Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation, Converting...
Sketch the phase portrait for each linear system below
Y. dt - (33) = (211) dY dt Y.
Here is the phase portrait of a homogeneous linear system of differential tions. 4. equa- (a) Classify the equilibrium (b) If λί is the eigenvalue with corresponding eigenvector (1,1) and A2 is the eigenvalue with corresponding eigenvector (-1,3), place the three numbers 0, λ, and λ2 in order frorn least to greatest. (c) If ((t), y(t) is the solution satisfying the initial condition (x(0),y(0)- (-2,2). Find i. lim r(t) i. lim rlt) ii. lim y(t) iv. lim y(t)
Here is...
Classify the critical point (0, 0) of the given linear system. Draw a phase portrait. dx/df 3x+ y a. dx/dt -x+ 2y dx/dt =-x +3y dy/dt -2x + y dy/dt x+ y Classify the stationary point (0, 0) of the given linear system. Draw a phase portrait. dy/dt -x+y b. dx/dt =-2x-y dx/dt-2x +5/7 y dx/dt 3x-y dx/dt 3x dy/dt 3x- y dy/dt 7x- 3y dy/dt x+y dy/dt 3y
dY 9. Consider the liner system dt Y. Determine which of the following phase portrait best represents the system. Justify your answer. Also, compute the general solution. (Sections 3.2, 3.3, and 3.4)
dY 9. Consider the liner system dt Y. Determine which of the following phase portrait best represents the system. Justify your answer. Also, compute the general solution. (Sections 3.2, 3.3, and 3.4)
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of differential equations dx dt 10x – x2 – 2xy, dy dt 5Y – 3y2 + xy. (a) Determine all of the critical points for the population model. (b) Determine the linearised system for each critical point in part (a) and discuss whether it can be used to approximate the behaviour of the non-linear system. (c) For the critical point at the origin: (i)...
Problem 6 (3 points) The general solution of the system of the linear system * = AY, Y)= ((0),y(t)), is given below. (1) Sketch the strait- line solutions and the phase portrait. DO NOT forget to use ARROWS. Make sure that your sketch shows ABSOLUTELY CLEAR slopes of Tangent line as t oot -oo. (2) Is the solution stable? Y(t) = kV1 + kye" V2; V. =(2,-1), V, = (1,3)
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...
1. (20 marks) This question is about the system of differential equations Y. dt=(k 1 (a) Consider the case k = 0. i. Determine the type of equilibrium at (0,0) (e.g., sink, spiral source). ii. Write down the general solution. iii. Sketch a phase portrait for the system. (b) Now consider the case k3 In this case, the matrix has an eigenvalue 2+V/2 with eigenvector i. -1+iv2 and an eigenvalue 2 iv2 with eigenvector . Determine the type of equilibrium...
1. (20 marks) This question is about the system of differential equations dY (3 1 (a) Consider the case k 0 i. Determine the type of equilibrium at (0,0) (e.g., sink, spiral source). i. Write down the general solution. ili Sketch a phase portrait for the system. (b) Now consider the case k -3. (-1+iv ) i. In this case, the matrix has an eigenvalue 2+i/2 with eigenvector and an eigenvalue 2-W2 with eigenvector Determine the type of equilibrium at...
1. The Duffing equation is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by -ax+3x3 = cos(wt) at medt dr. where function r = r(t) is the displacement at timet, is the velocity, and is the acceleration. The parameter 8 controls the amount of damping, a controls the linear stiffness, B controls the amount of non-linearity in the restoring force, and 7 and w are the amplitude and angular frequency of...