## plot phase space and extented phase space of ODEs...

Hi

I have the ODEs: x'=x

ode:=diff(x(t),t)-x(t);

sketch the phase space and extended phase space of previous ode.

## 3D phase space with directional field...

hello guys,

i have a system of autonomous equations which i want to plot its 3D phase space with directional field,

i have some problem with it :dy.mw , and i dont know how to command for add some directional field for 3D phase space .

thank you guys

## Analyzing a differential equation system with vary...

Hi there,
I have a set of differential equations whose solution, Jacobian matrix and its eigenvalues, direction field, phase portrait and nullclines, need to be computed.

Each of the equations has a varying parameter.

I know how to get the above for a single parameter value, but when I set a range of values for the parameters, Maple is not able to handle all cases as I would expect: solving the differential equation system:

eq1 := x*(1.6*(1-(1/100)*x)-phi*y)
eq2 := (x/(15+x)-0.3e-1*x-.4)*y+.6+theta
desys := [eq1, eq2];
vars := [x, y];

Error, (in unknown) invalid input: Utilities:-SetEquations expects its 2nd argument, equations, to be of type set({boolean, algebraic, relation}), but received {-600*y+(Array(1..2, {(1) = 8400, (2) = 15900})), Array(1..5, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 0})}

The equations are the following:
de1 := diff(x(t), t) = x(t)*(1.6*(1-(1/100)*x(t))-phi*y(t));
de2 := diff(y(t), t) = (x(t)/(15+x(t))-0.3e-1*x(t)-.4)*y(t)+.6+theta

the parameters being:
phi:=[0 0.5 1 1.5 2]
theta:=[5. 10.]

How can I handle the situation so that Maple computes each of the above for each combination of the parameters?

I would like to avoid using two for loops and having to store all results in increasingly bigger and complicated arrays.

The worksheet at issue is this: MaplePrimes_Tumour_model_phi_theta_variation.mw

Thanks,
jon

## Compute and display isoclines (nullclines) for an ...

Hi there,

I would like to compute and display the nullclines of a set of ordinary differential equations.

AFAIK, I can compute the nullclines in Maple by defining the equations and solving the system

e.g.:

# Define the equations
eq1 := u(t)*(1-u(t)/kappa)-u(t)*v(t) = 0;
eq2 := g*(u(t)-1)*v(t) = 0;

# Solve the system (i.e. compute the nullclines)
sol := solve({eq1, eq2}, {u(t), v(t)});

However, I am not quite able to imagine how to display them over a dfieldplot or a phaseportrait.

Attached is an example with some differential equations, and their vector field and trajectories: MaplePrimes_Predator_prey_model_nullclines.mw.

It can be use to illustrate how to (compute and) display the nullclines.

Thank you,

jon

## Frequency response from transfer function, Process...

Hi all,

I have a simple transfer function like

$G=\frac{0.1(s+0.2)\cdot e^{-4.5s}}{(s+0.5)(s+0.4)(s+0.1)(s+0.02)}$

I want to calculate the amplitude ratio and the phase shift.

Without maple I would set s=I*omega and solve for real and imaginary part to obtain the phase shift and amplitude ratio.

$AR=\sqrt{R^2+I^2}$

$\phi=arctan(I/R)$

Is there any possibility to do that in Maple more easily?

## my phaseportrait not working...

> with(DETools);
> eq1 := diff(y(t), t) = v(t);
d
--- y(t) = v(t)
dt
> eq2 := diff(v(t), t) = -3*v(t)+10*y(t);
d
--- v(t) = -3 v(t) + 10 y(t)
dt
> phaseportrait([eq1, eq2], [y, v], t = 0 .. 10, [[y(0) = 0, v(0) = 1]], y = -10 .. 10, v = -10 .. 10, linecolor = blue);

why my phase portrait not working

## Plotting a three dimensional phase plane...

Below is my code for solving a solution of three first order ODEs using the 4th-order Runge-Kutta method.

I have been able to successfully plot the solutions of each of the ODEs (x,y,z) against time t, however I am struggling to produce a plot in the three dimensional phase domain of x,y,z. Could anybody suggest what commands to use as everything I have tried (plot, plot3d, implicitplot3d etc) has produced an error.

h:= 0.01:
N:= 200:
x:=Vector(N+1):

## How to draw a phase curve of ODE system?...

Hey, I have a system of ODE and I can't draw it's phase curve. I tried to use DEplot and phaseportrait, but it doesn't work. Here is my system:

dx/dt=x

dy/dt=ky              (k is a constant)

Here is my piece of code:

DE := [diff(x(t), t) = x(t)];

DF := [diff(y(t), t) = k*y(t)];

with(DEtools);

phaseportrait([DE, DF], [y, x], t = -5 .. 5, y = -5 .. 5, x = -5 .. 5, k...

## phaseportrait shows me an error...

Hey, I'm trying to plot the phase curve of the simple system of ODE, BUT it shows me an error which one I don't understand at all :)

Here is my code:

DE := diff(x(t), t) = x(t);
DF := diff(y(t), t) = k*y(t);
with(DEtools);
phaseportrait([DE, D], [y, x], t = -5 .. 5, [[y(0) = 1, x(0)], [y(0) = 0, x(0) = 2], [y(0) = 0, z(0) = -2]], y = -5 .. 5, x = -5 .. 5, color = black, linecolor = red);

`Error, (in DEtools/phaseportrait) invalid initial...`

## DEplot: Plot not appearing when doing a phase plot...

Hi

I am trying to do a phase plot of an autonomous system using DEplot command. However, no phase plot appears. Could you tell me what am I doing wrong? The same code worked for my professor in class, but it's not working for me. I am using Maple 16. The code is posted below.

> with DEtools

> sys := diff(x(t), t) = y(t), diff(y(t), t) = x(t)*(1-x(t)*x(t))+y(t);

> DEplot([sys], [x(t), y(t)], t = 0 .. 0.1e-8, x = -3 .. 3, y = -3 .. 3, color = black)

## phase plots from a differential equation...

Hi

I want  to make the phase plot of d(Z(t))/dt (along y) and Z(t) (along x axis). The initial conditiona are arbitrary. I only have one equation:

d(dz/dt)/dz = - (sqrt(d(z^2)/dt + 10^10 * z^2)*dz/dt + 10^(10)*z)/dz/dt

Is it still possible to make a phase space plot or do i need another equation?

## False result even though having done the right man...

Hello,

I have the following problem:

My function is defined by the determinant of 2 Heun functions

If I plot the phase I get something which looks quite what I'm looking for.

To get a better result I thought I would manually carry out the Wronskian as far as possible...

Doing some manipulations I get another form of the Wronskian which in fact should give the same result...

the problem is it doesnt :-(

## Phase change...

Hi everybody,

Does somebody has an idea about how to model the temperature of water during freezing ?

I mean, if you consider a certain mass of water and you put it, for example on a fridge. I don't consider the convection but only the diffusion, and I would like to model, if possible with an animation the temperature of the water (as a cube for the shape).

Normally, during the freezing I should observe a decrease of the temperature, then a freezing...

## Animate two phase portrait in one...

Hi everyone,

I have this differential equations:

gamma1:=1;gamma2:=1;

l:=t->sqrt((x1(t)-x2(t))**2+(y1(t)-y2(t))**2);
2 2
l := t -> sqrt((x1(t) - x2(t)) + (y1(t) - y2(t)) )

DE:=[diff(x1(t),t)=(-1/(2*Pi))*gamma2*(y1(t)-y2(t))/l(t),diff(y1(t),t)=(-1/(2*Pi))*gamma1*(x1(t)-x2(t))/l(t),diff(x2(t),t)=(-1/(2*Pi))*gamma2*(y2(t)-y1(t))/l(t),diff(y2(t),t)=(-1/(2*Pi))*gamma1*(x2(t)-x1(t))/l(t)]:

I want to draw the phase portraits (x1(t...

## Plotting Phase Portrait for a four equation system...

Hi,

I'm trying to plot a few phase portraits for this four equation system but maple wont let me...

```> restart;

> s := 10; d[T] := 0.2e-1; k1 := 2.4*10^(-5); k2 := 3*10^(-3);
> N := 5; delt := .24; c := 2.4; T0 := 1; ```
`with(DEtools); `
`with(plots);`
` `

[INSERT YOUR CODE HERE BY COPY-PASTING IN 1-D FORMAT]

`Can someone please help me? Thanks`
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