Rössler Attractor

A Rössler attractor is a system of three nonlinear ordinary differential equations. The Rössler attractor is similar in nature to the Lorenz attractor. The coupled nonlinear equations can be solved using the Massless formulation available in the Mathematical Particle Tracing interface.

Model Definition

The governing differential equations for a Rössler attractor are

where a, b, and c are constants. This model uses the original parameters used by Rössler: a = 0.2, b = 0.2 and c = 5.7. The particles are released from an initial grid in the y direction at x = 0 and z = 0. In total, 31 particles are released uniformly between y = 3 and y = 8.

In this example, is selected from the list, and so the coordinates are dimensionless.

Results and Discussion

After the initial release of the particles, they travel counterclockwise in the xy-plane at z = 0. Once they cross to the positive side of the yz-plane the particles rise very sharply in the z direction. The particles with the largest radial coordinate in the xy-plane acquire the highest velocity and thus reach the highest point the z direction; see Figure 1. The region where the particles reach high elevations in a short period of time is called the upswing region. The particles with the outermost radial coordinate entering the upswing region end up at the innermost radial coordinate after the downswing region.

The average particle velocity is plotted in Figure 2. There is a double-peak in maximum velocity, corresponding to the upswing and downswing regions. A Poincaré map showing the intersection of the trajectories with the xz-plane is included in Figure 3. For x < 0 in this plane the particle z-coordinate is very small, while for x > 0 it can vary significantly.

Figure 1: Plot of the Rössler attractor after 10 seconds.

Figure 2: Plot of the average particle velocity versus time.

Figure 3: Particle trajectories with a Poincaré map superimposed (red dots) and Poincaré section shown (gray cut plane).

Reference

1. Wikipedia, http://en.wikipedia.org/wiki/R%C3%B6ssler_attractor

Particle_Tracing_Module/Tutorials/rossler_attractor

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

The geometry is designated to be large enough so that the particle trajectories do not make contact with the walls.

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 15.

In the text field, type 40.

Locate the section. In the text field, type -5.

In the toolbar, click

Component 1 (comp1)

In the window, click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Define the constants a, b and c as parameters. This means they could, in principle, be varied as part of a when solving.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
a	0.2	0.2	Model parameter
b	0.2	0.2	Model parameter
c	5.7	5.7	Model parameter

Mathematical Particle Tracing (pt)

The formulation allows a system of first order ordinary differential equations to be solved.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Particle Properties 1

In the window, under click .

In the window for , locate the section.

Specify the v vector as


-y-z	x
x+a*y	y
b+z*(x-c)	z

Release from Grid 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the qy,0 text field, type range(3,5/30,8).

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,0.02,10).

In the toolbar, click

Results

Particle Trajectories 1

Render the particle trajectories as ribbons. The default ribbon orientation is in the direction of the unit binormal, or the direction out of the plane tangent to the curved trajectory.

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. From the list, choose .

Select the check box.

In the associated text field, type 0.5.

Find the subsection. From the list, choose .

Particle Trajectories (pt)

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Click the

button in the toolbar. The plot should look like Figure 1.

Now plot the average particle velocity using the and a plot type.

Average Particle Velocity Magnitude

In the toolbar, click

and choose .

In the window for , type Average Particle Velocity Magnitude in the text field.

Locate the section. From the list, choose .

Particle 1

In the toolbar, click

and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

Color Expression 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

. The plot should look like Figure 2.

A dataset can be used to visualize how the particles cross a specific area of interest in the modeling domain. A is created using the intersection points of the trajectories with the plane.

Cut Plane 1

In the toolbar, click