Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 3.7, Problem 7E
Interpretation Introduction

Interpretation:

To sketch the graph of the nonlinear function g(p) = βpnKn+pn, for n >> 1 and to find the simple shape the function approaches as n, if the basal expression is given as p˙ = α +βpnKn+pn - δp. The phase portrait for the system for δK - α > β, δK - α = β2, and δK - α < 0 is to be plotted. To plot the bifurcation diagram for the system, assuming δK > β, and to indicate how the location and stability of the fixed points p* vary with respect to α. To discuss how p behaves if is very slowly

increased from α = 0 to α > δK, and then very slowlydecreased back to α = 0. To show that such a pulsed stimulation leads to hysteresis.

Concept Introduction:

The basal expression is

p˙ = α +βpnKn+pn - δp

Here α is the basal transcription rate, β is the maximal transcription rate, K is the activation coefficient, and δ is the decay rate of the protein.

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9. Needing a break from studying, you take a walk to the Pogonip koi pond, whereupon a wild-eyed stranger pops out from behind a redwood tree and directs the following polemic in your general direction: "The lies those so-called teachers at that university promulgate, let me tell you. I know the truth that they don't want you to know. As plain as day, " = 0 for all n ≥0. It's an easy induction proof, see?" He hands you a leaflet, where you see the proof that they don't want you to see: We proceed by strong induction on n. Base case: n = 0. We have 10: Induction step: Assume that d1 = = = 0. dx dxk dx = 0 for all kn. Then, by the product rule, nd dx da 1x+1 = 1/1(x²x²) = x²±²x² + x 11 x² d = x.0+x¹.0 0. dx This completes the induction. That derivative rule doesn't seem like the one you learned, but there's nothing obviously wrong with the proof. Is he right, are the math professors propping up the interests of Big Calculus? Or should he have paid better attention in CSE 16? What's going…
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