Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please show all work!

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).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 6 steps with 7 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS