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13. Use a Lyapunov function to show that the origin is globally asymptotically stable: x' =...
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using the Lyapunov function V(x, y, z) = ρ「2 + ơy2 + ơz?, show that the origin is globally a...
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using the Lyapunov function V(x, y, z) = ρ「2 + ơy2 + ơz?, show that the origin is globally asymptotically stable. (Hint. You may need to use the Invariance Principle as well.) στ
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using...
Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function (Lyapunov Direct) approach to; (5 Marks): Show that th...
Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function (Lyapunov Direct) approach to; (5 Marks): Show that the system is globally stable around the origin (5 Marks): The origin is globally asymptotically stable. (5 Marks): Only SKETCH a possible Phase Plan, as based on (a), (b). a. b. c.
Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov Function: Where p >0. Answer the following questions: . Is V(x) a good candidate Lyapunov function? Explain 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable.
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 a...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k> a. Consider the following Lyapunov Function Where p >0. Answer the following questions: 1. 1s V (x) a good candidate Lyapunov function? Explairn 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k>...
Show that the zero solution of y' =-y+y is asymptotically stable, but not globally, i.e. not...
Show that the zero solution of y' =-y+y is asymptotically stable, but not globally, i.e. not all solutions tend to zero as t + Sketch all solutions in the (ty) plane, taken to = 0. Also sketch all solutions in phase space. What can you conclude about the solution y = 1?
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, 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...
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its...
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
Use lagrange multipliers to find the point on the plane x-2y 3z-14=0 that is closet to the origin
Use lagrange multipliers to find the point on the plane x-2y 3z-14=0 that is closet to the origin?(try and minimize the square of the distance of a point (x,y,z) to the origin subject to the constraint that is on the plane) Help me please!
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk,...
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk, S is the part of the paraboloid z - x2 + y2, 251 (b) F(x, y, z) = (z, x-z, y), S is the triangle with vertices (1,0,0), (0, 1,0), and (0,0,1), oriented downward (c) F-(y. -x,z), S is the upward helicoid parametrized by r(u, v) = (UCOS v, usin v,V), osus 2, OSVS (Hint: Tu x Ty = (sin v, -cos v, u).)...
2nd pic is answer. show the work plz 13 Let X and Y have the joint...
2nd pic is answer.
show the work plz
13 Let X and Y have the joint probability density function ,흄.ru2 for 0 < x < y. < 2 f(x,y) = elsewhere What is the joint density function of U it is nonzero? 3X-2Y and V-X + 2Y where 687 Probability and Mathematical Statistics 32768 13° g(u,t) = 0 otherwise.
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using the Lyapunov function V(x, y, z) = ρ「2 + ơy2 + ơz?, show that the origin is globally asymptotically stable. (Hint. You may need to use the Invariance Principle as well.) στ
3 [15 pts Consider the Lorenz system given by xy-B2, z = where σ, ρ, β > 0 are constants. For ρ (0.1), using...
Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function (Lyapunov Direct) approach to; (5 Marks): Show that the system is globally stable around the origin (5 Marks): The origin is globally asymptotically stable. (5 Marks): Only SKETCH a possible Phase Plan, as based on (a), (b). a. b. c.
Consider the following nonlinear dynamic system, with a possible potential candidate function. Use the given Lyapunov function, us such function...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov Function: Where p >0. Answer the following questions: . Is V(x) a good candidate Lyapunov function? Explain 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable.
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k> a. Consider the following Lyapunov Function Where p >0. Answer the following questions: 1. 1s V (x) a good candidate Lyapunov function? Explairn 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k>...
Show that the zero solution of y' =-y+y is asymptotically stable, but not globally, i.e. not all solutions tend to zero as t + Sketch all solutions in the (ty) plane, taken to = 0. Also sketch all solutions in phase space. What can you conclude about the solution y = 1?
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...
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk, S is the part of the paraboloid z - x2 + y2, 251 (b) F(x, y, z) = (z, x-z, y), S is the triangle with vertices (1,0,0), (0, 1,0), and (0,0,1), oriented downward (c) F-(y. -x,z), S is the upward helicoid parametrized by r(u, v) = (UCOS v, usin v,V), osus 2, OSVS (Hint: Tu x Ty = (sin v, -cos v, u).)...
2nd pic is answer.
show the work plz
13 Let X and Y have the joint probability density function ,흄.ru2 for 0 < x < y. < 2 f(x,y) = elsewhere What is the joint density function of U it is nonzero? 3X-2Y and V-X + 2Y where 687 Probability and Mathematical Statistics 32768 13° g(u,t) = 0 otherwise.
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