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Problem 2: Consider the two-dimensional dynamical system given by F(x, y) = (x2 - y -...
3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical...
3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical system in two dimensions (here f: R2 + R and g: R² R are two functions with smooth partial derivatives). Draw the corresponding vector field in the case that f(x,y) = x2 - y2 8(x, y) = x+y+1 and argue that (x,y) = R2 such that f(x,y) = g(x,y) = 0 are fixed points of the dynamical system above.
Consider the discrete dynamical system given by the expression √1 + √1 + √1 + √1...
Consider the discrete dynamical system given by the expression
√1 + √1 + √1 + √1 + ⋯ where the " ⋯ " means the pattern continues
forever. (A) Find a recurrence equation that models this pattern.
(B) Instead of solving the recurrence equation, build a table of
values from the recurrence equation through 10 iterations. (C) Find
the nonnegative fixed point of this system and apply the Stability
and Oscillation theorem to determine the system’s behavior around
the fixed...
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b)...
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
Let f(x) = 10/x − x^2. Find all fixed points of f and determine 9 their...
Let f(x) = 10/x − x^2. Find all fixed points of f and determine
9
their stability. To where are orbits under f attracted?
Problem 3: Let f(x) = 10x – x? Find all fixed points of f and determine their stability. To where are orbits under f attracted? Problem 4: Let f(x) = 10 x – 23. Find all fixed points of f and determine their stability. To where are orbits under f attracted?
Consider the nonlinear system: - x + (x – 1) y y + 4x° (1 –...
Consider the nonlinear system: - x + (x – 1) y y + 4x° (1 – x). (a) Show that the system has a unique fixed point at the origin (0, 0). (b) Use a linear approximation to determine the stability of the fixed point. (c) Apply the Liapunov direct method to determine the stability of the fixed point. Is your conclusion different form that of Part (a)? Why? (d) Can the system have closed orbits (trajectories)? Explain.
Consider the function f(x, y) = -8 – 2y – x+y + x2 + 1972. How...
Consider the function f(x, y) = -8 – 2y – x+y + x2 + 1972. How many relative maxima, relative minima, and saddle points does f(x,y) have? NOTE: ONLY 3 ANSWER TRIES ON THIS PROBLEM. relative minima saddle point relative maxima Submit Answer Tries 0/3 This discussion is closed. Send Feedback
Problem 4. (Discrete time dynamical system ). Consider the following discrete time dynamical system: Assume xo...
Problem 4. (Discrete time dynamical system ). Consider the following discrete time dynamical system: Assume xo is given and 0.5 0.5 0.2 0.8 (a) Find eigenvalues of matrix A (b) For each eigenvalue find one eigenvector. (c) Let P be the matrix that has the eigenvectors as its columns. Find P-1 (d) Find P- AP (e) Use the answer from part (d) to find A" and xn-A"xo. (Your answers wl be in terms of n (f) Find xn and limn→ooXn...
? (c) (2 pts) Let f(x) = 4xe-*. Find the fixed points and their stability. (d)...
?
(c) (2 pts) Let f(x) = 4xe-*. Find the fixed points and their stability. (d) (2 pts) Let f(x) = 2.x2 – 3. Find the fixed points along with their stability and determine the 2-cycle along with its stability. (e) (2 pts) Let c > 0 and fc(x) = c (2 - V3x + 4). Find the interval of stability of 0.
Problem 1 (20 points) Given the following non-linear autonomous system, || x' = 2cy " ||...
Problem 1 (20 points) Given the following non-linear autonomous system, || x' = 2cy " || y' = 9-+ y2 : a) What are the equilibrium points? (2 points) b) Can you tell their stability via linearization? If you can, please determine their stability and if they are locally a sink, a port or a source. If you cannot, please explain why. (3 points) c) Please find a first integral of the form f (2,y) = xg (22 + y2)...
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt...
Problem 3. Consider the following continuous differential equation dx dt = αx − 2xy dy dt = 3xy − y 3a (5 pts): Find the steady states of the system. 3b (15 pts): Linearize the model about each of the fixed points and determine the type of stability. 3b (15 pts): Draw the phase portrait for this system, including nullclines, flow trajectories, and all fixed points. Problem 2 (25 pts): Two-dimensional linear ODEs For the following linear systems, identify the...
3. Continuous dynamical systems - Dimension 2 (a) Suppose the ODE system describes a continuous dynamical system in two dimensions (here f: R2 + R and g: R² R are two functions with smooth partial derivatives). Draw the corresponding vector field in the case that f(x,y) = x2 - y2 8(x, y) = x+y+1 and argue that (x,y) = R2 such that f(x,y) = g(x,y) = 0 are fixed points of the dynamical system above.
Consider the discrete dynamical system given by the expression
√1 + √1 + √1 + √1 + ⋯ where the " ⋯ " means the pattern continues
forever. (A) Find a recurrence equation that models this pattern.
(B) Instead of solving the recurrence equation, build a table of
values from the recurrence equation through 10 iterations. (C) Find
the nonnegative fixed point of this system and apply the Stability
and Oscillation theorem to determine the system’s behavior around
the fixed...
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
Let f(x) = 10/x − x^2. Find all fixed points of f and determine
9
their stability. To where are orbits under f attracted?
Problem 3: Let f(x) = 10x – x? Find all fixed points of f and determine their stability. To where are orbits under f attracted? Problem 4: Let f(x) = 10 x – 23. Find all fixed points of f and determine their stability. To where are orbits under f attracted?
Consider the nonlinear system: - x + (x – 1) y y + 4x° (1 – x). (a) Show that the system has a unique fixed point at the origin (0, 0). (b) Use a linear approximation to determine the stability of the fixed point. (c) Apply the Liapunov direct method to determine the stability of the fixed point. Is your conclusion different form that of Part (a)? Why? (d) Can the system have closed orbits (trajectories)? Explain.
Consider the function f(x, y) = -8 – 2y – x+y + x2 + 1972. How many relative maxima, relative minima, and saddle points does f(x,y) have? NOTE: ONLY 3 ANSWER TRIES ON THIS PROBLEM. relative minima saddle point relative maxima Submit Answer Tries 0/3 This discussion is closed. Send Feedback
Problem 4. (Discrete time dynamical system ). Consider the following discrete time dynamical system: Assume xo is given and 0.5 0.5 0.2 0.8 (a) Find eigenvalues of matrix A (b) For each eigenvalue find one eigenvector. (c) Let P be the matrix that has the eigenvectors as its columns. Find P-1 (d) Find P- AP (e) Use the answer from part (d) to find A" and xn-A"xo. (Your answers wl be in terms of n (f) Find xn and limn→ooXn...
?
(c) (2 pts) Let f(x) = 4xe-*. Find the fixed points and their stability. (d) (2 pts) Let f(x) = 2.x2 – 3. Find the fixed points along with their stability and determine the 2-cycle along with its stability. (e) (2 pts) Let c > 0 and fc(x) = c (2 - V3x + 4). Find the interval of stability of 0.
Problem 1 (20 points) Given the following non-linear autonomous system, || x' = 2cy " || y' = 9-+ y2 : a) What are the equilibrium points? (2 points) b) Can you tell their stability via linearization? If you can, please determine their stability and if they are locally a sink, a port or a source. If you cannot, please explain why. (3 points) c) Please find a first integral of the form f (2,y) = xg (22 + y2)...
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