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…heory'''是数学中研究给定族的定性或[[拓扑结构]]的改变，例如[[向量场]]中的一族[[积分曲线]]以及[[微分方程]]的一族解。'''分岔 Bifurcation'''常用于[[动力系统]]的数学研究中，是指当系统的参数值 *局部分岔 Local bifurcations是指可用平衡点的局部稳定性、周期轨道或其他不变集作

…which the function <math>f_c(z)=z^2+c</math> does not [[diverge (stability theory)|diverge]] when [[Iteration|iterated]] from <math>z=0</math>, i.e., for whi …Psychology, and Life Sciences, Vol. 12, No. 1 |publisher=Society for Chaos Theory in Psychology & Life Sciences }}</ref>

…Psychology, and Life Sciences, Vol. 12, No. 1 |publisher=Society for Chaos Theory in Psychology & Life Sciences }}</ref>且Benoît B. Mandelbrot在1980年发

[[Graph theory]]<br /> |list6title=Systems theory

…y [[Edward Norton Lorenz|Edward Lorenz]]. It is notable for having [[Chaos theory|chaotic]] solutions for certain parameter values and initial conditions. In A [[pitchfork bifurcation]] occurs at <math>\rho = 1</math>, and for <math>\rho > 1 </math> two addit

…74-0242ac1a0005 "Yet another chaotic attractor."] International Journal of Bifurcation and Chaos, 9 (7): 1465-1466, 1999. …9-0242ac1a0005 "A new chaotic attractor coined."] International Journal of Bifurcation and Chaos, 12(3): 659-661, 2002.

…/fractalfoundation.org/resources/what-is-chaos-theory/|title=What is Chaos Theory? – Fractal Foundation|language=en-US|access-date=2019-11-24}}</ref> [[蝴 …性。<ref>{{cite journal|date=March 2016|title=Introduction to Supersymmetric Theory of Stochastics|journal=Entropy|volume=18|issue=4|pages=108|doi=10.3390/e180

| [[File:225px-Pitchfork_bifurcation_left.png|thumb|upright=0.7|叉式分岔 a = 0，表面 b = 0]]

…学动力系统理论 mathematical dynamical systems theory、或动力学系统的数学理论 the mathematical theory of dynamical systems。

A '''dissipative system''' is a thermodynamically [[open system (systems theory)|open system]] which is operating out of, and often far from, [[thermodynam …breaking ([[anisotropy]]) and the formation of complex, sometimes [[Chaos theory|chaotic]], structures where interacting particles exhibit long range correl

…ilibrium in biological systems. The replicator equation, evolutionary game theory, Markov processes and chemical reaction networks all describe the dynamics …how it follows from a much broader philosophical framework, guided by the theory of individuation, which emphasizes the importance of relationships and inte

…。<ref name = " von Neumann2 " >von Neumann, John; Burks, Arthur W. (1966). Theory of Self-Reproducing Automata. University of Illinois Press.</ref>

…神经元网络。Wilson和 Cowan<ref name=":1">H R Wilson and J D Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kyberne …09</ref> 和 Bressloff <ref name=":6">P C Bressloff. Stochastic neural field theory and the system-size expansion. SIAM J. Applied Mathematics, 70:1488-1521,…

…l | vauthors = Tamagawa H, Funatani M, Ikeda K | title = Ling's Adsorption Theory as a Mechanism of Membrane Potential Generation Observed in Both Living and …51" group="lower-alpha">{{cite journal | vauthors = Rushton WA | title = A theory of the effects of fibre size in medullated nerve | journal = The Journal of

*'''[[分岔理论]] Bifurcation Theory''' *'''[[摄动理论]] Perturbation Theory'''（也可应用于代数方程）

…k|title = The Unity of Science and Economics: A New Foundation of Economic Theory|last = Chen|first = Jing|publisher = Springer|year = 2015|isbn = |location …型 Brusselator'''中。如果系统被驱动超过一定的阈值，振荡不再被阻尼，而是可能被放大。从数学上讲，这相当于一个'''霍普夫分岔 Hopf bifurcation'''，其中一个参数增加超过一定的值会导致'''极限环行为

…causal structure of bivariate time series|journal=International Journal of Bifurcation and Chaos|date=1 March 2007|volume=17|issue=3|pages=903–921|doi=10.1142/S

[[File:Hopfbifurcation.png|thumb|400px|'''<font color="#ff8000">霍普夫分岔 Hopf bifurcation</font>'''附近不动点分支的极限环例子。红色的轨迹，深蓝 * Solomon Lefschetz, "Differential Equations: Geometric Theory", Dover, 2005.

*2000-2011年 '''分叉与混乱 Int. Journal Bifurcation & Chaos''' *自2017年以来 '''先进理论与模拟 Advanced Theory and Simulations'''