2(a) Consider the one-parameter family of nonlinear ordinary differential equations -Ita-) where ...
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.
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...
1. Consider the family of differential equations dy/dx = y^3 + ky + k^3 . Please Help me with it, thanks so much 1. Consider the family of differential equations de set = y2 + ky + k3. (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...
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...
(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.
Find a first-order system of ordinary differential equations equivalent to the second-order nonlinear ordinary differential equation y ^-- = 3y 0 + (y 3 − y) (3 points) Find a first-order system of ordinary differential equations equivalent to the second-order nonlinear ordinary differential equation y" = 3y' +(y3 – y).
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,...
Consider the family of differential equations dy/dx=y^3+ky+k^2 Are there any equilibrium solutions when k=0? Draw bifurcation diagram Draw phase diagram for when k=1/2 Does limit exist when k=1/2 and y(0)=0
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...