Dynamical system 9: Saddle.

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Created this as a copy of Dynamical system 7: Stable spiral..

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 saddle.

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

Note that this question is purely formative and for experimenting with. No marks are given.

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			Name	Type	Generated Value

xr	integer	50
yr	integer	50
g	number	3
f	integer	-2

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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 saddle.

In order to achieve this you have to supply the diagonal elements of the matrix, the entries for $b,\;c$ are given. In this case the eigenvalues of $A$ are real and of opposite signs.

You are given that $b=\var{f},\;c=\var{g}$.

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

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 $b=\var{f},\;c=\var{g}$ we want to enter values for $a,\;d$ such that the system gives a saddle for its phase space.

For this to happen we need the eigenvalues of $A$ to be real with one positive and one negative.

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 real eigenvalues with opposite signs we need :

1. Real roots. $(a+d)^2 \gt 4(ad-bc) \Rightarrow (a-d)^2 \gt -4bc$.

Hence we have one condition is that $(a-d)^2 \gt \simplify[all,!collectNumbers]{-4*{f}*{g}={-4*f*g}}$

2. Opposite signs. The roots are given by $\displaystyle \frac{(a+d) \pm \sqrt{(a+d)^2-4(ad-bc)}}{2}$.

Hence if $\operatorname{det}(A)=ad-bc \lt 0$ we see that the roots are opposite in sign.

So this is the other condition: $ad-bc \lt 0 \Rightarrow ad \lt \var{f}\times \var{g}=\var{f*g}$

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This question is used in the following exams:

Simple dynamical systems. by Bill Foster in Bill's workspace.
Middlesex Workshop by Bill Foster in Bill's workspace.

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Bill Foster 8 years ago

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Bill Foster 10 years, 11 months ago

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Dynamical system 9: Saddle.

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 8 years ago

Bill Foster 8 years ago

Bill Foster 10 years, 11 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`