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For each problem, sketch all of the qualitatively different vector fields that occur as the parameter...
In the following exercises, sketch all the qualitatively different vector fields that occur as r is...
In the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a pitchfork bifurcation occurs at a critical value of r (to be determined) and classify the bifurcation as supercritical or subcritical. Finally, sketch the bifurcation diagram of x* vs. r. rx 3.4.4 * = x+- 1+x2
3.1.2 please :) EXERCISES FOR CHAPTER 3 3.1 Saddle-Node Bifurcation For each of the following exercises,...
3.1.2 please :)
EXERCISES FOR CHAPTER 3 3.1 Saddle-Node Bifurcation For each of the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. Finally, sketch the bifurcation diagram of fixed pointsversus 3.1.1 1+rx+ 3.1.2 ir-coshx 3.1.4 x=『竹x-x/(1+1) .1.sV (Unusual bifurcations) In discussing the nomal form of the saddle-node bi-
Nonlinear differential equations and Bifurcation theory. Given the ordinary differential equation =1+re+ where the parameter r...
Nonlinear differential equations and Bifurcation
theory.
Given the ordinary differential equation =1+re+ where the parameter r is a given real number. (a) Sketch all the qualitatively different vector fields that occur as r is varied. (b) Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. (c) Sketch the bifurcation diagram of fixed points r* versus r.
For each of the following equations you should locate all equilibrium points and investigate their stability...
For each of the following equations you should locate all equilibrium points and investigate their stability properties for different values of the parameter u ER. You should then draw a summary/bifurcation diagram, with arrows, indicating the qualitative behaviour of the non-equilibrium solutions for all values of y, including at the bifurcation point u = u*. Note any bifurcations that occur and determine which of the following terms can be used to describe the nature of the bifurcations you find: transcritical,...
Question #5 all parts thanks 5. Find the solution of the heat conduction problem for each...
Question #5 all parts thanks
5. Find the solution of the heat conduction problem for each initial condition given: 0<x <6, t> 0. (a) ux,0)-x)-4sin(x)-3sin(2x) +7sin(570:). (b) ux, 0)-x)-9t (c) In each of cases (a) and (b), find the limit of u(3,1) as t approaches oo. Are they different? Did you 45 expect them to be different?
The solution to the initial value problem below takes on different forms for different ranges of the parameter ζ, which is called the dimensionless damping coefficient. In each case, write the solutio...
The solution to the initial value problem below takes on
different forms for different ranges of the parameter ζ, which is
called the dimensionless damping coefficient. In each case, write
the solution in the suggested form. u'' + 6 ζ u' +9u = 0 u(0) = 0
u'(0) = 1 If ζ > 1 , uζ(t) = e( Incorrect: Your answer is
incorrect. ) sinh( Incorrect: Your answer is incorrect. )
Incorrect: Your answer is incorrect. Compute the limit as...
Problem 7. Given that each of the following vector fields F is conservative Find a potential...
Problem 7. Given that each of the following vector fields F is conservative Find a potential function f such that f = F and evaluate fe F dr along the given curve C 1. F(r,y) y C: F(t)(t3- 2t, t3 + 2t), 0 <t<1 2. F(x,y, ) yze"* i + e#* j + xye k C: F(t)(t2 1)i +(2 -1)( -2t)k, 0t 2
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
• • Show all of your work for each problem. Draw a line separating each problem....
• • Show all of your work for each problem. Draw a line separating each problem. 1) Find an equation of the sphere that passes through the point (6, -2,3) and has center (-1,2,1). 2) Find the values of x such that the vectors (3,2,x) and (2x, 4,x) are orthogonal 3) Find the velocity, speed, and acceleration of a particle moving with position function r(t) = (2+2 - 3)i + 2tj. Sketch the path of the particle and draw the...
23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health....
23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity a. Find all of the critical p there are no critical points i and two critical points if a O b. Draw the phase line each critical point is asy Consider...
In the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a pitchfork bifurcation occurs at a critical value of r (to be determined) and classify the bifurcation as supercritical or subcritical. Finally, sketch the bifurcation diagram of x* vs. r. rx 3.4.4 * = x+- 1+x2
3.1.2 please :)
EXERCISES FOR CHAPTER 3 3.1 Saddle-Node Bifurcation For each of the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. Finally, sketch the bifurcation diagram of fixed pointsversus 3.1.1 1+rx+ 3.1.2 ir-coshx 3.1.4 x=『竹x-x/(1+1) .1.sV (Unusual bifurcations) In discussing the nomal form of the saddle-node bi-
Nonlinear differential equations and Bifurcation
theory.
Given the ordinary differential equation =1+re+ where the parameter r is a given real number. (a) Sketch all the qualitatively different vector fields that occur as r is varied. (b) Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. (c) Sketch the bifurcation diagram of fixed points r* versus r.
For each of the following equations you should locate all equilibrium points and investigate their stability properties for different values of the parameter u ER. You should then draw a summary/bifurcation diagram, with arrows, indicating the qualitative behaviour of the non-equilibrium solutions for all values of y, including at the bifurcation point u = u*. Note any bifurcations that occur and determine which of the following terms can be used to describe the nature of the bifurcations you find: transcritical,...
Question #5 all parts thanks
5. Find the solution of the heat conduction problem for each initial condition given: 0<x <6, t> 0. (a) ux,0)-x)-4sin(x)-3sin(2x) +7sin(570:). (b) ux, 0)-x)-9t (c) In each of cases (a) and (b), find the limit of u(3,1) as t approaches oo. Are they different? Did you 45 expect them to be different?
The solution to the initial value problem below takes on
different forms for different ranges of the parameter ζ, which is
called the dimensionless damping coefficient. In each case, write
the solution in the suggested form. u'' + 6 ζ u' +9u = 0 u(0) = 0
u'(0) = 1 If ζ > 1 , uζ(t) = e( Incorrect: Your answer is
incorrect. ) sinh( Incorrect: Your answer is incorrect. )
Incorrect: Your answer is incorrect. Compute the limit as...
Problem 7. Given that each of the following vector fields F is conservative Find a potential function f such that f = F and evaluate fe F dr along the given curve C 1. F(r,y) y C: F(t)(t3- 2t, t3 + 2t), 0 <t<1 2. F(x,y, ) yze"* i + e#* j + xye k C: F(t)(t2 1)i +(2 -1)( -2t)k, 0t 2
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
• • Show all of your work for each problem. Draw a line separating each problem. 1) Find an equation of the sphere that passes through the point (6, -2,3) and has center (-1,2,1). 2) Find the values of x such that the vectors (3,2,x) and (2x, 4,x) are orthogonal 3) Find the velocity, speed, and acceleration of a particle moving with position function r(t) = (2+2 - 3)i + 2tj. Sketch the path of the particle and draw the...
23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity a. Find all of the critical p there are no critical points i and two critical points if a O b. Draw the phase line each critical point is asy Consider...
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