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https://github.com/shurimaskitten/homocliniccontinuation_coco
Enables the continuation and bifurcation analysis of homoclinic orbits in ordinary differential equations built on the COCO Continuation Toolbox in MATLAB.
https://github.com/shurimaskitten/homocliniccontinuation_coco
continuation differential-equations dynamical-systems homoclinic mathematical-modelling mathematics matlab
Last synced: about 9 hours ago
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Enables the continuation and bifurcation analysis of homoclinic orbits in ordinary differential equations built on the COCO Continuation Toolbox in MATLAB.
- Host: GitHub
- URL: https://github.com/shurimaskitten/homocliniccontinuation_coco
- Owner: ShurimasKitten
- License: mit
- Created: 2024-11-10T05:50:15.000Z (10 days ago)
- Default Branch: main
- Last Pushed: 2024-11-17T06:15:55.000Z (3 days ago)
- Last Synced: 2024-11-17T07:21:00.150Z (3 days ago)
- Topics: continuation, differential-equations, dynamical-systems, homoclinic, mathematical-modelling, mathematics, matlab
- Language: MATLAB
- Homepage:
- Size: 41.4 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# HomoclinicContinuation_COCO
Homoclinic connections are trajectories $\mathbf u(t)$ in the phase space of a dynamical system that leave a saddle equilibrium $\mathbf u_0$ and eventually return to the same equilibrium as time tends to positive and negative infinity. Such connections are important for understanding the dynamics of a system, as they often organize nearby bifurcations, making their computation a starting point in unraveling the bifurcation diagram.
This repository presents a continuation scheme for the continuation and bifurcation analysis of homoclinic connections in ordinary differential equations, utilizing the COCO Continuation Toolbox in MATLAB and largely following (Kuznetsov, Champneys, 1994). We do not aim to provide a detailed description of each bifurcation, but instead refer the reader to "Elements of Applied Bifurcation Theory" by Yuri Kuznetsov as an entry point to the literature.
We begin by formulating the boundary value problem (BVP) for path-following a homoclinic trajectory with respect to the parameters $\mu\in\mathbb{R}^2$, along with a description of detected codimension-two bifurcations. We then demonstrate the BVP with a four dimension climate model, identifying a codimension-two Belyakov point and a codimension-two resonance point with real eignvalues.
## The boundary value problem
We consider differential equations equation of the form
$$\frac{d\mathbf u}{dt} = f(\mathbf u(t),\mu),$$
with state variables $\mathbf u \in \mathbb{R}^n$ and bifurcation parameters $\mu \in \mathbb{R}^2$. A homoclinic orbit with $\mathbf u(t) \xrightarrow{} u_0$ for $t \xrightarrow{} \pm \infty$ may be continued with respect to the parameters $\mu\in\mathbb{R}^2$, and will generically trace out a codimension-one curve in the parameter space. We implement a boundary value problem (BVP), which employs projection boundary conditions and an integral phase condition, while simultaneously tracking the equilibrium point $\mathbf u_0$. The BVP is given by
$$f(\mathbf u_0,\mu)=0,$$
$$\frac{d\mathbf u}{dt} - Tf(\mathbf u, \mu) = 0,$$
$$J(\mathbf u_0, \mu) \mathbf v_{u,j} = \lambda_{u,j}\mathbf v_{u,j}, \quad i = 1,\dots, n_u,$$
$$J(\mathbf u_0, \mu) \mathbf v_{s,i} = \lambda_{s,i}\mathbf v_{s,i}, \quad i = 1, \dots, n_s,$$
$$\mathbf v^\dagger_{u,i}\mathbf v_{u,j} = 1, \quad j = 1, \dots, n_u,$$
$$\mathbf v^\dagger_{s,i}\mathbf v_{s,i} = 1, \quad i = 1, \dots, n_s,$$
$$\int^1_0 \left( \frac{d\tilde{\mathbf u}(t)}{dt} \right )^T \mathbf u(t) dt = 0,$$
$$L_s(\mu) \cdot (\mathbf w_s - \mathbf u_0) = 0,$$
$$L_u(\mu) \cdot (\mathbf w_u - \mathbf u_0) = 0,$$
where $J$ is the jacobian of $f$, $\mathbf u(t)$ represents the homoclinic solution at the current continuation step, and $\tilde{\mathbf u}(t)$ represents it at the previous step. The endpoints of the homoclinic connection are given by $\mathbf w_{s}$ and $\mathbf w_{u}$, which lie in the stable and unstable linear eignspaces of $\mathbf u_0$, respectively. These eigenspaces are spanned by the eignvectors $\mathbf v_{s,i}$ nad $\mathbf v_{u,j}$.
The projection operators, $L_s(\mu)$ and $L_u(\mu)$, are reconstructed at each continuation step and vary continuously with the parameters $\mu\in\mathbb R^2.$ More precisely, we solve the linear system
$$U_s(\mu)\left(V(\mu) V(\tilde{\mu})^T\right) = V(\tilde{\mu})V(\tilde{\mu})^T,$$
for $U_s$, where $V(\mu)$ is a matrix whose rows span the orthogonal complement of the unstable (linear) eignspace $E_u^\perp(\mu)$ at the current parameter value $\mu$, and similarly rows of $V(\tilde\mu)$ span the eignspace $E_u^\perp(\tilde\mu)$ at the previous parameter value $\tilde\mu$. We also solve the linear system
$$U_u(\mu)\left(W(\mu) W(\tilde{\mu})^T\right) = W(\tilde{\mu})W(\tilde{\mu})^T,$$
where the rows of $W(\mu)$ and $W(\tilde\mu)$ belong to the span of the orthogonal complement of the stable eigenspaces $E_s^\perp(\mu)$ and $E_s^\perp(\tilde\mu)$, respectively. The desired projection operators are then given by
$$L_s(\mu) = U_s(\mu)V(\alpha),$$
$$L_u(\mu) = U_u(\mu)W(\alpha),$$
varying continuiously with the parameters $\mu$ (Beyn 1990).
## Codimension-two homoclinic bifurcations
A homoclinic connection can become degenerate at isolated codimension-two points along a homoclinic bifurcation curve, acting as organisation centers for nearby bifurcations. We detect several codimension-two homoclinic bifurcations and also monitor the type of equilibrium involved in the homoclinic connection. The equilibrium type may be printed and has the following labels.
| EQtype | Name |
|:-------:|:-----------------------------------------:|
| 1 | Saddle with real leading eigenvalues |
| 2 | Saddle-focus with 2D leading stable manifold |
| 3 | Saddle-focus with 2D leading unstable manifold |
| 4 | Bi-focus with 2D stable and 2D unstable leading manifolds |
The codimension-two homoclinic bifurcations are classified as allows
| Symbol | Name |
|:-------:|:-----------------------------------------:|
| NSS | Neutral saddle |
| NSF | Neutral saddle-focus |
| DRS | Double real leading stable eigenvalues |
| DRU | Double real leading unstable eigenvalues |
| TLS | Three leading stable eigenvalues |
| TLR | Three leading unstable eigenvalues |
| NDS | Neutrally Divergent Saddle-Focus (Stable) |
| NDU | Neutrally Divergent Saddle-Focus (Unstable)|
| OFS | Orbit flip w.r.t stable manifold |
| OFU | Orbit flip w.r.t unstable manifold |
| H | Shilnikov-Hopf bifurcation |
| S | Non-central homoclinic saddle-node bifurcation |
| RES | Zero of the saddle value |
Note that, the continuation of a homoclinic bifurcation branch will typically end before the codimension-two points `H` and `S` are detected.
## Example one: a standard Belnikov transition
We now demonstrate the boundary value problem (BVP) using a four-dimensional climate model; however, the implementation is applicable to any system. The periodic solutions have already been computed and are stored in the `exampleData` file. Users only need to provide a COCO compatible vector field and ODE function handle within the structural array `probSettings`. For details on the implementation, refer to the function `loadDefaultSettings()`.
### Code
```markdown
# Load settings
[probSettings, thmEq, thmPO, thmHB, thmSN, thmHom, thmSNPst, thmSNPun, thmPDst] = loadDefaultSettings();
## HOM - part 1
# Settings
probSettings.corrSettings.TOL = 1e-4;
probSettings.collSettings.NTST = 250;
probSettings.contSettings.PtMX = [850 0];
probSettings.contSettings.h0 = 1e-2;
probSettings.contSettings.h_max = 1e-2;
# Construct COCO homoclinic problem
prob = proj_isol2hom('PO_example1', 80, probSettings);
# Run COCO
coco(prob, 'Hom_example1_part1', [], 1, {'mu', 'eta', 'EqType', 'x.coll.err', 'x.coll.err_TF'})
## HOM - part 2
# Settings
probSettings.contSettings.PtMX = [0 850];
probSettings.corrSettings.TOL = 1e-6;
probSettings.collSettings.NTST = 500;
probSettings.contSettings.h0 = 1e-1;
probSettings.contSettings.h_max = 2e-1;
# Construct COCO homoclinic problem
prob = proj_hom2hom('Hom_example1_part1', 1, probSettings);
# Run COCO
coco(prob, 'Hom_example1_part2', [], 1, {'mu', 'eta', 'EqType', 'x.coll.err', 'x.coll.err_TF'})
## Plot Hom
coco_plot_bd(thmHom, 'Hom_example1_part1')
coco_plot_bd(thmHom, 'Hom_example1_part2')
```
### Two-parameter bifurcation diagram
The bifurcation diagram features a homoclinic bifurcation branch labeled `Hom`. Additionally, it includes the curves `H` of Hopf bifurcation and `S` of saddle-node bifurcation of equilibria; however, these are not discussed. Along the curve `Hom`, one observes a codimension-two resonance homoclinic bifurcation with real eignvalues (i.e. `NSS`) and a codimension-two Belnikov transition involving the unstable eigenspace (i.e. `DRU`). The figure also illustrates representative homoclinic connections along each segment of `Hom`.
# Known bugs and TODO
- BUGS:
- Redundant labels sometimes printed. Its okay though.
- TODO:
- Computation of the adjoint problem.
- Following from the adjoint problem, the test functions for inclination flips.
- Orientability index.
- Continuation of eigenspaces using the Riccati equation approach as implemented in MATCONT.
# Contact and citation
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