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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>
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>现象。
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>现象。