Homework Answers
Note: In part (c), by Liapunov direct method to determine the stability of fixed point, we can say that the fixed point (0,0) is globally asymptotically stable.
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Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system...
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system t' =-y+px(x² + y) (4) y = 1+ y(x² + y2), where is a parameter. Obviously, the origin x* = (0,0) is a fixed point of (4). (e) The solution of the ODE for o(t) is obvious - the angle o increases at a constant rate. Without solving the ODE for r(t), explain how r(t) behaves when t o in the cases H<0,1 =...
Consider the nonlinear system ?x′ = ln(y^2 − x) and y'=x-y-1 (a)Find all the critical points...
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?
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...
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...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y =...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine...
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine the type of the equilibrium point at the origin. (35 points) (b) Find all straight-line solutions and draw the phase portrait for the system. (35 points) (c) What is the general solution to the system? (15 points) (d) Find the solution of the system with initial conditions: x(0) = 1 and y(0) = -1. (15 points)
4) Consider the nonlinear system: 43 Use the describing function method to find a linear approximation. 4) Consider the nonlinear system: 43 Use the describing function method to find a linear a...
4) Consider the nonlinear system: 43 Use the describing function method to find a linear approximation.
4) Consider the nonlinear system: 43 Use the describing function method to find a linear approximation.
Problem 2: Consider the two-dimensional dynamical system given by F(x, y) = (x2 - y -...
Problem 2: Consider the two-dimensional dynamical system given by F(x, y) = (x2 - y - 1, x + 2y). (a) (8 pts) Find its fixed points and determine their stability. (b) (8 pts) Find any period-2 orbits and determine their stability. If no such orbits exist, prove it.
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed...
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed point at (1,1) (a) Show that the following expression for K(x, y) is a constant of motion for this system: K(x, y)- 2 ln(ry) + Inl( 2)(y- 2)]-3In(2)(y+2)]. (b) Use the constant of motion to show that the fixed point is a centre of the nonlinear system.
Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt...
Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt 3 山 一一 and deduce that the critical point at the origin is asymptotically stable.
Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt 3 山 一一 and deduce that the critical point at the origin is asymptotically stable.
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(...
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system t' =-y+px(x² + y) (4) y = 1+ y(x² + y2), where is a parameter. Obviously, the origin x* = (0,0) is a fixed point of (4). (e) The solution of the ODE for o(t) is obvious - the angle o increases at a constant rate. Without solving the ODE for r(t), explain how r(t) behaves when t o in the cases H<0,1 =...
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...
2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).
dr Consider the system: = 4x – 2y dy = x + y dt (a) Determine the type of the equilibrium point at the origin. (35 points) (b) Find all straight-line solutions and draw the phase portrait for the system. (35 points) (c) What is the general solution to the system? (15 points) (d) Find the solution of the system with initial conditions: x(0) = 1 and y(0) = -1. (15 points)
4) Consider the nonlinear system: 43 Use the describing function method to find a linear approximation.
4) Consider the nonlinear system: 43 Use the describing function method to find a linear approximation.
Problem 2: Consider the two-dimensional dynamical system given by F(x, y) = (x2 - y - 1, x + 2y). (a) (8 pts) Find its fixed points and determine their stability. (b) (8 pts) Find any period-2 orbits and determine their stability. If no such orbits exist, prove it.
Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed point at (1,1) (a) Show that the following expression for K(x, y) is a constant of motion for this system: K(x, y)- 2 ln(ry) + Inl( 2)(y- 2)]-3In(2)(y+2)]. (b) Use the constant of motion to show that the fixed point is a centre of the nonlinear system.
Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt 3 山 一一 and deduce that the critical point at the origin is asymptotically stable.
Construct a Liapunov function on the form V(x,y) = ax2 + cy2 for the nonlinear system dx dt dy dt 3 山 一一 and deduce that the critical point at the origin is asymptotically stable.
5. Consider the nonlinear two dimensional Lotka-Volterra (predator-prey) system z'(t) = z(t)[2-2(t)-2y(t)l, y'(t) = y(t)12-y(t)--2(t)] (a) Find all critical points of this system, and at each determine whether or not the system is locally stable or unstable. (b) We proved in class, using the Bendixson-Dulac theorem, that this system has no periodic solution with trajectory in the first quadrant of the plane. Assuming this, use the Poincare-Bendixson theorem to prove that all trajectories (z(t),y(t)) of the system (2) with initial...
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