1 Sec. 8.1 8.2 Homework For each of the following systems, find all critical points (b)...
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. 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)...
The following system can be interpreted as a competition system describing the interaction of two species with populations x(t) and y(t) x' 40x – 22 – ry y' = 30y - y2 – 0.5xy This system has four critical points (0,0), (0, 30), (40,0), and (20, 20). (a) At critical point (20, 20), find the linearization of the system and its eigenvalues. Deter- mine the type and stability of the critical point (20, 20). Base on your work in part...
1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system. 185 Exercises (b) Describe the phase portrait for the nonlinear system (c) Does the linearized system accurately describe the local bchavior near the equilibrium points? (iii) x' = x+ y, y, 2y 1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system....
1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system. 185 Exercises (b) Describe the phase portrait for the nonlinear system (c) Does the linearized system accurately describe the local bchavior near the equilibrium points? x' = sin x, y, = cos y (i) 1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated...
1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system. 185 Exercises (b) Describe the phase portrait for the nonlinear system (c) Does the linearized system accurately describe the local bchavior near the equilibrium points? We were unable to transcribe this image 1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system....
Please a- c for non linear system b 3. For each of the given non-linear systems, (a) find the equilibrium points, (b) near each equilibrium point, sketch the phase portrait of the linearized system, (c) use the information in (a) and (b) to sketch the phase portrait of the system: x' = - 4x + 4xy Sx = 2x – 2x² + 5xy ly=2y-y² – ry ly' = y - 2y2 + 2xy
e and classify all the equilibriun points of each of the folowing nonlmear systems, given that the second system is Hamiltonian (aka con- servative). Sketch a phase portrait of each system. di = y(1-x2) dt e and classify all the equilibriun points of each of the folowing nonlmear systems, given that the second system is Hamiltonian (aka con- servative). Sketch a phase portrait of each system. di = y(1-x2) dt
Consider the nonlinear system ?x′ = ln(y^2 − x) and y'=x-y-1 (a)Find all the critical points (b)Find the corresponding linearized system near the critical points. (c) Classify the (i) type (node, saddle point, · · · ), and (ii) stability of the critical points for the corresponding linearized system. (d) What conclusion can you obtain for the type and stability of the critical points for the original nonlinear system?
1. For each of the following nonlinear systems, (a) Find all of the equilibrium points and describe the behavior of the associated linearized system. 185 Exercises (b) Describe the phase portrait for the nonlinear system (c) Does the linearized system accurately describe the local bchavior near the equilibrium points? We were unable to transcribe this image