5.2.2 The Van der Pol oscillator The Van der Pol oscillator is a nonlinear dynamical system with many applications, e.g., in physics, biology, sociology and economics. It is a non-conservative oscillator with nonlinear damping, modelled by the differential equation x" - µ(1 − x²)x' + x = 0, (5.9) where the parameter μ measures the strength of the nonlinear damping. (1) Rewrite Equation (5.9) as a system of two first-order ODEs for x and x'. Then, write a Python function vdp(x) which returns the vector f(x), as defined in (5.1), for the Van der Pol system. Here, x must be an array of “shape” [1, 2]. (2) Solve the Van der Pol system with µ = 1/2, using your Python functions rk4 and vdp, for t = [0,30], starting with initial conditions (i) x (0) = x'(0) = 0.1; and (ii) x(0) = −2, x'(0) = 2. (You should fix a suitable step size, h.) Then, plot time series, x(t), of the solutions (i) and (ii) in one figure, and plot phase portraits, (x(t), y(t)), for the two solutions, in another figure. Discuss your results. (3) Solve the Van der Pol system with μ = 5, now using your Python functions euler and vdp, fort [0, 30], starting with initial conditions x (0) = x'(0) = 0.1. First, plot a time series and phase portrait of the solution for h = 0.01. Then, do the same computations (using the Euler scheme) and plots for h = 0.08. Discuss you results.
5.2.2 The Van der Pol oscillator The Van der Pol oscillator is a nonlinear dynamical system with many applications, e.g., in physics, biology, sociology and economics. It is a non-conservative oscillator with nonlinear damping, modelled by the differential equation x" - µ(1 − x²)x' + x = 0, (5.9) where the parameter μ measures the strength of the nonlinear damping. (1) Rewrite Equation (5.9) as a system of two first-order ODEs for x and x'. Then, write a Python function vdp(x) which returns the vector f(x), as defined in (5.1), for the Van der Pol system. Here, x must be an array of “shape” [1, 2]. (2) Solve the Van der Pol system with µ = 1/2, using your Python functions rk4 and vdp, for t = [0,30], starting with initial conditions (i) x (0) = x'(0) = 0.1; and (ii) x(0) = −2, x'(0) = 2. (You should fix a suitable step size, h.) Then, plot time series, x(t), of the solutions (i) and (ii) in one figure, and plot phase portraits, (x(t), y(t)), for the two solutions, in another figure. Discuss your results. (3) Solve the Van der Pol system with μ = 5, now using your Python functions euler and vdp, fort [0, 30], starting with initial conditions x (0) = x'(0) = 0.1. First, plot a time series and phase portrait of the solution for h = 0.01. Then, do the same computations (using the Euler scheme) and plots for h = 0.08. Discuss you results.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
Related questions
Question
use python code
![5.2.2
The Van der Pol oscillator
The Van der Pol oscillator is a nonlinear dynamical system with many applications, e.g., in physics,
biology, sociology and economics. It is a non-conservative oscillator with nonlinear damping,
modelled by the differential equation
x" - µ(1 − x²)x' + x = 0,
(5.9)
where the parameter μ measures the strength of the nonlinear damping.
(1) Rewrite Equation (5.9) as a system of two first-order ODEs for x and x'. Then, write a
Python function vdp(x) which returns the vector f(x), as defined in (5.1), for the Van der
Pol system. Here, x must be an array of “shape” [1, 2].
(2) Solve the Van der Pol system with µ = 1/2, using your Python functions rk4 and vdp, for
t = [0,30], starting with initial conditions (i) x (0) = x'(0) = 0.1; and (ii) x(0) = −2, x'(0) = 2.
(You should fix a suitable step size, h.) Then, plot time series, x(t), of the solutions (i) and
(ii) in one figure, and plot phase portraits, (x(t), y(t)), for the two solutions, in another figure.
Discuss your results.
(3) Solve the Van der Pol system with
μ =
5, now using your Python functions euler and vdp,
fort [0, 30], starting with initial conditions x (0) = x'(0) = 0.1. First, plot a time series and
phase portrait of the solution for h = 0.01. Then, do the same computations (using the Euler
scheme) and plots for h = 0.08. Discuss you results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3011c556-643e-4a01-b0e0-55d8cf24eddf%2F0db09648-39f9-4ec3-baed-daa5d13bc9fa%2Frd3dv0f_processed.png&w=3840&q=75)
Transcribed Image Text:5.2.2
The Van der Pol oscillator
The Van der Pol oscillator is a nonlinear dynamical system with many applications, e.g., in physics,
biology, sociology and economics. It is a non-conservative oscillator with nonlinear damping,
modelled by the differential equation
x" - µ(1 − x²)x' + x = 0,
(5.9)
where the parameter μ measures the strength of the nonlinear damping.
(1) Rewrite Equation (5.9) as a system of two first-order ODEs for x and x'. Then, write a
Python function vdp(x) which returns the vector f(x), as defined in (5.1), for the Van der
Pol system. Here, x must be an array of “shape” [1, 2].
(2) Solve the Van der Pol system with µ = 1/2, using your Python functions rk4 and vdp, for
t = [0,30], starting with initial conditions (i) x (0) = x'(0) = 0.1; and (ii) x(0) = −2, x'(0) = 2.
(You should fix a suitable step size, h.) Then, plot time series, x(t), of the solutions (i) and
(ii) in one figure, and plot phase portraits, (x(t), y(t)), for the two solutions, in another figure.
Discuss your results.
(3) Solve the Van der Pol system with
μ =
5, now using your Python functions euler and vdp,
fort [0, 30], starting with initial conditions x (0) = x'(0) = 0.1. First, plot a time series and
phase portrait of the solution for h = 0.01. Then, do the same computations (using the Euler
scheme) and plots for h = 0.08. Discuss you results.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education