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Bifurcations occur at the points on a continuation diagram where a major change takes place. The speciﬁc value of the parameter where the change occurs is the bifurcation point. In these cases, the continuation diagram is known as a bifurcation diagram. 8) as a function of the parameter q. In addition to the position of the equilibrium, the stability type is indicated by the line style: a stable (attracting) steady state is indicated by the solid line from q ¼ 1 to q A 2:86; an unstable (repelling) ﬁxed point, by the dashed line from that point up until q ¼ 5.
Then SWX ¼ ÀSw0 ; and SW S þ SW T S T þ S diagðWX þ w0 ÞS T ¼ 0: The latter is an algebraic Lyapunov equation. Such equations arise in many applications in control theory, and e‰cient numerical techniques exist for their solution. Example: Application to Gene Expression The above stationary covariance equations can be applied to compute exact expressions for the mean and coe‰cient of variation of a simple gene expression circuit. 3. Let X1 ðtÞ and X2 ðtÞ be random variables describing the number of mRNA and protein molecules, respectively.
To deﬁne an inputoutput system, we consider the parameter k2 as the input by setting uðtÞ ¼ k2 ðtÞ and the concentration of S2 as output: yðtÞ ¼ s2 ðtÞ À s2 . We linearize about the equilibrium with input set to u ¼ 20 nMhÀ1 . The corresponding steady state is ðs1 ; s2 Þ A ð5:53; 3:37Þ. a11 a12 We now compute the Jacobian A ¼ : a21 a22 a11 ¼ qf1 ¼ ÀðkÀ1 þ kÀ3 Þ; qs1 a12 ¼ qf1 ¼ k3 ; qs2 a21 ¼ k4 uqs1qÀ1 qf2 ¼À þ kÀ3 ; qs1 ð1 þ k4 s1q Þ 2 a22 ¼ qf2 ¼ ÀðkÀ2 þ k3 Þ; qs2 and the input matrix B ¼ b1 ¼ qf1 ¼ 0; qu b2 ¼ !