Test Functions.

Codimension-two homoclinic orbits are detected along branches of codim 1 homoclinics by locating zeroes of certain test functions $\psi_i$. The test functions that are ``switched on'' during any continuation are given by the choice of the labels $i$, and are specified by the parameters NPSI,(/,I,IPSI(I)),I=1,NPSI) in h.xxx. Here NPSI gives the number of activated test functions and IPSI(1),$\ldots$,IPSI(NPSI) give the labels of the test functions (numbers between 1 and 16). A zero of each labeled test function defines a certain codimension-two homoclinic singularity, specified as follows. The notation used for eigenvalues is the same as that in ChKu:94 or ChKuSa:95.

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$i=1$: Resonant eigenvalues (neutral saddle); $\mu_1=-\lambda_1$.

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$i=2$: Double real leading stable eigenvalues (saddle to saddle-focus transition); $\mu_1=\mu_2$.

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$i=3$: Double real leading unstable eigenvalues (saddle to saddle-focus transition);
$\lambda_1=\lambda_2$.

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$i=4$: Neutral saddle, saddle-focus or bi-focus (includes $i=1$); Re$(\mu_1) = -$   Re$(\lambda_1)$.

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$i=5$: Neutrally-divergent saddle-focus (stable eigenvalues complex);
Re$(\lambda_1) = -$   Re$(\mu_1) -$   Re$(\mu_2)$.

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$i=6$: Neutrally-divergent saddle-focus (unstable eigenvalues complex);
Re$(\mu_1) = -$   Re$(\lambda_1) -$   Re$(\lambda_2)$.

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$i=7$: Three leading eigenvalues (stable); Re$(\lambda_1) = -$   Re$(\mu_1) -$   Re$(\mu_2)$.

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$i=8$: Three leading eigenvalues (unstable); Re$(\mu_1) = -$   Re$(\lambda_1) -$   Re$(\lambda_2)$.

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$i=9$: Local bifurcation (zero eigenvalue or Hopf): number of stable eigenvalues decreases; Re$(\mu_1)=0$.

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$i=10$: Local bifurcation (zero eigenvalue or Hopf): number of unstable eigenvalues decreases; Re$(\lambda_1)=0$.

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$i=11$: Orbit flip with respect to leading stable direction (e.g., 1D unstable manifold).

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$i=12$: Orbit flip with respect to leading unstable direction, (e.g., 1D stable manifold).

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$i=13$: Inclination flip with respect to stable manifold (e.g., 1D unstable manifold).

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$i=14$: Inclination flip with respect to unstable manifold (e.g., 1D stable manifold).

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$i=15$: Non-central homoclinic to saddle-node (in stable manifold).

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$i=16$: Non-central homoclinic to saddle-node (in unstable manifold).

Expert users may wish to add their own test functions by editing the function PSIHO in autlib5.f.

It is important to remember that, in order to specify activated test functions, it is required to also add the corresponding label $+20$ to the list of continuation parameters and a zero of this parameter to the list of user-defined output points. Having done this, the corresponding parameters are output to the screen and zeros are accurately located.