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Consider the nondimensional differential equation du where u is an unknown parameter (constant) (a) Determine the equil...
2(a) Consider the one-parameter family of nonlinear ordinary differential equations -Ita-) where ...
2(a) Consider the one-parameter family of nonlinear ordinary differential equations -Ita-) where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them
2(a) Consider the one-parameter family of nonlinear ordinary differential equations -Ita-) where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordin...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordinary differential equations dr where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them. 10 Marks (b) Consider the second order differential equation i. Show that (1) can be written as the system of ordinary differential equations (y R for...
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.
(5) Consider the following differential equation: z' = (12-4)(p-r2) Here p is a parameter. Sketch a...
(5) Consider the following differential equation: z' = (12-4)(p-r2) Here p is a parameter. Sketch a bifurcation diagram for this differential equation which indicates the type of equilibria in each branch of the diagram. Also indicate all bifurcation values and classify the type of each bifurca tion.
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by...
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by examining dt the equilibrium solutions of the equation for various values of the parameter 8 1. Find the equilibrium solutions of the equation for 8 = -4,-2, 0, 2, 4 and make a sketch of the phase line for each value. Determine the stability of each equilibria. 2. Use a computer or some other means to sketch some solution curves for each value of...
3. Consider the following stiff system of autonomous ordinary differential equations du f(u, u) =-3u +3,...
3. Consider the following stiff system of autonomous ordinary differential equations du f(u, u) =-3u +3, u(0)2 = ' dt de g(u, v) -2000u - 1000, v(0)-3 Note that 1 u<2 and -4 <v < 3 for all t. (a) Find the Jacobian matrix for the system of equa tions (b) Find the eigenvalues of the Jacobian matrix. (c) In the figure the shaded region shows the region of absolute stability, in the complex h plane, for third order explicit...
Problem 4: Suppose that the movement of rush-hour traffic on a typical expresswa be modeled using the differential equation du du where u(x) is the density of cars (vehicles per mile), and a is dista...
Problem 4: Suppose that the movement of rush-hour traffic on a typical expresswa be modeled using the differential equation du du where u(x) is the density of cars (vehicles per mile), and a is distance miles) in the direction of traffic flow. We w to the boundary conditions ant to solve this equation subject u(0) 300, u(5) 400. a) Use second-order accurate, central-difference approximations to discretize the differential equation and write down the finite-difference equation for a typical point zi...
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where...
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
4) Given the first order equation x'= [lx-x where x = x(t) and u is a...
4) Given the first order equation x'= [lx-x where x = x(t) and u is a real parameter. a) Find all fixed points for u0 and determine whether they are stable or unstable. b) Find all fixed points for u<0 and determine whether they are stable or unstable. c) Sketch the bifurcation diagram for this equation. Be sure to indicate the direction of flow on your diagram (with arrows). This is called a "pitchfork bifurcation" for obvious reasons.
Question 5 (5 marks) Consider the differential equation dy2-r (0) dr where r R is a constant. (a) Find the value(s) of...
Question 5 (5 marks) Consider the differential equation dy2-r (0) dr where r R is a constant. (a) Find the value(s) of r such that the differential equation has at least one equilibrium solution C (b) Determine the equilibrium solution(s) for the value(s) of r found in (a) (c) Use an in (b). Explain your answer. appropriate diagram to determine the stability of the equilibrium solution(s) found
Question 5 (5 marks) Consider the differential equation dy2-r (0) dr where r...
2(a) Consider the one-parameter family of nonlinear ordinary differential equations -Ita-) where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them
2(a) Consider the one-parameter family of nonlinear ordinary differential equations -Ita-) where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordinary differential equations dr where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them. 10 Marks (b) Consider the second order differential equation i. Show that (1) can be written as the system of ordinary differential equations (y R for...
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.
(5) Consider the following differential equation: z' = (12-4)(p-r2) Here p is a parameter. Sketch a bifurcation diagram for this differential equation which indicates the type of equilibria in each branch of the diagram. Also indicate all bifurcation values and classify the type of each bifurca tion.
Bifurcation dy Consider the autonomous differential equation =y? - 2y + 8. We will begin by examining dt the equilibrium solutions of the equation for various values of the parameter 8 1. Find the equilibrium solutions of the equation for 8 = -4,-2, 0, 2, 4 and make a sketch of the phase line for each value. Determine the stability of each equilibria. 2. Use a computer or some other means to sketch some solution curves for each value of...
3. Consider the following stiff system of autonomous ordinary differential equations du f(u, u) =-3u +3, u(0)2 = ' dt de g(u, v) -2000u - 1000, v(0)-3 Note that 1 u<2 and -4 <v < 3 for all t. (a) Find the Jacobian matrix for the system of equa tions (b) Find the eigenvalues of the Jacobian matrix. (c) In the figure the shaded region shows the region of absolute stability, in the complex h plane, for third order explicit...
Problem 4: Suppose that the movement of rush-hour traffic on a typical expresswa be modeled using the differential equation du du where u(x) is the density of cars (vehicles per mile), and a is distance miles) in the direction of traffic flow. We w to the boundary conditions ant to solve this equation subject u(0) 300, u(5) 400. a) Use second-order accurate, central-difference approximations to discretize the differential equation and write down the finite-difference equation for a typical point zi...
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
4) Given the first order equation x'= [lx-x where x = x(t) and u is a real parameter. a) Find all fixed points for u0 and determine whether they are stable or unstable. b) Find all fixed points for u<0 and determine whether they are stable or unstable. c) Sketch the bifurcation diagram for this equation. Be sure to indicate the direction of flow on your diagram (with arrows). This is called a "pitchfork bifurcation" for obvious reasons.
Question 5 (5 marks) Consider the differential equation dy2-r (0) dr where r R is a constant. (a) Find the value(s) of r such that the differential equation has at least one equilibrium solution C (b) Determine the equilibrium solution(s) for the value(s) of r found in (a) (c) Use an in (b). Explain your answer. appropriate diagram to determine the stability of the equilibrium solution(s) found
Question 5 (5 marks) Consider the differential equation dy2-r (0) dr where r...
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