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# E figure 24 shows different phase portraits for the

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(e) Figure 24 shows different phase portraits for the system for various values of μ above and below the critical value μ = - 0 . 8645. 27

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mu=–0.55 –4 –2 0 2 4 y –4 –2 2 4 x mu=–0.75 –4 –2 0 2 4 y –4 –2 2 4 x mu=–0.8 –4 –2 0 2 4 y –4 –2 2 4 x mu=–0.95 –4 –2 0 2 4 y –4 –2 2 4 x Figure 23: Phase portraits for Exercise 2c for various values of μ between - 1 and - 0 . 5. 28
mu=–0.855 –4 –2 0 2 4 y –4 –2 2 4 x mu=–0.8645 –4 –2 0 2 4 y –4 –2 2 4 x mu=–0.866 –4 –2 0 2 4 y –4 –2 2 4 x mu=–0.87 –4 –2 0 2 4 y –4 –2 2 4 x Figure 24: Phase portraits for Exercise 2e for various values of μ above and below the critical value of μ c = - 0 . 8645. 29

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3. Purpose: To determine the type of bifurcation. From [Str94], Problem 8.1.6 . Exercise: Consider the system ˙ x = y - 2 x, ˙ y = μ + x 2 - y . (a) Sketch the nullclines. (b) Find and classify the bifurcations that occur as μ varies. (c) Sketch the phase portrait as a function of μ . Solution: (a) The nullclines for this system are given by y = 2 x and y = μ + x 2 , however the number of fixed points varies depending on the value of μ . If μ > 1, the there are no fixed points for the system (see Figure 25). For μ = 1, one fixed point occurs (see Figure 26), and for μ < 1, there are 2 fixed points for the system (see Figure 27). –4 –2 0 2 4 y –2 –1 1 2 3 x Figure 25: The nullclines for the μ > 1 case –4 –2 0 2 4 y –2 –1 1 2 3 x Figure 26: The nullclines for the μ = 1 case (b) The bifurcations of the system occur at the critical value of μ c = 1, and is a saddle-node bifurcation. This can be found by evaluating the Jacobian, finding the eigenvalues and classifying the fixed points for μ 1. 30
–4 –2 0 2 4 y –2 –1 1 2 3 x Figure 27: The nullclines for the μ < 1 case 31

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References [Gri71] J.S. Griffith. Mathematical Neurobiology . Academic Press, 1971. [Hea02] Michael T. Heath. Scientific Computing: An Introductory Survey . McGraw Hill, second edition, 2002. [KMTP94] Vladmir A. Kuznetsov, Iliya A. Makalkin, Mark A. Taylor, and Alan S. Perelson. Nonlinear dy- namics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bulletin of Mathematical Biology , 56(2), 1994. [Pan96] John Carl Panetta. A mathematical model of periodically pulsed chemotherapy: Tumor re- currence and metastasis in a competitive environment. Bulletin of Mathematical Biology , 58(3):425–227, 1996. [Str94] Steven H. Strogatz. Nonlinear Dynamics and Chaos . Addison-Wesley, 1994. 32
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