## Overview

I provide a procedure to construct a specified invariant manifold for a specified system of ordinary differential equations or delay differential equations. The invariant manifold may be any of a centre manifold, a slow manifold, an un/stable manifold, a sub-centre manifold, a nonlinear normal form, or any spectral submanifold. Thus the procedure may be used to analyse pitchfork bifurcations, or oscillatory Hopf bifurcations, or any more complicated superposition. In the cases when the neglected spectral modes all decay, the constructed invariant manifold supplies a faithful, large time, emergent, model of the dynamics of the differential equations. Further, in the case of a slow manifold, this procedure now derives vectors defining the projection onto the invariant manifold along the isochrons: this projection is needed for initial conditions, forcing, system modifications, and uncertainty quantification.

The procedure now also empowers one to account for sinusoidal time dependence in the ODEs, such as to derive spectral sub-manifold models of forced nonlinear normal modes.

**Execute on your computer?** The procedure uses computer
algebra, the package
Reduce, to construct approximations to the invariant
manifolds.

- So download and install Reduce, and then download InvariantManifold.zip.
- Startup Reduce in the unzipped
`InvariantManifold`folder. - Execute
`in_tex "invariantManifold.tex"$`to load the procedure. - Test by executing
`exampleslowman();`and confirm the output is as in`invariantManifold.pdf` - See examples in
`diverseExamples.pdf`and then try for systems of your interest.

**Maybe use the web service**
Alternatively, click this link to expand this
page. Then via the web form below you may obtain an
invariant manifold of your specified system of ordinary
differential equations (ODE) or delay
differential equations (DDE), when the
ODE/DDE has fast and centre modes. The
modes may be slow, as in a pitchfork bifurcation, or
oscillatory, as in a Hopf bifurcation, or some more
complicated superposition. In the case when the fast modes
all decay, the centre manifold supplies you with a faithful
large time model of the dynamics.

For example, this web page could help you analyse the long time dynamics of the system \[ \frac{d\vec u}{dt} =\left[\begin{array}{ccc} 2&1&2\\ 1&-1&1\\ -3&-1&-3 \end{array}\right] \vec u +\varepsilon \left[\begin{array}{c}u_2u_3\\ -u_1u_3\\ -u_1u_2\end{array}\right]. \] As this system is already entered for you, just enter the magic word, then click on the Submit button to see.

The bottom of the web form lists further examples.

For example, you can obtain the equivalent modulation equations corresponding to a given set of ODE/DDEs that have one or more oscillatory modes (a Hopf bifurcation for example). The analysis provides you with equations for the evolution of the complex amplitudes of the oscillators. This approach is better than averaging/homogenisation.

FYI: the source code is now available for collaborative
development via the folder `CentreManifold` of a
Github repository

#### Non-autonomous ODEs?

Systems with slowly-varying time-dependence, or with sinusoidal time-dependence may be analysed here. Systems with more general time-dependence are significantly more difficult, but are analysed via the web page Normal form of stochastic or deterministic multiscale differential equations.## Submit your system of ODE/DDEs for analysis

- the n variables of the system must be denoted u1,u2,...,un;
- any time tau delayed variables must be denoted u1(tau),...,un(tau) for any delay tau;
- specify the ODE/DDEs of the dynamical system \(\frac{d\vec u}{dt}=\vec f(\vec u)\) by specifying the `nonlinear' function\(~\vec f(\vec u)\) which should have an equilibrium at the origin;
- the invariant manifold is constructed to be tangential to the invariant subspace of the origin which is defined by you specifying the m eigenvalues and m eigenvectors,\(~\vec e_j\), of the m zero and/or pure imaginary eigenvalues of the matrix\(~L\) of the linearisation about the origin;
- the invariant manifold is parametrised by you also specifying m vectors,\(~\vec z_j\), which are to be orthogonal to updates to the invariant manifold---these vectors MAY be the eigenvectors of the m zero and/or pure imaginary eigenvalues of the adjoint of the matrix\(~L\);
- for the moment, the ODE/DDEs must be multinomial in form;
- Use the syntax of Reduce for the algebraic expressions

## Wait a minute or two

### In the results

- The centre manifold is parametrised by m slow variables
`s(j)`. - Each such slow variable
`s(j)`is either- a 'real' slow variable (specified eigenvalue zero), or
- a complex amplitude of an oscillatory mode\(~e^{i\omega t}\) for some frequency\(~\omega\) corresponding to specified eigenvalue \(\lambda=i\omega\), or
- most generally, a (complex) amplitude of the exponential mode\(~e^{\lambda t}\) for each specified eigenvalue\(~\lambda\).

- In a real system of ODE/DDEs, the complex amplitudes occur in complex conjugate pairs.
- I use the variable
`small`(also appearing as\(~\varepsilon\)) to control and order the asymptotic expansion: introducing`small`into your definition of the `nonlinear' function empowers finer control of the asymptotics. For example, in the delay ODE example the parameter\(~\delta\) is made 'small'. Analogously, cubic terms may best be made `small' when added to quadratic terms. - The code does cater for degenerate cases involving
generalised eigenvectors. But the code does this by
modifying\(~L\), the matrix of the linearisation at the
origin. The code attempts to make the smallest modification
it can to remove the degeneracy, and flags the change with
variable 'small' so you recover the original with
\(\verb|small|=1\).
*Be careful*that the results are relevant to what you want. - Similarly, the code tries to make all, except the
nominated eigenvalues, of\(~\vec z^*L\vec e\) to be 'small',
and also makes\(~\vec f(\vec 0)\) `small'.
*Be very careful*that the results are relevant to what you want. - For explanations and relevant theory, see my book Modelling emergent dynamics in complex systems, or Low- dimensional modelling of dynamics via computer algebra, or the classic Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations.
- In the case of a slow manifold (all specified
eigenvalues are zero), the code also computes a basis of
normal vectors to the isochrons (actually normal to their
tangent space at the slow manifold). Use these basis
vectors:
- to project initial conditions onto the slow manifold;
- to project non-autonomous forcing onto the slow evolution;
- to predict the consequences of modifying the original system; and
- in uncertainty quantification to quantify effects on the model of uncertainties in the original system.

## Other example systems of ODEs/DDEs

### Double Hopf bifurcation in a delay DE

This example models the double Hopf bifurcation that occurs in the coupled delay differential equations \(\frac{dx}{dt}=-4(1+\delta)^2 \left[\frac38y(t) +\frac58 y(t-\pi) \right]\), and \(\frac{dy}{dt} =\left[1+y(t)\right] x(t)\) as parameter\(~\delta\) crosses zero. Define \(u_1(t)=x(t)\) and \(u_2(t)=y(t)\); denoted as`u1`and

`u2`. The time delayed variable\(~y(t-\pi)\) is denoted

`u2(pi)`. Copy and paste the following entries.

Description | Delay ODE example |
---|---|

RHS function | (-4*(1+small*delta)^2*(5/8*u2 +3/8*u2(pi)),
+u1*(1+u2)) |

Centre eigenvalues | i,2*i,-i,-2*i |

Centre subspace | (1,-i), (1,-i/2), (1,+i), (1,+i/2) |

Adjoint subspace | (1,-i), (1,-2*i), (1,+i), (1,+2*i) |

Order of error | 3 |

small,delta |

### Metastability in a four state Markov chain

Variable\(~\varepsilon\) characterises the rate of exchange between metastable states. \[\begin{array}{l} &\dot u_{1}=-\varepsilon u_{1}+u_{2} \\&\dot u_{2}=\varepsilon \big(u_{3}-u_{2}+u_{1}\big)-u_{2} \\&\dot u_{3}=\varepsilon \big(u_{4}-u_{3}+u_{2}\big)-u_{3} \\&\dot u_{4}=-\varepsilon u_{4}+u_{3} \end{array} \] The linear perturbation terms gets multiplied by`small`again. Copy and paste the following.

Description | ODE example |
---|---|

RHS function | (u2,-u2,-u3,u3)
+small*(-u1, +u1-u2+u3, +u2-u3+u4, -u4) |

Slow eigenvalues | 0,0 |

Slow subspace | (0,0,0,1), (1,0,0,0) |

Adjoint subspace | (0,0,1,1), (1,1,0,0) |

Order of error | 5 |

small |

### Nonlinear normal modes

Renson (2012) explored finite element construction of the nonlinear normal modes of a pair of coupled oscillators. Defining two new variables one of their example systems is \[\begin{array}{rcl} &&\dot x_1=x_3\,, \\&&\dot x_2=x_4\,, \\&&\dot x_3=-2x_1+x_2-\frac12x_1^3+\frac3{10}(-x_3+x_4), \\&&\dot x_4=x_1-2x_2+\frac3{10}(x_3-2x_4). \end{array}\] Copy and paste the following code which makes the linear damping to be effectively small (which then makes it small squared); consequently scale the smallness of the cubic nonlinearity.Description | ODE example |
---|---|

RHS function | (
u3,
u4,
-2*u1 +u2 -small*u1^3/2 +small*3/10*(-u3+u4),
u1 -2*u2 +small*3/10*(u3 -2*u4)
) |

All eigenvalues | i, -i, i*sqrt(3), -i*sqrt(3) |

All space basis | (1,1,+i,+i), (1,1,-i,-i),
(1,-1,i*sqrt(3),-i*sqrt(3)),
(1,-1,-i*sqrt(3),i*sqrt(3)) |

Adjoint subspace | (1,1,+i,+i), (1,1,-i,-i),
(-i*sqrt(3),+i*sqrt(3),1,-1),
(+i*sqrt(3),-i*sqrt(3),1,-1) |

Order of error | 3 |

small |

### Harmonically forced nonlinear normal mode

Let's apply periodic forcing to the previous example, both direct and parametric. For example, here derive the effect on the mode with frequency one. Defining two new variables one of their example systems is \[\begin{array}{rcl} &&\dot x_1=x_3\,, \\&&\dot x_2=x_4\,, \\&&\dot x_3=-2x_1+x_2-\frac12x_1^3+\frac3{10}(-x_3+x_4)+f_1\cos(t), \\&&\dot x_4=x_1-2x_2+\frac3{10}(x_3-2x_4)f_2\sin(t/2), \end{array}\] where \(f_1\) is the strength of the direct forcing, and \(f_2~\)is the strength of the parametric oscillation in the last ODE. Copy and paste the following code which makes both the linear damping and the direct forcing to be effectively small (which then makes it small squared); consequently scale the smallness of the cubic nonlinearity.Description | ODE example |
---|---|

RHS function | (
u3,
u4,
-2*u1 +u2 -small*u1^3/2 +small*3/10*(-u3+u4)+small*f_1*sin(t),
u1 -2*u2 +small*3/10*(u3 -2*u4)*f_2*cos(t/2)
) |

Modal eigenvalues | i, -i |

Modal space basis | (1,1,+i,+i), (1,1,-i,-i) |

Adjoint subspace | (1,1,+i,+i), (1,1,-i,-i) |

Order of error | 5 |

f_1,f_2,small |

### Slow manifold among fast oscillations

Lorenz (1986) explored a five equation toy model to illustrate the quasi-geostrophic approximation and its slow manifold. \[\begin{array}{rcl} &&\dot u=-vw+bvz\,, \\&&\dot v=uw-buz\,, \\&&\dot w=-uv\,, \\&&\dot x=-z \\&&\dot z=x+buv\,. \end{array}\] Copy and paste the following code to find the 3D slow manifold among the rapid oscillations of\(~x,z\).Description | ODE example |
---|---|

RHS function | (-u2*u3+b*u2*u5,
+u1*u3 -b*u1*u5,
-u1*u2,
-u5,
u4 +b*u1*u2) |

Slow eigenvalues | 0,0,0 |

Slow subspace | (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0) |

Adjoint subspace | (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0) |

Order of error | 5 |

small,b |

Description | ODE example |
---|---|

RHS function | (-u2*u3 +b*u2*u5,
+u1*u3 -b*u1*u5,
-u1*u2,
-u5,
u4 +b*u1*u2) |

All eigenvalues | 0, 0, 0, i, -i |

All space basis | (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0),
(0,0,0,1,-i), (0,0,0,1,+i) |

Adjoint subspace | (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0),
(0,0,0,1,-i), (0,0,0,1,+i) |

Order of error | 4 |

small,b |

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