Dynamical system 6:Centre.

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Bill Foster

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Bill Foster 10 years, 11 months ago

Created this as a copy of Dynamical system 5.

Name	Status	Author	Last Modified
Dynamical system 4	Ready to use	Bill Foster	16/11/2020 10:02
Dynamical system 6:Centre.	draft	Bill Foster	16/11/2020 10:02
Dynamical system 7: Stable spiral.	Ready to use	Bill Foster	16/11/2020 10:02
Dynamical system 9: Saddle.	Ready to use	Bill Foster	16/11/2020 10:02
Bill's copy of Dynamical system 6:Centre.	Ready to use	Bill Foster	16/11/2020 10:02

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Question statement

Consider the following two-dimensional dynamical system . You have to find values for $a,\;b\;c,\;d$ such that the system's phase space is a centre.

\[\begin{align}\dot{x}&=\simplify[std]{a*x+b*y},\\\dot{y}&=\simplify[std]{c*x+d*y}.\end{align}\]

Give any introductory information the student needs.

			Name	Type	Generated Value

xr	integer	50
yr	integer	50
v	integer	3
f	integer	6

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50

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Generated value: `integer`

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The system can be written in the form $\dot{\boldsymbol{x}}=\mathsf{A}\boldsymbol{x}$, where $\boldsymbol{x}=\pmatrix{x,y}^\mathsf{T}$.

Input the components of the matrix $\mathsf{A}$ in order to obtain a centre where the eigenvalues of $A$ are $\simplify{{v}*i}$ and $\simplify{{-v}*i}$.

You are given that $a=\var{f}$.

$\mathsf{A}=\Bigg($	$\var{f}$	Gap 0.	$\Bigg)$
	Gap 1.	Gap 2.

Once you have input appropriate values into the matrix, the diagram below shows the plot of $(x(t),y(t))$.

At $t=0$ we have initally $x=-5,\;\;y=5$. Moving the point gives phase diagrams for the following initial values at $t=0$:

$x=\;$ $y=\;$

You can click on Steps to see the solutions for $x(t),\;y(t)$ after you have input values into the matrix.

Advice

Given the system of differential equations:

\[\begin{align}\dot{x}&=\simplify[std]{a*x+b*y},\\\dot{y}&=\simplify[std]{c*x+d*y}.\end{align}\]

It can be written in the form $\dot{\boldsymbol{x}}=\mathsf{A}\boldsymbol{x}$, where $\boldsymbol{x}=\pmatrix{x,y}^\mathsf{T}$ and

\[\mathsf{A}=\pmatrix{a& b\\c & d}\]

In this case, with $a=\var{f}$ we want to enter values for $b,\;c,\;d$ such that the system gives a centre.

For this to happen we need the eigenvalues of $A$ to be purely imaginary and you are given that they are $\simplify{{v}*i}$ and $\simplify{{-v}*i}$.

The characteristic polynomial for $A$ is given by

\[\det\left(\mathsf{A}-\lambda\mathsf{I}\right)=0,\]

i.e. $(a-\lambda)(d-\lambda)-bc=0$. This leads to $\lambda^2-(a+d)\lambda+(ad-bc)=0$.

So in order to get purely imaginary eigenvalues $\pm \var{abs(v)*i}$ we need :

1. $a+d=0 \Rightarrow d=\var{-f}$ as $a=\var{f}$.

2. $ad-bc=\var{v^2} \Rightarrow bc=-\var{v^2}-\var{f^2}$.

Hence if you set $d=\var{-f}$ and you choose values of $b,\;c$ such that $bc=-\var{f^2+v^2}$, this will give the required system of differential equations with phase space a centre and the required eigenvalues.

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This question is used in the following exams:

Simple dynamical systems. by Bill Foster in Bill's workspace.
Interactive by Bill Foster in Bill's workspace.
Sorcha's copy of Interactive by TEAME UCC in TEAME's workspace.
Rob's copy of Demo exam by Rob Wilson in Rob's workspace.
Paula's copy of Demo exam by Paula Ruiz-Hincapie in Paula's workspace.
Ricardo's copy of Interactive by Ricardo Monge in Ricardo's workspace.
Review Assessment #1 by Stephen Bowlzer in Stephen's workspace.
Middlesex Workshop by Bill Foster in Bill's workspace.
Demo exam by Christian Lawson-Perfect in Demos.

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Taxonomy: mathcentre

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Bill Foster 10 years, 11 months ago

Error in variable testing condition

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Generated value: `integer`

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Dynamical system 6:Centre.

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 10 years, 11 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`