Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 8.5, Problem 3E
Interpretation Introduction
Interpretation:
To show that the system
Concept Introduction:
The mapping from
If there is a point
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, an autonomous system is expressed in polar coordinates. Determine all periodic solutions, all limit cycles, and the stability characteristics of all periodic solutions.
1.drdt=r2(1−r2),dθdt=1
an autonomous system is expressed in polar coordinates. Determine all periodic solutions, all limit cycles, and the stability characteristics of all periodic solutions.3.drdt=r(r−1)(r−3),dθdt=1
a.Find all the critical points (equilibrium solutions).
b.Use an appropriate graphing device to draw a direction field and phase portrait for the system.
c.From the plot(s) in part b, determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type.
b.Describe the basin of attraction for each asymptotically stable critical point. 5.dx/dt=1+2y,dy/dt=1−3x2
Chapter 8 Solutions
Nonlinear Dynamics and Chaos
Ch. 8.1 - Prob. 1ECh. 8.1 - Prob. 2ECh. 8.1 - Prob. 3ECh. 8.1 - Prob. 4ECh. 8.1 - Prob. 5ECh. 8.1 - Prob. 6ECh. 8.1 - Prob. 7ECh. 8.1 - Prob. 8ECh. 8.1 - Prob. 9ECh. 8.1 - Prob. 10E
Ch. 8.1 - Prob. 11ECh. 8.1 - Prob. 12ECh. 8.1 - Prob. 13ECh. 8.1 - Prob. 14ECh. 8.1 - Prob. 15ECh. 8.2 - Prob. 1ECh. 8.2 - Prob. 2ECh. 8.2 - Prob. 3ECh. 8.2 - Prob. 4ECh. 8.2 - Prob. 5ECh. 8.2 - Prob. 6ECh. 8.2 - Prob. 7ECh. 8.2 - Prob. 8ECh. 8.2 - Prob. 9ECh. 8.2 - Prob. 10ECh. 8.2 - Prob. 11ECh. 8.2 - Prob. 12ECh. 8.2 - Prob. 13ECh. 8.2 - Prob. 14ECh. 8.2 - Prob. 15ECh. 8.2 - Prob. 16ECh. 8.2 - Prob. 17ECh. 8.3 - Prob. 1ECh. 8.3 - Prob. 2ECh. 8.3 - Prob. 3ECh. 8.4 - Prob. 1ECh. 8.4 - Prob. 2ECh. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 7ECh. 8.4 - Prob. 8ECh. 8.4 - Prob. 9ECh. 8.4 - Prob. 10ECh. 8.4 - Prob. 11ECh. 8.4 - Prob. 12ECh. 8.5 - Prob. 1ECh. 8.5 - Prob. 2ECh. 8.5 - Prob. 3ECh. 8.5 - Prob. 4ECh. 8.5 - Prob. 5ECh. 8.6 - Prob. 1ECh. 8.6 - Prob. 2ECh. 8.6 - Prob. 3ECh. 8.6 - Prob. 4ECh. 8.6 - Prob. 5ECh. 8.6 - Prob. 6ECh. 8.6 - Prob. 7ECh. 8.6 - Prob. 8ECh. 8.6 - Prob. 9ECh. 8.7 - Prob. 1ECh. 8.7 - Prob. 2ECh. 8.7 - Prob. 3ECh. 8.7 - Prob. 4ECh. 8.7 - Prob. 5ECh. 8.7 - Prob. 6ECh. 8.7 - Prob. 7ECh. 8.7 - Prob. 8ECh. 8.7 - Prob. 9ECh. 8.7 - Prob. 10ECh. 8.7 - Prob. 11ECh. 8.7 - Prob. 12E
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- Using support, a long taut string is fixed along the positive x-axis. At time t = 0, the support is removed and gravity is permitted to act on the string. Assuming that the end x = 0 is fixed, the initial, boundary-value problem describing displacements y(x, t) of points in the string is: ∂^2?/∂?^2 = ?^2*∂^2?/∂?^2 − ?, ? > 0, ? > 0 ?(0, ?) = 0, ? > 0 ?(?, 0) = 0, ? > 0 ??(?,0)/dt = 0,? > 0 Where g is gravity and is equal to 9.81. Use Laplace transforms to solve this problem.Q1. What is the value of y(x,t) at x=c and t=0.5? Submit your answer to one decimal place.arrow_forwarda.Find all the critical points (equilibrium solutions). b.Use an appropriate graphing device to draw a direction field and phase portrait for the system. c.From the plot(s) in part b, determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. b.Describe the basin of attraction for each asymptotically stable critical point.11.dx/dt=(2−x)(y−x),dy/dt=y(2−x−x2)arrow_forwardUsing the Hertz rule to test the stability of the system defined by the following functionarrow_forward
- Find all the critical points (equilibrium solutions). b.Use an appropriate graphing device to draw a direction field and phase portrait for the system. c.From the plot(s) in part b, determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. b.Describe the basin of attraction for each asymptotically stable critical point. 4.dx/dt=x−xy,dy/dt=y+2xyarrow_forwarda.show that (0,0) is the only critical point of the DE x''+∈(1/3(x')^3 -x')+x=0 b. show that (0,0) is unstable when ∈>0. When is (0,0) an unstable spiral point.arrow_forwardLinear DE of Order 1arrow_forward
- (a) Solve the initial value problem in explicit form: t ∂u ∂t + (t + u) ∂u ∂x = x − t, u(x, 1) = 1 + x, |x| ≤ 2, sketch typical base characteristics in the xt-plane. Show the subset of R2 in which the solution u(x, t) is defined.arrow_forward8.A certain vibrating system satisfies the equation u″ + γu′ + u = 0. Find the value of the damping coefficient γ for which the quasi-period of the damped motion is 50% greater than the period of the corresponding undamped motion.arrow_forward14. A tank contains 500 L of water with 10 kg of dissolved salt. Salt water containing 0.05 kg of salt per literof water enters the tank at a rate of 5 liters per minute. Pure water enters the tank at a rate of 10 litersper minute. At the same time, the well-stirred salt water drains from the tank at a rate of 15 liters perminute. Find the amount of salt in the tank at any time. What is the equilibrium solution of the modelingdifferential equation? What is the long term behavior?arrow_forward
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