4.3.14.3. Limit Cycle operator¶

The Limit Cycle Operator detects limit cycles within a vector field. Integral curves are seeded at a Poincaré section and integrated through the vector field. Curves that return to the Poincaré section indicate a limit cycle, and the integration of the curve will stop. Those integral curves that do not return to the Poincaré section are terminated according to separate termination criteria.

A signed return distance is calculated for the integral curves that return to the Poincaré section. Curves with a return distance below the cycle tolerance are considered to be limit cycles. If a curve does not satisfy the tolerance, then its return distance is compared to its neighboring integral curves. If a zero crossing is found, then a binary search is conducted. The binary search is also limited by a maximum number of iterations.

4.3.14.3.1. Source¶

The set of points that seed the integral curves that reveal the Limit Cycles. In addition to the Source attributes common to all ICS operators, the Limit Cycle operator supports the following attributes:

4.3.14.3.1.1. Source Type¶

The source type controls how the seeds for curves are created. The Limit Cycle operator only supports uniform samples on a line.

4.3.14.3.2. Integration¶

Specify settings for numerical integrators. In addition to the Integration attributes common to all ICS operators, the Limit Cycle operator supports the following attributes.

4.3.14.3.2.1. Integration Direction¶

Sets the integration direction through time. The user can choose from a combination of forward, backward, and directionless. Eigen vectors are an example of a directionless vector field. In order to integrate using a directionless field, any orientation discontinuity must be corrected prior to linear interpolation. That is, all vectors must be rotated to match the orientation of the trajectory. The ICS code will do this processing for standard fields (e.g non-higher order elements).

4.3.14.3.2.2. Termination¶

Integral curve termination can be controlled in several different ways. The termination is based on the most conservative criteria, so only one criteria must be met for termination. The options are:

Maximum number of steps

The maximum number of integration steps that will be allowed.

4.3.14.3.3. Appearance¶

The appearance tab specifies how the integral curve will be rendered. In addition to the Appearance attributes common to all ICS operators, the Integral Curve operator supports the following attributes.

4.3.14.3.3.1. Cycle tolerance¶

The smallest return distance for classifying an integral curve as a limit cycle.

4.3.14.3.3.2. Maximum iterations¶

The maximum numbers of iterations when performing the bi-section method.

4.3.14.3.3.3. Show partial results¶

If the maximum number of bi-section iterations has been reached without finding a limit cycle, show the integral curves still in the queue.

4.3.14.3.3.4. Show the signed return distances for the first iteration¶

Instead of plotting the limit cycles, plot the return distances along the Poincaré section.

4.3.14.3.3.5. Coloring¶

There are various coloring options, the names of which are self-descriptive such as coloring the curves with a solid color or according to a seed. Only those options that require further clarification are described further here.

Average Distance from seed

Each curve is colored according to the average distance of all the points in the curve from the seed.

Variable

Each curve’s color varies by the value of a scalar variable.