(5) Consider the following differential equation: z' = (12-4)(p-r2) Here p is a parameter. Sketch a...
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...
Consider the nondimensional differential equation du where u is an unknown parameter (constant) (a) Determine the equilibrium solutions in terms of μ. (b) Draw the bifurcation diagram and clearly identify the bifurcation point. (c) Classify the stability of the branches in your bifurcation diagram using the process in class where we assumed u(t)uilibrium +u(t) where uequilibrium is the constant(s) you determined in (a) Repeat the steps in exercise (2) for the nondimensional differential equation given by du_2 dt where u...
For each problem, sketch all of the qualitatively different vector fields that occur as the parameter u is varied. Find the values of u at which bifurcation occur, and classify the bifurcations. Finally, sketch the bifurcation diagram or the steady states x* vs the parameter u. 1. ** = 5 – ļe-x? 2. espe= ux - T H > 0.
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...
Consider the following differential equation date = y2(y2 – 4). (a) Find all critical values. (b) Draw the phase diagram to classify each as stable, semi-stable or unstable.
Question 3 (20 points) Consider the following differential equation = y(y2 - 4). (a) Find all critical values. (6) Draw the phase diagram to classify each as stable, semi-stable or unstable.
3. Consider the differential equation where p is an undefned parameter a. Find the general solution for any p. b. For what values ofp does the solution stay bounded (that is, stays finite) as x -> o. c. Where is your solution discontinuous? Explain your answer
III) Consider the following differential equation: 5•(t) + 3x(t) - 4 = 0, x(0) = 2. 1. Find the backward solution. 2. Is this solution convergent or divergent? Justify your answer. 3. Determine the stationary solution and indicate whether it is stable or unstable. 4. Sketch a phase diagram and a time-path diagram.
1. Consider the family of differential equations done = y2 + ky + kº. (a) Are there any equilibrium solutions when k =0? If so, what are they? (b) Draw the bifurcation diagram. That is, sketch a graph of the critical values as a function of the parameter k. Clearly label the axes. (You may use Mathematica for this problem, but your final answer must be drawn by hand.) (c) Draw the phase diagram for when k = -1. For...