## Bifurcations of limit cycles

On a limit cycle curve the following bifurcations can occur

- Branch Point of Cycles, this will be denoted as `BPC`
- Period Doubling, denoted as `PD`
- Fold, also known as Limit Point of Cycles, this will be denoted as `LPC`
- Neimark-Sacker, this will be denoted as `NS`

The test function for the Period Doubling bifurcation is defined by the following system

|  |  | (48)

here and are so-called bordering vector-functions [25], see [15] for details on the implementation. The system is discretized using orthogonal collocation and solved using the standard MATLAB sparse system solver. The solution component of this system is the test function and equals zero when there is a Period Doubling bifurcation.

The Fold bifurcation is detected in the same way as the Fold bifurcation of equilibria, the last component of the tangent vector (the component) is used as the test function.

The Neimark-Sacker bifurcation is detected by monitoring the eigenvalues of the monodromy matrix for the cycle. The monodromy matrix is computed like in AUTO by a block elimination in the discretized form of the Jacobian of (47).

BPC cycles are not generic in families of limit cycles, but they are common in the case of symmetries, if the branch parameter is also the continuation parameter. CL_MATCONT uses a strategy that requires only the solution of linear systems; it is based on the fact that in a symmetry-breaking BPC cycle has rank defect two, where is the square matrix , obtained from the discretized form of the Jacobian of (47). To be precise, if , then

and

where and denote discretization in mesh points and in collocation points, respectively. Therefore we border with two additional rows and columns to obtain

so that is nonsingular in the BPC cycle. Then we solve the systems

where have components, and , , , and , are scalar test functions for the BPC. In the BPC cycle they all vanish.

The singularity matrix is

|  |  | (49)

The first row corresponds to the BPC. It contains zeros which indicates that , , , and should vanish. The last row corresponds to the NS. Because we have to exclude that all four testfunctions of the BPC are zeros, we introduce an extra testfunction which corresponds to the norm of these four testfunctions. A NS is detected if this norm is nonzero,the testfunction for the fold is nonzero and the testfunction for the NS is equal to zero.

Subsections

- Branch Point Locator
- Normal form coefficients