For each of the following systems, find a > 0 and b > 0 such that L(x, y) = ax^2 + by^2 obeys d/dt(L) not = 0 whenever (x, y) 6= (0, 0). (This makes L a Liapounov function.) State whether the origin is a stable or unstable equilibrium in each case.
(a) x' = −x^3 + 7xy^2 , y' = −3x^2y + y^3 .
(b) x' = x^3 − y^3 , y' = 3xy^2 + 4x^2 y + 5y^3 .
Homework Answers
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For each of the following systems, find a > 0 and b > 0 such that L(x, y) = ax^2 + by^2 obe...
For each of the following systems: (i) Find the general solution by using eigenvalues and eigenvectors....
For each of the following systems:
(i) Find the general solution by using eigenvalues and
eigenvectors.
(ii) State whether the origin is stable, asymptotically stable,
or unstable.
(iii) State whether the origin is a node, saddle, center, or
spiral.
For each of the following systems: (i) Find the general solution by using eigenvalues and eigenvectors. (ii) State whether the origin is stable, asymptotically stable, or unstable. |(iii) State whether the origin is a node, saddle, center, or spiral. Problem 1:...
1. Consider the system x˙ = xy2 + x 2 y + x 3 , y˙...
1. Consider the system x˙ = xy2 + x 2 y + x 3 , y˙ = −x 3 + y 3
. Is (0, 0) attracting/stable or unstable? Give a proof.
1. Consider the system = ry® + ²y + x", j=-" + y. Is (0,0) attracting/stable or unstable? Give a proof.
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y...
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y (a) Find the general solution. (b) Without a computer, sketch a phase diagram that shows four linear solution trajectories and that shows one solution trajectory in each of the four regions between the separatrices. (c) Determine the solution that satisfies x(0) = 1 and y(0) = 0. x(t) = yt) = (d) The point (0,0) is a ... Osaddle point stable node unstable node...
Consider an LTI system whose input x[n] and output y[n] are related by the difference equation...
Consider an LTI system whose input x[n] and output y[n] are related by the difference equation y[n – 1] + 3 y[n] + $y[n + 1] = x[n]. Determine the three possible choices for the impulse response that makes this system 1) causal, 2) two-sided and 3) anti-causal. Then for each case, determine if the system is stable or not. Causality Impulse Response Stability Causal Unstable v two-sided Unstable anti-Causal Unstable y In your answers, enter z(n) for a discrete-time...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find and sketch the nullclines and give the vectorfield on the nullelines. Also find all steady states and perform a linear stability analysis for each of them to find whether they are stable or unstable and give the type (saddle, node, focus).
Problems: (1) Answer True or False to each of the following. You must substantiate your answers. (A) A differentiable function is always globally Lipschitz. (B) The trajectory of the system , r(0) is...
Problems: (1) Answer True or False to each of the following. You must substantiate your answers. (A) A differentiable function is always globally Lipschitz. (B) The trajectory of the system , r(0) is bounded for all t 0 (C) A linear tine-varying system á(t) A(t)a(t) is asymptotically stable around the origin if and only if it is uniformly exponentially stable around the origin. (D) Given the equation x f(x), and suppose that xe 0 is an exponentially stable equilibrium point...
8. Find the Fourier transform of the following signal. (5 points) x(0) 2 1 9. Determine...
8. Find the Fourier transform of the following signal. (5 points) x(0) 2 1 9. Determine whether or not the following signals are periodic, and if periodic, give their periods in seconds and frequency in hertz. a. X(t) = 12.8 Cos (320xt - . (3 points). b. x(n) = 11.6 Cos (3n). (3 points). 6. x(n) = 1.45 sinn). (3 points). 10. Determine whether or not the LTI systems with the following impulse responses are causal and stable. Note that...
(ii) R= [0, 1] x [0, 1] C R2 olsun. f: RR fonksiyonu f(x,y) = 2-Y...
(ii) R= [0, 1] x [0, 1] C R2 olsun. f: RR fonksiyonu f(x,y) = 2-Y eğer (2, y) + (0,0) ise (x+y)3 0 eğer (x,y) = (0,0) ise şeklinde tanımlansın. f fonksiyonunun Rüzerinde integrallenebilir olup/olmadığını ispatlayıp, eğer integrallenebilir ise SR fdA integralini hesaplayınız. Prove whether the f function is integrable on R. if it can be integrated; calculate the integral SR fdA.
please do only 5 and 7 PART II. Manually solve each of these diagonal systems. 5. Y'(x)=10-5 6, Y'(x)=| 0-5 0 |Yu), Y(0)=| 0-2 3 1 -1 0 Y(0)=| 0-4 3 0 0-2 7. Y'(x)=10-7 |Y(x), 0 0 0 3...
please do only 5 and 7
PART II. Manually solve each of these diagonal systems. 5. Y'(x)=10-5 6, Y'(x)=| 0-5 0 |Yu), Y(0)=| 0-2 3 1 -1 0 Y(0)=| 0-4 3 0 0-2 7. Y'(x)=10-7 |Y(x), 0 0 0 3
PART II. Manually solve each of these diagonal systems. 5. Y'(x)=10-5 6, Y'(x)=| 0-5 0 |Yu), Y(0)=| 0-2 3 1 -1 0 Y(0)=| 0-4 3 0 0-2 7. Y'(x)=10-7 |Y(x), 0 0 0 3
(1 point) Consider the function defined by ?(?,?)=??(9?2+5?2)?2+?2F(x,y)=xy(9x2+5y2)x2+y2 except at (?,?)=(0,0)(x,y)=(0,0) where ?(0,0)=0F(0,0)=0. Then we have...
(1 point) Consider the function defined by
?(?,?)=??(9?2+5?2)?2+?2F(x,y)=xy(9x2+5y2)x2+y2
except at (?,?)=(0,0)(x,y)=(0,0) where ?(0,0)=0F(0,0)=0.
Then we have
∂∂?∂?∂?(0,0)=∂∂y∂F∂x(0,0)=
∂∂?∂?∂?(0,0)=∂∂x∂F∂y(0,0)=
Note that the answers are different. The existence and continuity
of all second partials in a region around a point guarantees the
equality of the two mixed second derivatives at the point. In the
above case, continuity fails at (0,0)(0,0).
(1 point) Consider the function defined by F(x, y) = xy(9x2 + 5y2) x2 + y2 except at (x, y) = (0,0)...
For each of the following systems:
(i) Find the general solution by using eigenvalues and
eigenvectors.
(ii) State whether the origin is stable, asymptotically stable,
or unstable.
(iii) State whether the origin is a node, saddle, center, or
spiral.
For each of the following systems: (i) Find the general solution by using eigenvalues and eigenvectors. (ii) State whether the origin is stable, asymptotically stable, or unstable. |(iii) State whether the origin is a node, saddle, center, or spiral. Problem 1:...
1. Consider the system x˙ = xy2 + x 2 y + x 3 , y˙ = −x 3 + y 3
. Is (0, 0) attracting/stable or unstable? Give a proof.
1. Consider the system = ry® + ²y + x", j=-" + y. Is (0,0) attracting/stable or unstable? Give a proof.
Consider the following differential equation system: x' = 16x + 8y y = -24x – 12y (a) Find the general solution. (b) Without a computer, sketch a phase diagram that shows four linear solution trajectories and that shows one solution trajectory in each of the four regions between the separatrices. (c) Determine the solution that satisfies x(0) = 1 and y(0) = 0. x(t) = yt) = (d) The point (0,0) is a ... Osaddle point stable node unstable node...
Consider an LTI system whose input x[n] and output y[n] are related by the difference equation y[n – 1] + 3 y[n] + $y[n + 1] = x[n]. Determine the three possible choices for the impulse response that makes this system 1) causal, 2) two-sided and 3) anti-causal. Then for each case, determine if the system is stable or not. Causality Impulse Response Stability Causal Unstable v two-sided Unstable anti-Causal Unstable y In your answers, enter z(n) for a discrete-time...
Problem 3: Study the planar system x' = (y – 1)(y – 2), - . Find and sketch the nullclines and give the vectorfield on the nullelines. Also find all steady states and perform a linear stability analysis for each of them to find whether they are stable or unstable and give the type (saddle, node, focus).
Problems: (1) Answer True or False to each of the following. You must substantiate your answers. (A) A differentiable function is always globally Lipschitz. (B) The trajectory of the system , r(0) is bounded for all t 0 (C) A linear tine-varying system á(t) A(t)a(t) is asymptotically stable around the origin if and only if it is uniformly exponentially stable around the origin. (D) Given the equation x f(x), and suppose that xe 0 is an exponentially stable equilibrium point...
8. Find the Fourier transform of the following signal. (5 points) x(0) 2 1 9. Determine whether or not the following signals are periodic, and if periodic, give their periods in seconds and frequency in hertz. a. X(t) = 12.8 Cos (320xt - . (3 points). b. x(n) = 11.6 Cos (3n). (3 points). 6. x(n) = 1.45 sinn). (3 points). 10. Determine whether or not the LTI systems with the following impulse responses are causal and stable. Note that...
(ii) R= [0, 1] x [0, 1] C R2 olsun. f: RR fonksiyonu f(x,y) = 2-Y eğer (2, y) + (0,0) ise (x+y)3 0 eğer (x,y) = (0,0) ise şeklinde tanımlansın. f fonksiyonunun Rüzerinde integrallenebilir olup/olmadığını ispatlayıp, eğer integrallenebilir ise SR fdA integralini hesaplayınız. Prove whether the f function is integrable on R. if it can be integrated; calculate the integral SR fdA.
please do only 5 and 7
PART II. Manually solve each of these diagonal systems. 5. Y'(x)=10-5 6, Y'(x)=| 0-5 0 |Yu), Y(0)=| 0-2 3 1 -1 0 Y(0)=| 0-4 3 0 0-2 7. Y'(x)=10-7 |Y(x), 0 0 0 3
PART II. Manually solve each of these diagonal systems. 5. Y'(x)=10-5 6, Y'(x)=| 0-5 0 |Yu), Y(0)=| 0-2 3 1 -1 0 Y(0)=| 0-4 3 0 0-2 7. Y'(x)=10-7 |Y(x), 0 0 0 3
(1 point) Consider the function defined by
?(?,?)=??(9?2+5?2)?2+?2F(x,y)=xy(9x2+5y2)x2+y2
except at (?,?)=(0,0)(x,y)=(0,0) where ?(0,0)=0F(0,0)=0.
Then we have
∂∂?∂?∂?(0,0)=∂∂y∂F∂x(0,0)=
∂∂?∂?∂?(0,0)=∂∂x∂F∂y(0,0)=
Note that the answers are different. The existence and continuity
of all second partials in a region around a point guarantees the
equality of the two mixed second derivatives at the point. In the
above case, continuity fails at (0,0)(0,0).
(1 point) Consider the function defined by F(x, y) = xy(9x2 + 5y2) x2 + y2 except at (x, y) = (0,0)...
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