Teorija haosa

Grafik Lorencovog atraktora za ulazne vrednosti r = 28, σ = 10, b = 8/3

Animacija dvoštapnog klatna na srednjoj energiji koja pokazuje haotično ponašanje. Pokretanje klatna iz malo drugačijeg početnog stanja rezultiralo bi u znatno drugačijoj putanji. Klatno sa dvostrukom šipkom je jedan od najjednostavnijih dinamičkih sistema sa haotičnim rešenjima.

U matematici, teorija haosa opisuje ponašanje određenih dinamičkih sistema (sistema čije stanje evoluira tokom vremena), koji mogu da ispolje dinamiku koja je veoma osetljiva na početne uslove (popularno, efekat leptira). Kao rezultat ove osetljivosti, koja se manifestuje eksponencijalnim rastom perturbacija u početnim uslovima, ponašanje haotičnih sistema izgleda slučajno. Ovo se događa čak i ako su ti sistemi deterministički, što znači da je njihova dalja dinamika u potpunosti određena početnim uslovima, bez slučajnih faktora. Ovo ponašanje je poznato kao deterministički haos, ili prosto haos.

Haotično ponašanje se takođe javlja u prirodnim sistemima, kao što su meteorološke prilike. Ono se može objasniti haos-teoretskom analizom matematičkog modela takvog sistema, koji oslikava zakone fizike koji su relevantni za odgovarajući prirodni sistem.

Pregled

Haotično ponašanje je uočeno u laboratoriji kod mnoštva sistema, uključujući električna kola, lasere, oscilujuće hemijske reakcije, dinamiku fluida, i mehaničke i magnetno-mehaničke uređaje. Posmatranja haotičnog ponašanja u prirodi se vrše i kod dinamike satelita Sunčevog sistema, vremena evolucije magnetnog polja nebeskih tela, rasta populacije u ekologiji, dinamici akcionih potencijala kod neurona i molekularnih vibracija. Primer haotičnih sistema je i ponašanje vremena i klime. Postoje određene kontroverze oko toga da li se haotična dinamika javlja kod dinamike tektonskih ploča i u ekonomiji.

Sistemi koji ispoljavaju matematički haos su deterministički i zbog toga u nekom smislu pokazuju uređenost; ovakva stručna upotreba termina haos nije u skladu sa svakodnevnom upotrebom, koja podrazumeva potpuni nered. Srodno polje fizike, teorija kvantnog haosa proučava sisteme koji prate zakone kvantne mehanike. Nedavno se pojavilo i novo polje, relativistički haos,, koje se trudi da opiše sisteme koji prate zakone opšte relativnosti.

Ovaj članak se trudi da opiše granice stepena nereda koji računari mogu da modeluju sa jednostavnim pravilima, koja daju kompleksne rezultate. Na primer, prikazani Lorencov sistem je haotičan, iako ima jasno definisanu strukturu Ograničeni haos je koristan izraz za opisivanje modela nereda.

Literatura

Publikacije

• Sharkovskii, A.N. (1964). „Co-existence of cycles of a continuous mapping of the line into itself”. Ukrainian Math. J. 16: 61—71. 
• Li, T.Y.; Yorke, J.A. (1975). „Period Three Implies Chaos” (PDF). American Mathematical Monthly. 82 (10): 985—92. Bibcode:1975AmMM...82..985L. CiteSeerX 10.1.1.329.5038 . JSTOR 2318254. doi:10.2307/2318254. Arhivirano iz originala (PDF) 2009-12-29. g. Pristupljeno 2009-08-12. 
• Alemansour, Hamed; Miandoab, Ehsan Maani; Pishkenari, Hossein Nejat (mart 2017). „Effect of size on the chaotic behavior of nano resonators”. Communications in Nonlinear Science and Numerical Simulation. 44: 495—505. Bibcode:2017CNSNS..44..495A. doi:10.1016/j.cnsns.2016.09.010. CS1 održavanje: Format datuma (veza)
• Crutchfield; Tucker; Morrison; J.D. Farmer; Packard; N.H.; Shaw; R.S (decembar 1986). „Chaos”. Scientific American. 255 (6): 38—49 (bibliography p.136). Bibcode:1986SciAm.255d..38T. doi:10.1038/scientificamerican1286-46. CS1 održavanje: Format datuma (veza) Online version (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).
• Kolyada, S.F. (2004). „Li-Yorke sensitivity and other concepts of chaos”. Ukrainian Math. J. 56 (8): 1242—57. S2CID 207251437. doi:10.1007/s11253-005-0055-4. 
• Day, R.H.; Pavlov, O.V. (2004). „Computing Economic Chaos”. Computational Economics. 23 (4): 289—301. S2CID 119972392. SSRN 806124 . doi:10.1023/B:CSEM.0000026787.81469.1f. 
• Strelioff, C.; Hübler, A. (2006). „Medium-Term Prediction of Chaos” (PDF). Phys. Rev. Lett. 96 (4): 044101. Bibcode:2006PhRvL..96d4101S. PMID 16486826. doi:10.1103/PhysRevLett.96.044101. 044101. Arhivirano iz originala (PDF) 2013-04-26. g. 
• Hübler, A.; Foster, G.; Phelps, K. (2007). „Managing Chaos: Thinking out of the Box” (PDF). Complexity. 12 (3): 10—13. Bibcode:2007Cmplx..12c..10H. doi:10.1002/cplx.20159. Arhivirano iz originala (PDF) 2012-10-30. g. Pristupljeno 2011-07-17. 
• Motter, Adilson E.; Campbell, David K. (2013). „Chaos at 50”. Physics Today. 66 (5): 27. Bibcode:2013PhT....66e..27M. S2CID 54005470. arXiv:1306.5777 . doi:10.1063/PT.3.1977. 

Udžbenici

• Alligood, K.T.; Sauer, T.; Yorke, J.A. (1997). Chaos: an introduction to dynamical systems. Springer-Verlag. ISBN 978-0-387-94677-1. 
• Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 978-0-521-39511-3. 
• Badii, R.; Politi, A. (1997). Complexity: hierarchical structures and scaling in physics. Cambridge University Press. ISBN 978-0-521-66385-4. 
• Bunde; Havlin, Shlomo, ur. (1996). Fractals and Disordered Systems. Springer. ISBN 978-3642848704.  and Bunde; Havlin, Shlomo, ur. (1994). Fractals in Science. Springer. ISBN 978-3-540-56220-7. 
• Collet, Pierre, and Eckmann, Jean-Pierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 978-0-8176-4926-5. CS1 održavanje: Višestruka imena: spisak autora (veza)
• Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems (2nd izd.). Westview Press. ISBN 978-0-8133-4085-2. 
• Robinson, Clark (1995). Dynamical systems: Stability, symbolic dynamics, and chaos. CRC Press. ISBN 0-8493-8493-1. 
• Feldman, D. P. (2012). Chaos and Fractals: An Elementary Introduction. Oxford University Press. ISBN 978-0-19-956644-0. Arhivirano iz originala 31. 12. 2019. g. Pristupljeno 24. 07. 2021. 
• Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 978-0-521-47685-0. 
• Guckenheimer, John; Holmes, Philip (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag. ISBN 978-0-387-90819-9. 
• Gulick, Denny (1992). Encounters with Chaos. McGraw-Hill. ISBN 978-0-07-025203-5. 
• Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. Springer-Verlag. ISBN 978-0-387-97173-5. 
• Hoover, William Graham (2001) [1999]. Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 978-981-02-4073-8. 
• Kautz, Richard (2011). Chaos: The Science of Predictable Random Motion. Oxford University Press. ISBN 978-0-19-959458-0. 
• Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 978-0-472-08472-2. 
• Moon, Francis (1990). Chaotic and Fractal Dynamics. Springer-Verlag. ISBN 978-0-471-54571-2. 
• Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press. ISBN 978-0-521-01084-9. 
• Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 978-0-7382-0453-6. 
• Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 978-0-19-850840-3. 
• Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge University Press. ISBN 978-0-521-83912-9. 
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. 
• Thompson JM, Stewart HB (2001). Nonlinear Dynamics And Chaos. John Wiley and Sons Ltd. ISBN 978-0-471-87645-8. 
• Tufillaro; Reilly (1992). An experimental approach to nonlinear dynamics and chaos. American Journal of Physics. 61. Addison-Wesley. str. 958. Bibcode:1993AmJPh..61..958T. ISBN 978-0-201-55441-0. doi:10.1119/1.17380. 
• Wiggins, Stephen (2003). Introduction to Applied Dynamical Systems and Chaos. Springer. ISBN 978-0-387-00177-7. 
• Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. ISBN 978-0-19-852604-9. 

Semitehnička i popularna dela

• Christophe Letellier (2012). Chaos in Nature. World Scientific Publishing Company. ISBN 978-981-4374-42-2. 
• Abraham, Ralph; et al. (2000). Abraham, Ralph H.; Ueda, Yoshisuke, ur. The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory. World Scientific Series on Nonlinear Science Series A. 39. World Scientific. Bibcode:2000cagm.book.....A. ISBN 978-981-238-647-2. doi:10.1142/4510. 
• Barnsley, Michael F. (2000). Fractals Everywhere. Morgan Kaufmann. ISBN 978-0-12-079069-2. 
• Bird, Richard J. (2003). Chaos and Life: Complexity and Order in Evolution and Thought. Columbia University Press. ISBN 978-0-231-12662-5. 
• John Briggs and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness, Harper Perennial 1990, 224 pp.
• John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper Perennial 2000, 224 pp.
• Cunningham, Lawrence A. (1994). „From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis”. George Washington Law Review. 62: 546. 
• Predrag Cvitanović, Universality in Chaos, Adam Hilger 1989, 648 pp.
• Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press 1988, 272 pp.
• James Gleick, Chaos: Making a New Science, New York: Penguin, 1988. 368 pp.
• John Gribbin (2005-01-27). Deep Simplicity. Penguin Press Science. Penguin Books. 
• L Douglas Kiel, Euel W Elliott (ed.), Chaos Theory in the Social Sciences: Foundations and Applications, University of Michigan Press, 1997, 360 pp.
• Arvind Kumar, Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature , National Book Trust, 2003.
• Hans Lauwerier, Fractals, Princeton University Press, 1991.
• Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996.
• Marshall, Alan (2002). The Unity of Nature - Wholeness and Disintegration in Ecology and Science. ISBN 9781860949548. doi:10.1142/9781860949548. 
• David Peak and Michael Frame, Chaos Under Control: The Art and Science of Complexity, Freeman, 1994.
• Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp.
• Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991.
• Clifford A. Pickover, Chaos in Wonderland: Visual Adventures in a Fractal World, St Martins Pr 1994.
• Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984.
• Peitgen, Heinz-Otto; Richter, Peter H. (1986). The Beauty of Fractals. ISBN 978-3-642-61719-5. doi:10.1007/978-3-642-61717-1. 
• David Ruelle, Chance and Chaos, Princeton University Press 1993.
• Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993.
• Roulstone, Ian; Norbury, John (2013). Invisible in the Storm: the role of mathematics in understanding weather. Princeton University Press. ISBN 978-0691152721. 
• Ruelle, D. (1989). Chaotic Evolution and Strange Attractors. ISBN 9780521362726. doi:10.1017/CBO9780511608773. 
• Manfred Schroeder, Fractals, Chaos, and Power Laws, Freeman, 1991.
• Smith, Peter (1998). Explaining Chaos. ISBN 9780511554544. doi:10.1017/CBO9780511554544. 
• Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990.
• Steven Strogatz, Sync: The emerging science of spontaneous order, Hyperion, 2003.
• Yoshisuke Ueda, The Road To Chaos, Aerial Pr, 1993.
• M. Mitchell Waldrop, Complexity : The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, 1992.
• Antonio Sawaya, Financial Time Series Analysis : Chaos and Neurodynamics Approach, Lambert, 2012.

Spoljašnje veze

• Hazewinkel Michiel, ur. (2001). „Chaos”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Nonlinear Dynamics Research Group with Animations in Flash
• The Chaos group at the University of Maryland
• The Chaos Hypertextbook. An introductory primer on chaos and fractals
• ChaosBook.org An advanced graduate textbook on chaos (no fractals)
• Society for Chaos Theory in Psychology & Life Sciences
• Nonlinear Dynamics Research Group at CSDC, Florence Italy
• Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum
• Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.
• Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
• Gleick's Chaos (excerpt) Arhivirano 2007-02-02 na sajtu Wayback Machine
• Systems Analysis, Modelling and Prediction Group at the University of Oxford
• A page about the Mackey-Glass equation
• High Anxieties — The Mathematics of Chaos (2008) BBC documentary directed by David Malone
• The chaos theory of evolution – article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos.
• Jos Leys, Étienne Ghys et Aurélien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
• "Chaos Theory", BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (In Our Time, May 16, 2002)
• Chaos: The Science of the Butterfly Effect (2019) an explanation presented by Derek Muller