectory of this system is some smooth function x t displaystyle x t nbsp with values in R 2 displaystyle mathbb R 2 nbsp which satisfies this differential equation Such a trajectory is called closed or periodic if it is not constant but returns to its starting point i e if there exists some t 0 gt 0 displaystyle t 0 gt 0 nbsp such that x t t 0 x t displaystyle x t t 0 x t nbsp for all t R displaystyle t in mathbb R nbsp An orbit is the image of a trajectory a subset of R 2 displaystyle mathbb R 2 nbsp A closed orbit or cycle is the image of a closed trajectory A limit cycle is a cycle which is the limit set of some other trajectory Properties editBy the Jordan curve theorem every closed trajectory divides the plane into two regions the interior and the exterior of the curve Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching displaystyle infty nbsp then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching displaystyle infty nbsp The corresponding statement holds for a trajectory in the interior that approaches the limit cycle for time approaching displaystyle infty nbsp and also for trajectories in the exterior approaching the limit cycle Stable unstable and semi stable limit cycles editIn the case where all the neighboring trajectories approach the limit cycle as time approaches infinity it is called a stable or attractive limit cycle w limit cycle If instead all neighboring trajectories approach it as time approaches negative infinity then it is an unstable limit cycle a limit cycle If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity and another one which spirals into it as time approaches negative infinity then it is a semi stable limit cycle There are also limit cycles that are neither stable unstable nor semi stable for instance a neighboring trajectory may approach the limit cycle from the outside but the inside of the limit cycle is approached by a family of other cycles which wouldn t be limit cycles Stable limit cycles are examples of attractors They imply self sustained oscillations the closed trajectory describes the perfect periodic behavior of the system and any small perturbation from this closed trajectory causes the system to return to it making the system stick to the limit cycle Finding limit cycles editEvery closed trajectory contains within its interior a stationary point of the system i e a point p displaystyle p nbsp where V p 0 displaystyle V p 0 nbsp The Bendixson Dulac theorem and the Poincare Bendixson theorem predict the absence or existence respectively of limit cycles of two dimensional nonlinear dynamical systems Open problems editFinding limit cycles in general is a very difficult problem The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert s sixteenth problem It is unknown for instance whether there is any system x V x displaystyle x V x nbsp in the plane where both components of V displaystyle V nbsp are quadratic polynomials of the two variables such that the system has more than 4 limit cycles Applications edit nbsp Examples of limit cycles branching from fixed points near Hopf bifurcation Trajectories in red stable structures in dark blue unstable structures in light blue The parameter choice determines the occurrence and stability of limit cycles Limit cycles are important in many scientific applications where systems with self sustained oscillations are modelled Some examples include Aerodynamic limit cycle oscillations 1 The Hodgkin Huxley model for action potentials in neurons The Sel kov model of glycolysis 2 The daily oscillations in gene expression hormone levels and body temperature of animals which are part of the circadian rhythm 3 4 although this is contradicted by more recent evidence 5 The migration of cancer cells in confining micro environments follows limit cycle oscillations 6 Some non linear electrical circuits exhibit limit cycle oscillations 7 which inspired the original Van der Pol model The control of respiration and hematopoiesis as appearing in the Mackey Glass equations 8 See also editAttractor Hyperbolic set Periodic point Self oscillation Stable manifoldReferences edit Thomas Jeffrey P Dowell Earl H Hall Kenneth C 2002 Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence Flutter and Limit Cycle Oscillations PDF AIAA Journal American Institute of Aeronautics and Astronautics 40 4 638 Bibcode 2002AIAAJ 40 638T doi 10 2514 2 1720 retrieved December 9 2019 Sel kov E E 1968 Self Oscillations in Glycolysis 1 A Simple Kinetic Model European Journal of Biochemistry 4 1 79 86 doi 10 1111 j 1432 1033 1968 tb00175 x ISSN 1432 1033 PMID 4230812 Leloup Jean Christophe Gonze Didier Goldbeter Albert 1999 12 01 Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora Journal of Biological Rhythms 14 6 433 448 doi 10 1177 074873099129000948 ISSN 0748 7304 PMID 10643740 S2CID 15074869 Roenneberg Till Chua Elaine Jane Bernardo Ric Mendoza Eduardo 2008 09 09 Modelling Biological Rhythms Current Biology 18 17 R826 R835 doi 10 1016 j cub 2008 07 017 ISSN 0960 9822 PMID 18786388 S2CID 2798371 Meijer JH Michel S Vanderleest HT Rohling JH December 2010 Daily and seasonal adaptation of the circadian clock requires plasticity of the SCN neuronal network The European Journal of Neuroscience 32 12 2143 51 doi 10 1111 j 1460 9568 2010 07522 x PMID 21143668 S2CID 12754517 Bruckner David B Fink Alexandra Schreiber Christoph Rottgermann Peter J F Radler Joachim Broedersz Chase P 2019 Stochastic nonlinear dynamics of confined cell migration in two state systems Nature Physics 15 6 595 601 Bibcode 2019NatPh 15 595B doi 10 1038 s41567 019 0445 4 ISSN 1745 2481 S2CID 126819906 Ginoux Jean Marc Letellier Christophe 2012 04 30 Van der Pol and the history of relaxation oscillations Toward the emergence of a concept Chaos An Interdisciplinary Journal of Nonlinear Science 22 2 023120 arXiv 1408 4890 Bibcode 2012Chaos 22b3120G doi 10 1063 1 3670008 ISSN 1054 1500 PMID 22757527 S2CID 293369 Mackey M Glass L 1977 07 15 Oscillation and chaos in physiological control systems Science 197 4300 287 289 Bibcode 1977Sci 197 287M doi 10 1126 science 267326 ISSN 0036 8075 PMID 267326 Further reading editSteven H Strogatz 2014 Nonlinear Dynamics and Chaos With Applications to Physics Biology Chemistry and Engineering Avalon ISBN 9780813349114 M Vidyasagar 2002 Nonlinear Systems Analysis Second ed SIAM ISBN 9780898715262 Philip Hartman Ordinary Differential Equation Society for Industrial and Applied Mathematics 2002 Witold Hurewicz Lectures on Ordinary Differential Equations Dover 2002 Solomon Lefschetz Differential Equations Geometric Theory Dover 2005 Lawrence Perko Differential Equations and Dynamical Systems Springer Verlag 2006 Arthur Mattuck Limit Cycles Existence and Non existence Criteria MIT Open Courseware http videolectures net mit1803s06 mattuck lec32 External links edit limit cycle planetmath org Retrieved 2019 07 06 Retrieved from https en wikipedia org w index php title Limit cycle amp oldid 1170226048, wikipedia, wiki, book, books, library,
