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第33行： − In the mathematical language of [[Dynamical systems theory|dynamic systems analysis]], one of the simplest bistable systems is+ − − In the mathematical language of dynamic systems analysis, one of the simplest bistable systems is − − 在动态系统分析的数学语言中，最简单的双稳系统之一是 − − + − </math> − − − This system describes a ball rolling down a curve with shape <math>\frac{y^4}{4} - \frac{y^2}{2}</math>, and has three equilibrium points: <math> y = 1 </math>, <math> y = 0 </math>, and <math> y = -1</math>. The middle point <math>y=0</math> is unstable, while the other two points are stable. The direction of change of <math>y(t)</math> over time depends on the initial condition <math>y(0)</math>. If the initial condition is positive (<math>y(0)>0</math>), then the solution <math>y(t)</math> approaches 1 over time, but if the initial condition is negative (<math>y(0)< 0</math>), then <math>y(t)</math> approaches −1 over time. Thus, the dynamics are "bistable". The final state of the system can be either <math> y = 1 </math> or <math> y = -1 </math>, depending on the initial conditions.<ref name="Chong">{{cite journal | author = Ket Hing Chong | author2 = Sandhya Samarasinghe | author3 = Don Kulasiri | author4 = Jie Zheng | name-list-style = amp | year = 2015| title = Computational techniques in mathematical modelling of biological switches | journal = Modsim2015 | pages = 578–584 }} For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in {{cite journal |last1=Collins |first1=James J. |last2=Gardner |first2=Timothy S. |last3=Cantor |first3=Charles R. |title=Construction of a genetic toggle switch in Escherichia coli |journal=Nature |volume=403 |issue=6767 |pages=339–42 |year=2000 |pmid=10659857 |doi=10.1038/35002131 |bibcode=2000Natur.403..339G |s2cid=345059 }}. The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'.</ref> − − This system describes a ball rolling down a curve with shape \frac{y^4}{4} - \frac{y^2}{2}, and has three equilibrium points: y = 1 , y = 0 , and y = -1. The middle point y=0 is unstable, while the other two points are stable. The direction of change of y(t) over time depends on the initial condition y(0). If the initial condition is positive (y(0)>0), then the solution y(t) approaches 1 over time, but if the initial condition is negative (y(0)< 0), then y(t) approaches −1 over time. Thus, the dynamics are "bistable". The final state of the system can be either y = 1 or y = -1 , depending on the initial conditions. For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in . The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'. − − 这个系统描述了一个球沿曲线滚动的形状为 y ^ 4}{4}-frac { y ^ 2}{2} ，有三个平衡点: y = 1，y = 0，y =-1。中点 y = 0是不稳定的，而其他两点是稳定的。Y (t)随时间的变化方向取决于初始条件 y (0)。如果初始条件是正的(y (0) > 0) ，那么解 y (t)随时间接近1，但如果初始条件是负的(y (0) < 0) ，那么 y (t)随时间接近 -1。因此，动力学是“双稳态”的。系统的最终状态可以是 y = 1或 y =-1，这取决于初始条件。有关双稳态数学建模的详细技术，请参阅 Chong 等人的教程。(2015) http://www.mssanz.org.au/modsim2015/c2/chong.pdf 教程提供了一个简单的双稳态示例，使用了一个合成切换开关建议在。本教程还使用动力系统的软件 XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html 来实际展示如何看到双稳态捕获的鞍结分岔图和滞后行为当分叉参数增加或减少的临界点和蛋白质得到打开或关闭。 − − The appearance of a bistable region can be understood for the model system − <math> − \frac{dy}{dt} = y (r-y^2) − which undergoes a supercritical [[pitchfork bifurcation]] with [[Bifurcation theory|bifurcation parameter]] <math> r </math>. − + − \frac{dy}{dt} = y (r-y^2) − which undergoes a supercritical pitchfork bifurcation with bifurcation parameter r . − 模型系统 frac { dy }{ dt } = y (r-y ^ 2)经历具有参数 r 的超临界叉式分岔时，可以理解双稳区的出现。+
→数学模型
== 数学模型 ==
== 数学模型 ==
双稳性动力学系统的最简单的样例用数学语言描述如下
:<math>
:<math>
\frac{dy}{dt} = y (1-y^2).
\frac{dy}{dt} = y(1-y^2).
</math>
</math>
The appearance of a bistable region can be understood for the model system
该方程有三个平衡点: <math>y=1</math>, <math>y=0</math>, and <math>y=-1</math>。中点 <math>y=0</math> 不稳定，而其他两点是稳定的。<math>y(t)</math>的演化方向和最终状态取决于初始条件 <math>y(0)</math>。若 <math>y(0)>0</math>，则 <math>y(t)</math> 趋向于1，若 <math>y(0)<0</math>，则 <math>y(t)</math> 趋向-1。<ref name="Chong">{{cite journal | author = Ket Hing Chong | author2 = Sandhya Samarasinghe | author3 = Don Kulasiri | author4 = Jie Zheng | name-list-style = amp | year = 2015| title = Computational techniques in mathematical modelling of biological switches | journal = Modsim2015 | pages = 578–584 }} For detailed techniques of mathematical modelling of bistability, see the tutorial by Chong et al. (2015) http://www.mssanz.org.au/modsim2015/C2/chong.pdf The tutorial provides a simple example illustration of bistability using a synthetic toggle switch proposed in {{cite journal |last1=Collins |first1=James J. |last2=Gardner |first2=Timothy S. |last3=Cantor |first3=Charles R. |title=Construction of a genetic toggle switch in Escherichia coli |journal=Nature |volume=403 |issue=6767 |pages=339–42 |year=2000 |pmid=10659857 |doi=10.1038/35002131 |bibcode=2000Natur.403..339G |s2cid=345059 }}. The tutorial also uses the dynamical system software XPPAUT http://www.math.pitt.edu/~bard/xpp/xpp.html to show practically how to see bistability captured by a saddle-node bifurcation diagram and the hysteresis behaviours when the bifurcation parameter is increased or decreased slowly over the tipping points and a protein gets turned 'On' or turned 'Off'.</ref>
更复杂的双稳性系统 <math>\frac{dy}{dt} = y (r-y^2)</math> 具有超临界的<font color="#ff8000">叉分岔pitchfork bifurcation</font>[[pitchfork bifurcation]]现象。
==In biological and chemical systems==
==In biological and chemical systems==