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ore generally, we allow a and I to be nonzero, but still rest i. Show that in this case, the system still has only one critic (Hint: It is enough to show that the set of solutions for 11 L

ore generally, we allow a and I to be nonzero, but still rest i. Show that in this case, the system still has only one critic (Hint: It is enough to show that the set of solutions for 11 L

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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2. Stability properties of almost linear systems. The Fitzhugh-Nagumo system models the transmission of neural impulses along an axon. The system of equations is given by 1³ x = F(x,y) = y + x· +I 3 y' = G(x,y) = −x+a − by where a, b, and I are constant parameters (I is the external stimulus that leads to excitation of the system, which is like an applied current). In these equations x is similar to the voltage and represents the excitability of the system; the variable y represents a combination of other forces that tend to return the system to rest. This system captures the essential behavior of nerve impulses. (a) First consider a special case where a = I = 0, 0 < b < 1. Find the critical point and describe its type and stability.
Transcribed Image Text:2. Stability properties of almost linear systems. The Fitzhugh-Nagumo system models the transmission of neural impulses along an axon. The system of equations is given by 1³ x = F(x,y) = y + x· +I 3 y' = G(x,y) = −x+a − by where a, b, and I are constant parameters (I is the external stimulus that leads to excitation of the system, which is like an applied current). In these equations x is similar to the voltage and represents the excitability of the system; the variable y represents a combination of other forces that tend to return the system to rest. This system captures the essential behavior of nerve impulses. (a) First consider a special case where a = I = 0, 0 < b < 1. Find the critical point and describe its type and stability.
(b) More generally, we allow a and I to be nonzero, but still restrict to 0 < b < 1. i. Show that in this case, the system still has only one critical point regardless of the value of I. (Hint: It is enough to show that the set of solutions for F(x,y)=0 and G(x,y)=0 consists of a single point. Recall that a monotone cubic polynomial has a single real root. ) ii. For the case a = 1, b = 3,1 = 0, the only critical point is (xo, Yo) = (1, -3). Obtain the linear system satisfied by u = x -xo, w = y - yo near the critical point, and determine the stability near (xo, yo).
Transcribed Image Text:(b) More generally, we allow a and I to be nonzero, but still restrict to 0 < b < 1. i. Show that in this case, the system still has only one critical point regardless of the value of I. (Hint: It is enough to show that the set of solutions for F(x,y)=0 and G(x,y)=0 consists of a single point. Recall that a monotone cubic polynomial has a single real root. ) ii. For the case a = 1, b = 3,1 = 0, the only critical point is (xo, Yo) = (1, -3). Obtain the linear system satisfied by u = x -xo, w = y - yo near the critical point, and determine the stability near (xo, yo).
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