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Overview
Via this page you obtain a centre 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 centre 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 \[
\frac{d\vec u}{dt}=\vec f(\vec u) =\left[\begin{array}{ccc}
2&1&2\\ 1&-1&1\\ -3&-1&-3 \end{array}\right] \vec u
+\varepsilon \left[\begin{array}{c}u_2u_3\\ -u_1u_3\\
-u_1u_2\end{array}\right]. \] As this system is already
entered for you, just enter the magic word, then click on
the Submit button to see.
The bottom of this web page 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.
For example, you can construct sub-centre manifolds, such as slow manifolds among fast oscillations.
Submit your system of ODE/DDEs for analysis
Fill in the fields below for your ODE/DDE system:
- the n variables of the system must be denoted u(1),...,u(n);
- any time tau delayed variables must be denoted
u(1,tau),...,u(n,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 centre manifold is constructed to be tangential to
the centre 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 centre manifold is parametrised by you also
specifying m vectors, \(\vec z_j\), which are to be
orthogonal to updates to the centre manifold---these vectors
are to 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 algebraic expressions
(general examples)
Wait a minute or two
The analysis may take minutes after submitting. Be patient.
Read the following. Please inform me of any problems.
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 or a complex amplitude of an oscillatory mode
\(e^{i\omega t}\) for some frequency \(\omega\).
- 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 frequencies, 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.
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 u(1) and u(2). The time
delayed variable \(y(t-\pi)\) is denoted u(2,pi).
Copy and paste the following entries.
| Description | Delay ODE example |
| RHS function |
(-4*(1+small*delta)^2*(5/8*u(2)+3/8*u(2,pi)),
+u(1)*(1+u(2))) |
| centre frequencies |
1,2,-1,-2 |
| 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 |
| Print | small,delta |
The web service
finds that in terms of complex amplitudes \(s_1(t)\),
\(s_2(t)\) and their complex conjugates \(\bar s_1(t)\) and
\(\bar s_2(t)\), the centre manifold \[ \vec
u=\left[\begin{array}{c} e^{it}s_1 +e^{i2t}s_2 +e^{-it}\bar
s_1 +e^{-i2t}\bar s_2\\ \textstyle -ie^{it}s_1
-\frac12ie^{i2t}s_2 +ie^{-it}\bar s_1 +\frac12ie^{-i2t}\bar
s_2 \end{array}\right] \] to error \({\cal
O}(\varepsilon)\). The corresponding evolution on the
centre manifold is \[\begin{array}{l} ds_1/dt=
\varepsilon(-0.04-0.09i)s_2\bar s_1 +{\cal
O}(\varepsilon^2)\\ ds_2/dt= \varepsilon(0.42-0.49i)s_1^2
+{\cal O}(\varepsilon^2). \end{array}\]
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 |
(u(2),-u(2),-u(3),u(3)))
+small*tp mat((-u(1),+u(1)-u(2)+u(3),+u(2)-u(3)+u(4),-u(4)) |
| centre frequencies |
0,0 |
| centre subspace |
(0,0,0,1),(1,0,0,0) |
| adjoint subspace |
(0,0,1,1),(1,1,0,0) |
| Order of error |
5 |
| Print | small |
Then \(s_1\) and \(s_2\) represent the probabilities of being in states one and two, and in three and four, respectively.
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 |
(
u(3),
u(4),
-2*u(1)+u(2)-small*u(1)^3/2+small*3/10*(-u(3)+u(4)),
u(1)-2*u(2)+small*3/10*(u(3)-2*u(4))
) |
| centre frequencies |
1,-1,sqrt(3),-sqrt(3) |
| centre subspace |
(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 |
| Print | small |
The square root frequencies do not cause any trouble (although one may need to reformat the LaTeX of the cis operator).
In the model, observe that \(s_1=s_2=0\) is invariant, as is \(s_3=s_4=0\).
These are the nonlinear normal modes.
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 |
(-u(2)*u(3)+b*u(2)*u(5),
+u(1)*u(3)-b*u(1)*u(5),
-u(1)*u(2),
-u(5),
u(4)+b*u(1)*u(2)) |
| centre frequencies |
0,0,0 |
| centre 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 |
| Print | small,b |
Alternatively, find that there is not a coordinate transform that separates the system into a normal form of separated fast and slow modes (Cox and Roberts. Initialisation and the quasi-geostrophic slow manifold. http://arXiv.org/abs/nlin.CD/0303011, 1994.)
Copy and paste the following to find that, as well as the slow modes modulating the fast oscillations, the fast oscillations drive a slow response in\(~w\).
| Description | ODE example |
| RHS function |
(-u(2)*u(3)+b*u(2)*u(5),
+u(1)*u(3)-b*u(1)*u(5),
-u(1)*u(2),
-u(5),
u(4)+b*u(1)*u(2)) |
| centre frequencies |
0,0,0,1,-1 |
| centre 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) |
| 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 |
| Print | small,b |
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