洛伦兹振荡器,一个动态系统的研究中出现洛伦茨吸引子

动态系统dynamical system)是数学上的一个概念。动态系统是一种固定的规则,它描述一个给定空间(如某个物理系统的状态空间)中所有点随时间的变化情况。例如描述钟摆晃动、管道中水的流动,或者湖中每年春季鱼类的数量,凡此等等的数学模型都是动态系统。

在动态系统中有所谓状态的概念,状态是一组可以被确定下来的实数。状态的微小变动对应这组实数的微小变动。这组实数也是一种流形的几何空间坐标。动态系统的演化规则是一组函数固定规则,它描述未来状态如何依赖于当前状态的。这种规则是确定性的,即对于给定的时间间隔内,从现在的状态只能演化出一个未来的状态。

若只是在一系列不连续的时间点考察系统的状态,则这个动态系统为离散动态系统;若时间连续,就得到一个连续动态系统。如果系统以一种连续可微的方式依赖于时间,我们就称它为一个光滑动态系统

参见

参考书籍

  • 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.
  • Introduction to Social Macrodynamics. Mathematical models of the World System development
  • Differential Equations, Dynamical Systems, and an Introduction to Chaos 微分方程、动力系统与混沌导论

延伸阅读

Works providing a broad coverage:

  • Ralph Abraham and Jerrold E. Marsden. Foundations of mechanics. Benjamin–Cummings. 1978. 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. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective. Springer. 2005. ISBN 3-540-22066-6. 
  • Stephen Smale. Differentiable dynamical systems. Bulletin of the American Mathematical Society. 1967, 73 (6): 747–817. doi:10.1090/S0002-9904-1967-11798-1. 

Introductory texts with a unique perspective:

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

Textbooks

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

Popularizations:

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

外部链接

  • 动态系统介绍
  • Dynamical systems at SUNY:纽约州立大学石溪校区研究小组的网站,有会议、研究者和未解决问题等资料(英)
  • Oliver Knill:以JavaScript说明一些动态系统的例子(英)
  • Arxiv preprint server:关于此范畴每日的新论文(英)
  • Chaos @ UMD:主攻应用层面(英)
  • 动态系统的稳定性分析[永久失效链接]