Consider the following nonlinear, forced ODE: ï+di+ ³x + ax³ = cos(wt) For 3 = -1, a = 1 and y = 0, we may write this as a system of first order differential equations as x = v v = −dv + x − x³ You may assume that > 0. (a) Write down all of the fixed points of the system of equations. (b) For each fixed point, write down the linearized equations near the fixed point. (c) For each linearized system, set 8 = 0 and determine what type of fixed point it is (source, sink, center, spiral, saddle, etc.) and what the stability is.
Consider the following nonlinear, forced ODE: ï+di+ ³x + ax³ = cos(wt) For 3 = -1, a = 1 and y = 0, we may write this as a system of first order differential equations as x = v v = −dv + x − x³ You may assume that > 0. (a) Write down all of the fixed points of the system of equations. (b) For each fixed point, write down the linearized equations near the fixed point. (c) For each linearized system, set 8 = 0 and determine what type of fixed point it is (source, sink, center, spiral, saddle, etc.) and what the stability is.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the following nonlinear, forced ODE:
\[ \ddot{x} + \delta \dot{x} + \beta x + \alpha x^3 = \gamma \cos(\omega t) \]
For \(\beta = -1\), \(\alpha = 1\), and \(\gamma = 0\), we may write this as a system of first order differential equations as
\[ \dot{x} = v \]
\[ \dot{v} = -\delta v + x - x^3 \]
You may assume that \(\delta \geq 0\).
(a) Write down all of the fixed points of the system of equations.
(b) For each fixed point, write down the linearized equations near the fixed point.
(c) For each linearized system, set \(\delta = 0\) and determine what type of fixed point it is (source, sink, center, spiral, saddle, etc.) and what the stability is.
(d) Describe in words how these fixed points will change if \(\delta\) is a small positive number.
(e) Please sketch the phase portrait (i.e., trajectories in the \(x - v\) plane) for the system with a small positive \(\delta\). Pick one of the stable fixed points and sketch the set of initial conditions that will eventually end up near this fixed point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6e2f997-9120-4975-9388-a1bc7e4c3a16%2F48981832-eedd-4b89-8236-52883db167cc%2Fk21iltb_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following nonlinear, forced ODE:
\[ \ddot{x} + \delta \dot{x} + \beta x + \alpha x^3 = \gamma \cos(\omega t) \]
For \(\beta = -1\), \(\alpha = 1\), and \(\gamma = 0\), we may write this as a system of first order differential equations as
\[ \dot{x} = v \]
\[ \dot{v} = -\delta v + x - x^3 \]
You may assume that \(\delta \geq 0\).
(a) Write down all of the fixed points of the system of equations.
(b) For each fixed point, write down the linearized equations near the fixed point.
(c) For each linearized system, set \(\delta = 0\) and determine what type of fixed point it is (source, sink, center, spiral, saddle, etc.) and what the stability is.
(d) Describe in words how these fixed points will change if \(\delta\) is a small positive number.
(e) Please sketch the phase portrait (i.e., trajectories in the \(x - v\) plane) for the system with a small positive \(\delta\). Pick one of the stable fixed points and sketch the set of initial conditions that will eventually end up near this fixed point.
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