Sistem dinamis

Dalam matematika, sistem dinamikal adalah sebuah sistem dimana sebuah fungsi mendeskripsikan ketergantungan waktu dari sebuah titik dalam sebuah ruang geometri. Contoh-contohnya meliputi model matematika yang mendeskripsikan gerak pendulum jam, aliran air dalam sebuah pipa, dan jumlah iklan setiap musim semi di danau.

Pada waktu manapun yang diberikan, sistem dinamikal memiliki keadaan yang diberikan oleh serangkaian angkata nyata (sebuah cektor) yang dapat diwakili oleh sebuah poin dalam sebuah ruang keadaan (sebuah manifold geometri). Aturan evolusi dari sistem dinamikal adalah sebuah fungsi yang menyebut apakah keadaan-keadaan mendatang diikuti dari keadaan saat ini. Seringkali, fungsi tersebut bersifat [[sistem deterministik (matematika)}deterministik]], yang selama waktu yang diberikan hanya terdiri dari satu keadaan mendatang dari keadaan saat ini.^[1]^[2] Namun, beberapa sistem bersifat stokastik, dalam peristiwa-peristiwa acak yang juga berdampak pada evolusi keadaan yang beragam.

Dalam fisika, sistem dinamikal dideskripsikan sebagai sebuah "partikel atau kelompok dari partikel yang keadaannya beragam sepanjang waktu dan kemudian menunjukkan persamaan diferensial yang melibatkan derivatif waktu."^[3] Dalam rangkaian untuk membuat sebuah prediksi tentang perilaku mendatang dari sistem tersebut, sebuah solusi analitik dari persamaan semacam itu atau integrasi mereka sepanjang waktu melalui simulasi komputer direalisasikan.

Studi sistem dinamikal adalah fokus teori sistem dinamikal, yang memiliki aplikasi kepada serangkaian besar bidang seperti matematika, fisika,^[4]^[5] biologi,^[6] kimia, teknik,^[7] ekonomi,^[8] dan kedokteran. Sistem dinamikal adalah sebuah bagian fundamental dari teori kekacauan, dinamika peta logistik, teori bifurkasi, proses majelis diri, dan konsep tepi kekacauan.

Referensi

^ Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.
^ Katok, A.; Hasselblatt, B. (1995). Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press. ISBN 0-521-34187-6.
^ "Nature". Springer Nature. Diakses tanggal 17 February 2017.
^ Melby, P.; et.al. (2005). "Dynamics of Self-Adjusting Systems With Noise". Chaos 15. Bibcode:2005Chaos..15c3902M. doi:10.1063/1.1953147.
^ Gintautas, V.; et.al. (2008). "Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics". J. Stat. Phys. 130. arXiv:0705.0311  . Bibcode:2008JSP...130..617G. doi:10.1007/s10955-007-9444-4.
^ Jackson, T.; Radunskaya, A. (2015). Applications of Dynamical Systems in Biology and Medicine. Springer.
^ Kreyszig, Erwin (2011). Advanced Engineering Mathematics. Hoboken: Wiley. ISBN 978-0-470-64613-7.
^ Gandolfo, Giancarlo (2009) [1971]. Economic Dynamics: Methods and Models (edisi ke-Fourth). Berlin: Springer. ISBN 978-3-642-13503-3.

Bacaan tambahan

Karya yang menyediakan sorotan besar:

Ralph Abraham and Jerrold E. Marsden (1978). Foundations of mechanics. Benjamin–Cummings. ISBN 0-8053-0102-X. (available as a reprint: ISBN 0-201-40840-6)
Encyclopaedia of Mathematical Sciences (ISSN 0938-0396) has a sub-series on dynamical systems with reviews of current research.
Christian Bonatti; Lorenzo J. Díaz; Marcelo Viana (2005). Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer. ISBN 3-540-22066-6.
Stephen Smale (1967). "Differentiable dynamical systems". Bulletin of the American Mathematical Society. 73 (6): 747–817. doi:10.1090/S0002-9904-1967-11798-1.

Teks-teks pengenalan dengan sudut pandang unik:

V. I. Arnold (1982). Mathematical methods of classical mechanics. Springer-Verlag. ISBN 0-387-96890-3.
Jacob Palis and Welington de Melo (1982). Geometric theory of dynamical systems: an introduction. Springer-Verlag. ISBN 0-387-90668-1.
David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press. ISBN 0-12-601710-7.
Tim Bedford, Michael Keane and Caroline Series, eds. (1991). Ergodic theory, symbolic dynamics and hyperbolic spaces. Oxford University Press. ISBN 0-19-853390-X.
Ralph H. Abraham and Christopher D. Shaw (1992). Dynamics—the geometry of behavior, 2nd edition. Addison-Wesley. ISBN 0-201-56716-4.

Buku teks

Kathleen T. Alligood, Tim D. Sauer and James A. Yorke (2000). Chaos. An introduction to dynamical systems. Springer Verlag. ISBN 0-387-94677-2.
Oded Galor (2011). Discrete Dynamical Systems. Springer. ISBN 978-3-642-07185-0.
Morris W. Hirsch, Stephen Smale and Robert L. Devaney (2003). Differential Equations, dynamical systems, and an introduction to chaos. Academic Press. ISBN 0-12-349703-5.
Anatole Katok; Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN 0-521-57557-5.
Stephen Lynch (2010). Dynamical Systems with Applications using Maple 2nd Ed. Springer. ISBN 0-8176-4389-3.
Stephen Lynch (2007). Dynamical Systems with Applications using Mathematica. Springer. ISBN 0-8176-4482-2.
Stephen Lynch (2014). Dynamical Systems with Applications using MATLAB 2nd Edition. Springer International Publishing. ISBN 3319068199.
James Meiss (2007). Differential Dynamical Systems. SIAM. ISBN 0-89871-635-7.
David D. Nolte (2015). Introduction to Modern Dynamics: Chaos, Networks, Space and Time. Oxford University Press. ISBN 978-0199657032.
Julien Clinton Sprott (2003). Chaos and time-series analysis. Oxford University Press. ISBN 0-19-850839-5.
Steven H. Strogatz (1994). Nonlinear dynamics and chaos: with applications to physics, biology chemistry and engineering. Addison Wesley. ISBN 0-201-54344-3.
Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
Stephen Wiggins (2003). Introduction to Applied Dynamical Systems and Chaos. Springer. ISBN 0-387-00177-8.

Popularisasi:

Florin Diacu and Philip Holmes (1996). Celestial Encounters. Princeton. ISBN 0-691-02743-9.
James Gleick (1988). Chaos: Making a New Science. Penguin. ISBN 0-14-009250-1.
Ivar Ekeland (1990). Mathematics and the Unexpected (Paperback). University Of Chicago Press. ISBN 0-226-19990-8.
Ian Stewart (1997). Does God Play Dice? The New Mathematics of Chaos. Penguin. ISBN 0-14-025602-4.

Pranala luar

MATLAB files http://uk.mathworks.com/matlabcentral/profile/authors/63144-stephen-lynch
Interactive applet for the Standard and Henon Maps by A. Luhn
A collection of dynamic and non-linear system models and demo applets (in Monash University's Virtual Lab)
Arxiv preprint server has daily submissions of (non-refereed) manuscripts in dynamical systems.
DSWeb provides up-to-date information on dynamical systems and its applications.
Encyclopedia of dynamical systems A part of Scholarpedia — peer reviewed and written by invited experts.
Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
Oliver Knill has a series of examples of dynamical systems with explanations and interactive controls.
Sci.Nonlinear FAQ 2.0 (Sept 2003) provides definitions, explanations and resources related to nonlinear science

Buku maya atau catatan ceramah:

Geometrical theory of dynamical systems. Nils Berglund's lecture notes for a course at ETH at the advanced undergraduate level.
Dynamical systems. George D. Birkhoff's 1927 book already takes a modern approach to dynamical systems.
Chaos: classical and quantum. An introduction to dynamical systems from the periodic orbit point of view.
Learning Dynamical Systems. Tutorial on learning dynamical systems.
Ordinary Differential Equations and Dynamical Systems. Lecture notes by Gerald Teschl

Kelompok riset:

Dynamical Systems Group Groningen, IWI, University of Groningen.
Chaos @ UMD. Concentrates on the applications of dynamical systems.
[1], SUNY Stony Brook. Lists of conferences, researchers, and some open problems.
Center for Dynamics and Geometry, Penn State.
Control and Dynamical Systems, Caltech.
Laboratory of Nonlinear Systems, Ecole Polytechnique Fédérale de Lausanne (EPFL).
Center for Dynamical Systems, University of Bremen
Systems Analysis, Modelling and Prediction Group, University of Oxford
Non-Linear Dynamics Group, Instituto Superior Técnico, Technical University of Lisbon
Dynamical Systems, IMPA, Instituto Nacional de Matemática Pura e Applicada.
Nonlinear Dynamics Workgroup, Institute of Computer Science, Czech Academy of Sciences.

Perangkat lunak simulasi yang berdasarkan pada kesepakatan Sistem Dinamikal:

FyDiK
iDMC, simulation and dynamical analysis of nonlinear models

[1] Strogatz, S. H. (2001). Nonlinear Dynamics and Chaos: with Applications to Physics, Biology and Chemistry. Perseus.

[2] Katok, A.; Hasselblatt, B. (1995). Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press. ISBN 0-521-34187-6.

[3] "Nature". Springer Nature. Diakses tanggal 17 February 2017.

[4] Melby, P.; et.al. (2005). "Dynamics of Self-Adjusting Systems With Noise". Chaos 15. Bibcode:2005Chaos..15c3902M. doi:10.1063/1.1953147.

[5] Gintautas, V.; et.al. (2008). "Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics". J. Stat. Phys. 130. arXiv:0705.0311  . Bibcode:2008JSP...130..617G. doi:10.1007/s10955-007-9444-4.

[6] Jackson, T.; Radunskaya, A. (2015). Applications of Dynamical Systems in Biology and Medicine. Springer.

[7] Kreyszig, Erwin (2011). Advanced Engineering Mathematics. Hoboken: Wiley. ISBN 978-0-470-64613-7.

[8] Gandolfo, Giancarlo (2009) [1971]. Economic Dynamics: Methods and Models (edisi ke-Fourth). Berlin: Springer. ISBN 978-3-642-13503-3.

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