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2. Assume x = F(x) and assume F is two times differentiable...

2. Assume x = F(x) and assume F is two times differentiable. Assume Flxol =0 and F'LXO) 70. - Perform a sensitivity analysis at x = xo. Determine the signat F'cxo) such that x=Xo is a stable equilebrium solution of x =F(x). 4 Find a general solution of in = u(1-u) and then solve each initial va...

Let |x| denote the euclidean norm of a vector x € Rn, m...

Let |x| denote the euclidean norm of a vector x € Rn, meaning that |x²² = Xixi (in Einstein notation). Let 1/21/2 denote the L² norm of a function u(x) on a specified domain n C Rd, meaning that : n The L² norm of a vector function u(x) is defined as ||ulll2, so 11ullz...

3. [5 points] We consider the following SIR model, St+1 = S...

3. [5 points] We consider the following SIR model, St+1 = St - aStit It+1 = It + astIt - 4It R1+1 = Rt + 411, with S0 = 400 and a is a positive constant. If an outbreak does not happen, what is the largest value that a have to be smaller than? Why?

2icy x 8(x1-x)y4y? - = + y I((x - - - 8(-x1 - x)yty? ...

2icy x 8(x1-x)y4y? - = + y I((x - - - 8(-x1 - x)yty? + [(x + x1)2 - y = y2 x2 + - ((x - x1)2 + y? - y2) y [(x - x1)2 + (y - y1) ²) ((x - x1)2 + (y + y )²)]³/² - y2) + I((x + x1² + (y - y1) ² ((x + x12 + (y + y1)²)]3/2 1) Write the system of four 1...

2. (a) A dynamical written canonical form gives as SY/ with...

2. (a) A dynamical written canonical form gives as SY/ with (:) and k # real constant Identify the type and stability of the critical point at the origin, depending on all possible values giving the ranges values of for which they occur. Justify your ariswer. (b) Consider the dynamical system ...

6. Describe the phase portrait of the map of the circle give...

6. Describe the phase portrait of the map of the circle given by f(θ) = θ + 2π/n + ϵ sin(n θ) for 0 < ϵ < 1/n. 7. Prove that a homeomorphism of R can have no periodic points with prime period greater than 2. Give an example of a homeomorphism that has a perio...

1. Determine for what values of lambda, f(x) = lambda x (1-x...

1. Determine for what values of lambda, f(x) = lambda x (1-x) has an attracting fixed point p. (See the cheat-sheet on piazza for definitions). For those parameters lambda, what are the intervals containing p on which f is a uniform contraction? 2. Let f(x) = 2 x (1-x). Graph f¹°(x) on [1/...

Find the general solution of the following DISCRETE dynamica...

Find the general solution of the following DISCRETE dynamical systems and then find the particular solution for the given initial conditions. State the nature and location of the singular point. 1.) X(n+1) = [ 2 1 -1; 0 -1 2; 0 0 -1] * X(n) ; with X(0) = [ 1 2 -3 ] T 2.) X(n+1) = [ 12 -15; 4 -...

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