Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
expand_more
expand_more
format_list_bulleted
Question
Chapter 7.6, Problem 7E
Interpretation Introduction
Interpretation:
To calculate the averaged equations and to analyze the long-term behavior of the system
Concept Introduction:
The averaged or slow-time equations are given by
The common solution of the system is given by
The explicit solution is specified as
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Consider the following SEL. Calculate the solution for the system using the Gauss-Jacob iterative method
(Consider k=3 and initial values equal to zero)
Consider the following.
x1'
=
3x1 − 2x2,
x1(0)
=
3
x2'
=
2x1 − 2x2,
x2(0)
=
1
2
(a) Transform the given system into a single equation of second order by solving the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for x1.'
(b) Find x1 and x2 that also satisfy the initial conditions.
The metal plate has the constant te mperatures shown on its boundaries. Find the equilibrium temperature at each of the indicated interior points by setting up a system of linear equations and applying either the Jacobi or the Gauss-Seidel method. Obtain a solution that is accurate to within 0. 001
Chapter 7 Solutions
Nonlinear Dynamics and Chaos
Ch. 7.1 - Prob. 1ECh. 7.1 - Prob. 2ECh. 7.1 - Prob. 3ECh. 7.1 - Prob. 4ECh. 7.1 - Prob. 5ECh. 7.1 - Prob. 6ECh. 7.1 - Prob. 7ECh. 7.1 - Prob. 8ECh. 7.1 - Prob. 9ECh. 7.2 - Prob. 1E
Ch. 7.2 - Prob. 2ECh. 7.2 - Prob. 3ECh. 7.2 - Prob. 4ECh. 7.2 - Prob. 5ECh. 7.2 - Prob. 6ECh. 7.2 - Prob. 7ECh. 7.2 - Prob. 8ECh. 7.2 - Prob. 9ECh. 7.2 - Prob. 10ECh. 7.2 - Prob. 11ECh. 7.2 - Prob. 12ECh. 7.2 - Prob. 13ECh. 7.2 - Prob. 14ECh. 7.2 - Prob. 15ECh. 7.2 - Prob. 16ECh. 7.2 - Prob. 17ECh. 7.2 - Prob. 18ECh. 7.2 - Prob. 19ECh. 7.3 - Prob. 1ECh. 7.3 - Prob. 2ECh. 7.3 - Prob. 3ECh. 7.3 - Prob. 4ECh. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - Prob. 8ECh. 7.3 - Prob. 9ECh. 7.3 - Prob. 10ECh. 7.3 - Prob. 11ECh. 7.3 - Prob. 12ECh. 7.4 - Prob. 1ECh. 7.4 - Prob. 2ECh. 7.5 - Prob. 1ECh. 7.5 - Prob. 2ECh. 7.5 - Prob. 3ECh. 7.5 - Prob. 4ECh. 7.5 - Prob. 5ECh. 7.5 - Prob. 6ECh. 7.5 - Prob. 7ECh. 7.6 - Prob. 1ECh. 7.6 - Prob. 2ECh. 7.6 - Prob. 3ECh. 7.6 - Prob. 4ECh. 7.6 - Prob. 5ECh. 7.6 - Prob. 6ECh. 7.6 - Prob. 7ECh. 7.6 - Prob. 8ECh. 7.6 - Prob. 9ECh. 7.6 - Prob. 10ECh. 7.6 - Prob. 11ECh. 7.6 - Prob. 12ECh. 7.6 - Prob. 13ECh. 7.6 - Prob. 14ECh. 7.6 - Prob. 15ECh. 7.6 - Prob. 16ECh. 7.6 - Prob. 17ECh. 7.6 - Prob. 18ECh. 7.6 - Prob. 19ECh. 7.6 - Prob. 20ECh. 7.6 - Prob. 21ECh. 7.6 - Prob. 22ECh. 7.6 - Prob. 23ECh. 7.6 - Prob. 24ECh. 7.6 - Prob. 25ECh. 7.6 - Prob. 26E
Knowledge Booster
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
- Apply the method of undetermined coefficients to find a particular solution of each of the following systems. If initial conditions are given, find the particular solution that satisfies these conditions. Primes denote derivatives with respect to t . x’ = 2x + y +1 , y’ = 4x + 2y + e4tarrow_forwardConsider a 6-meter metal bar with a uniform initial temperature (across the bar) of 35°C . Suppose it is in thermal contact with an external source of heat given by h(x)= 3−x, 0 ≤ x ≤ 6. So the temperature u(x,t) had the ut=uxx+h(x) permission. Suppose further that the temperature of the ends are kept constant, being at x=0 of 5°C , while at x=6 of 30°C . Under such conditions: Find the steady-state temperature distribution of the bar and the boundary value problem that determines the transient distribution. (no need to solve the problem).arrow_forwardconsider the nonlinear second-order ODE x''+2x'+4x/(1+x^2)=0 a. convert to DE to a system b. Find all critical points of the resulting systemarrow_forward
- 10.The average sizes of the prey and predator populations are defined as ¯x=1T∫A+TAx(t)dt,¯y=1T∫A+TAy(t)dt,x¯=1T∫AA+Txtdt,y¯=1T∫AA+Tytdt, respectively, where T is the period of a full cycle, and A is any nonnegative constant. a.Using the approximation (24), which is valid near the critical point, show that ¯x=c/γx¯=c/γ and ¯y=a/αy¯=a/α. b.For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate ¯xx¯ and ¯yy¯ as well as you can. Try to determine whether ¯xx¯ and ¯yy¯ are given by c/γ and a/α, respectively, in this case. Hint: Consider how you might estimate the value of an integral even though you do not have a formula for the integrand. c.Calculate other solutions of the system (2)—that is, solutions satisfying other initial conditions—and determine ¯xx¯ and ¯yy¯ for these solutions. Are the values of ¯xx¯ and ¯yy¯ the same for all solutions? In Problems 11 and 12, we consider the effect of modifying the equation for the prey x by including a term −σx2 so that…arrow_forwardFor the following scenarios for a population: 1) Construct a dynamic model by writing down a difference equation both in the updating function form (N(t+1)=f(N(t))N(t+1)=f(N(t))) and the increment form (N(t+1)−N(t)=g(N(t))N(t+1)−N(t)=g(N(t))) and specify the time step; 2) find the solution of the linear difference equation with a generic initial value and check that it satisfies the difference equation by plugging the solution into your equation (in either form); 3) plug in the given initial value and predict the future. Zombies have appeared in Chicago. Every day, each zombie produces 3 new zombies. Suppose that initially there is only one zombie, how many zombies will there be in 7 days? Suppose hunters kill 50 deer in a national forest every hunting season, while the deer by themselves have equal birth and death rates. If there are initially 500 deer in the forest, predict how many there will be in 5 years. The number of infected people in a population grows by 8% per day and…arrow_forwardFind values of C1 and C2 so that the given functions will satisfy the prescribed initial conditions.arrow_forward
- Apply the method of undetermined coefficients to find a particular solution of each of the following systems. If initial conditions are given, find the particular solution that satisfies these conditions. Primes denote derivatives with respect to t . x’ = x + 2y + 3, y’ = 2x + y - 2arrow_forwardFind the steady-state solution qp(t) and the steady-state current in an LRC-series circuit when the impressed voltage is E(t) = E0 sin γt.arrow_forwardGiven is a population of wolves (W) and rabbits (R). R[t+1] = R[t]+ g*R[t] * (1 – R[t]/K) - sR[t]W[t] W[t+1] = (1-u)W[t] + vR[t]W[t] Where the carrying capacity of rabbits is 1 million. The growth rate of rabbits is 10% a year and s is equal to 0.00001, v is 0.0000001, and u is equal to 0.01. How many wolves and how many rabbits exist in the equilibrium? Use the equations for the co-existence equilibrium. Implement the model into Google Sheets with the initial populations of 200,000 rabbits and 10,000 wolves. Show the dynamics over time. It may take a few hundred steps to get to an equilibrium Look at the effect of the carrying capacity of rabbits. Suppose environmental pollution decrease the carrying capacity for the rabbits. After trying different values of K, what can you conclude is the minimum carrying capacity needed to have a population of wolves surviving in this environment?arrow_forward
- a) Derive the system of differential equations and initial conditions representing this system. b) Solve for x(t) and y(t), assuming r=10, x(0)=10, y(0)=5, V(tank 1)=25, and V(tank 2)=50. c) Find the limiting amount as t goes to infinity of salt in each tankarrow_forwardThe average number of customers in the system in the single-channel, single-phase model described in Section 12.4 is L=λ/(μ – λ) Show that for m = 1 server, the multichannel queuing model in Section 12.5, is identical to the single-channel system. Note that the formula for P0 (Equation 12-13) must be utilized in this highly algebraic exercise.arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
UG/ linear equation in linear algebra; Author: The Gate Academy;https://www.youtube.com/watch?v=aN5ezoOXX5A;License: Standard YouTube License, CC-BY
System of Linear Equations-I; Author: IIT Roorkee July 2018;https://www.youtube.com/watch?v=HOXWRNuH3BE;License: Standard YouTube License, CC-BY