1. Consider the system described by i(t) = u(t) – sin (x(t)) y(t) = u(t) + cos (x(t)) a) Find all equilibrium points of the system. b) For each equilibrium point, determine whether or not the equilibrium point is (i) stable in the sense of Lyapunov; (ii) asymptotically stable; (iii) globally asymptotically stable. Explain your answers. c) Determine whether or not the system is bounded-input bounded-output stable. Explain your answer.

1. Consider the system described by i(t) = u(t) – sin (x(t)) y(t) = u(t) + cos (x(t)) a) Find all equilibrium points of the system. b) For each equilibrium point, determine whether or not the equilibrium point is (i) stable in the sense of Lyapunov; (ii) asymptotically stable; (iii) globally asymptotically stable. Explain your answers. c) Determine whether or not the system is bounded-input bounded-output stable. Explain your answer.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Classify the equilibrium is x= [0,0] for the first order system 12 d[x (1) dt (1) [8] [13] [28] 10 6…

A: The given first order system is ddtx1(t)x2(t)=12106x1(t)x2(t) That is,…

Q: 1. Find and classify (as stable, unstable, or semi-stable) the equilibrium points of(y-2)(25-²).

Q: Consider the following system of equations (Odell, 1980) which are said to describe a predator-prey…

Q: Determine the local stability of equilibrium of the ODE d dt x(t) = x(1−xy) dt³ (t) = y=x².

A: Given, a system of ordinary differential equations:It is required to find the local stability of the…

Q: d.S dt dI dt = TI - PSI, pSI – TI, where I represents individuals who are infected with some virus…

A: The system of differential equations given describes a simplified model of disease spread.The…

Q: Q2: Consider the equation u, = (u-u), + au with u=0, as x→∞0. From the steady xxxx state of this…

Q: Find the solution of the linear system corresponding to the initial conditions x(0) = 1, y(0) = 0.…

A: As per the question we are given the initial condition for a trajectory of a linear system as : x(0)…

Q: A second order system with a single input and a single output is the state equation given below It…

A: As per the question we are given the state space representation of a dynamic system. And we have to…

Q: Describe each of the following equations by giving their order, telling whether they are linear r…

Q: Find the general solution of the fourth order linear equation y(4) + 4y" + 4y = 0 O a. y(t) Ciev2…

Q: Let U(x) = 4 x² – 2 g³ be a potential energy in one dimension. Find and classify the equilibria of…

Q: Solve the system Sa' (t) = x(t) + y(t) ly(t) = y(t) Then what is æ(t)? O x(t) = 2(0)e O a(t) =…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: c) Determine the BIBO and asymptotic stability for the system described by the differential equation…

A: Given: D+1D2+5D+5y(t)=D+1xt

Q: Suppose (t + 3)y{ (t – 6)y; 8tyi + 2y2, У. (1) — 0, 4y1+ 4ty2, У2(1) %3 2. = a. This system of…

Q: Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system with…

Q: H1. Find the equilibria and determine their stability using linear stability analysis for the…

A: The objective of the question is to find the equilibrium points and determine their stability for…

Q: a) Use the Laplace transform to derive the transfer function of a system which is described by the…

A: Note:- As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: (a) Verify that *₁ = x₂(x3 + U₁), *₂ = x₁(x3 + U₁), x3 = U₂ (x₁(t) = sin(2t), ₂(t) = cos(2t), 3(t) =…

A: As per the question we are given a nonlinear dynamic system as : ẋ1 = x2(x3 + u1) ẋ2 = -x1(x3 +…

Q: Let P Enter your answers in terms of the variable t. = (t) 2t-2 L-21-2 = [t2-2t 2t 1-2t ¹+2t…

Q: b) Examine f(x, y)= 2x² – 4.x+ xy – 1 for relative extrema.

A: b) Examine fx,y=2x2-4x+xy2-1 for relative extrema fx,y=2x2-4x+xy2-1 fx=∂f∂x=4x-4+y2 fy=∂f∂y=2xy…

Q: 1. Write down the system of differential equations for the gradient flow associated to V (x, y) : xy…

Q: 2.31. (a) Determine whether the system described by is (i) memoryless, (iii) causal, (v) time…

A: (a) y(t) = cos[x(t-2)]i) y(t) depends on other values of input x(t) other than the value at…

Q: 3). Consider the system 2 ()' - (²₂3) (1). = -2 - (a) Plot the solution curve passing through the…

A: Matrix differentiation is done row or column wise.

Q: B2. Suppose the non-linear system dx/dt = f(x, y), dy/dt = g(x,y) has a fixed point at (x*,*).…

Q: the system of linear differential equations for two real valued functions x(t) and y(t): Sx'(t) =…

Q: a.Sketch the graph of the forcing function on an appropriate interval. b.Find the solution of the…

Q: Which of the following functions V(y1, y2) is a Lyapunov function for the dynamical system with…

Q: Let X = X(t) be the response of the linear dynamical system x' = ax - By y' = Bx + ay that satisfies…

Q: 2. Find the particular solution of the given system of equations. Sketch a picture and describe the…

Q: Consider the nonhomogeneous linear equation y"- y= g(t) where g is a continuous function. Let y,()…

Q: (iii) For the linearised system in part (i): find all straight line orbits, determine the behaviour…

A: note :As per the instructions we will only solve parts (b.iii) and (b.iv) of the question.

Q: Consider the system of non-linear differential equations: dx/dt = x cos y, dy/dt = y cos x.…

Q: Use variation of parameters to solve the given nonhomogeneous system. (-₁ + ) x + (cot (1) 10 X(t):…

Q: We know that: Theorem (*) If y₁, y₂ are linearly dependent on some interval, and their derivatives…

A: Theorem (*) : If y1 , y2 are linearly dependent on some interval and derivatives y'1 and y'2 exist…

Q: Which of the following functions V(y₁, y2) is a Lyapunov function for the dynamical system with…

A: Solution:- The accurate answer is a.V(y1,y2)=e(y1y2)2.Here's why:A Lyapunov function have to be…

Q: 2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain…

A: The objective of the question is to find a non-constant function v(t) such that y₂(t) = v(t)y₁(t) is…

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

1. Consider the system described by
i(t) = u(t) – sin (x(t))
y(t) = u(t) + cos (x(t))
%3D
a) Find all equilibrium points of the system.
b) For each equilibrium point, determine whether or not the equilibrium point is (i) stable in the
sense of Lyapunov; (ii) asymptotically stable; (iii) globally asymptotically stable. Explain
your answers.
c) Determine whether or not the system is bounded-input bounded-output stable. Explain your
answer.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Consider the system - sin(x) dt (a) Find all the critical points (equilibrium solutions). (b) Find an equation of the form H(x, y) = c satisfied by the solution curves.
b) Using Lyapunov's method to dtermine the stability of the equilibrium point (0, 0) for the following systems. dx | dt dy dt = -2y - x5, = = 3x - 2y³ 1
Let U(x) = 3x² - 4x be a potential energy in one dimension. Find and classify the equilibria of the system. 3 If one equlibrium is stable and the other unstable, enter the stable equilibrium first. If both equilibria have the same stability, enter them in either order. x= Stable O Unstable O Stable Unstable 41
3. The populations x(t) and y(t) of two species satisfy the equations x' x(4-3y), y = y (4-3x), (x, y ≥ 0). Find the steady states of the system and characterise them.
1. Consider the system X(t) = AX(t) where A = (82) and B = (a) For which values of a is the system controllable? (b) For a = 1, i. Calculate et ii. Write the integral formula for the solution X(t) in terms of X(0) = Xo = iii. Extract a separate formula for each component of X(t) = ((1))- and a a constant. Io Yo and u.
(a) Find all second partial derivatives of f(x, y) = sin(x + 2y) + e³² (b) Find and classify any critical points of the function f(x, y) = −4xy + y² + 5x − 4y + 5. (c) Use the chain rule to find the derivative of the function z = x¹y³ 3xy along the curve given by x = cos 0, y = sin 20. Give your answer in terms of 0. (d) Determine if the function f(z) = cos z is complex differentiable (analytic).
Consider the system described by i(t) = u(t) – sin (r(t)) y(t) = u(t) + cos (r(t)) a) Find all equilibrium points of the system. b) For each equilibrium point, determine whether or not the equilibrium point is (i) stable in the sense of Lyapunov; (ii) asymptotically stable; (iii) globally asymptotically stable. Explain your answers. c) Determine whether or not the system is bounded-input bounded-output stable. Explain your answer.
A.) find the general solution B.) Find the initial value where y(2) = 8
1. Given z = e' tan(x) + (2x + 3y)5 x=71² y = 5t +9 (a) Draw a tree diagram showing which variables depend on which. The variable z should be at the top. (b) Calculate the partial derivatives of z. Əz dx (c) Calculate the derivatives of x(t) and y(t). dx dt dy dt dz dt = (d) Use the multivariable chain rule to calculate || dz dt || z(t) = əz ду (e) Now plug in for x and y in terms of t to find z as a function of t, then differentiate that function. =
(a) Find and evaluate the first-order partial derivatives of the function. f(x, y) = 2e³ ln(ty) (b) (c) (d) fz(x, y) = fay(x, y) = fry (1, e) =
The velocity of a particle at position r = (x, y), in two-dimensional Cartesian coordinates, is given by v = (αy,αx), where α ∈ R is a constant. What are the physical dimensions of the parameter α? Find the general form of r(t), the position of the particle, as a function of time t (hint: write v = (αy,αx) as a system of first order ODEs). What is the critical point and its nature in this system? Suppose at t = 0, r(0) = (2,0). By eliminating the variable t, find the nature of the particle’s trajectory and give a sketch of it.