Heureka.

The solution is much simpler and more straightforward than we all thought:

The ODE system determines its Jacobian matrix. This, as any matrix, has a rank (the number of linearly independent row vectors). And this number is the effective number of equations. If the determinant of the Jacobian is different from zero, then we are sure that all equations are non-redundant, i.e. none of tham can be transformed away. Of course, the value of the Jacobian's rank may be different in different regions of the variable space. Then these regions need to be considered in isolation.

Computing the rank of a matrix is a routine task in linear algebra and is implemented in any decent linear algebra package. How things look in Mathematica can be seen in the appended file.

I had posted this result over the weekend and Monday morning I had to observe that it had disappeared. Strange things happen.

1. Do you have a special equation? One might have a look at it.

2. According to the Cauchy-Arnold-Newton theorem

(all the ODEs can be transformed locally into x1'=0,x2'=0,...,xn-1'=0,xn'=1)

theroetically all equations are 1D.

3. So you have to give a more precise definiton, and provide a concrete example, otherwise the question is a bit fuzzy.

János

If you find a conservation law for the variables, you will get an algebraic equation and you can use it to reduce the dimension of differenctial equations.

A dimension-reduction method is to find a quantity (e.g. total energy) based on physical considerations which is constant in your case.

One approach is to consider the network represented by your dynamical system. If one of the variables does not influence the others, then it does not participate in the dynamics but is only "along for the ride." x in your example does not appear on the right-hand side of any of the equations, and hence it represents an extraneous dimension. Similarly, if a variable is not influenced by the others or is not an explicit function of time (such as dx/dt = constant) then its effect can be included in the initial conditions (if constant = 0) or it makes the remaining system non-autonomous (x = constant*t). There are more complicated cases where the network breaks into two non-interacting networks or where a portion of the network is just a slave to the rest of the master network.

A very simple way to know the dimension of the system is to compute the correlation dimension of the reconstructed signal of the attractor. Whenever this signal is greater than 3, you must have a 4-D system.

András gave the answer that came to my mind first. Whenever the trajectories of a system don't make use of their full statespace but remain restricted to some sum-manifold one has to suspect that there is a conserved quantity. Experience from physics shows that finding conserved quantities is not a routine task with clear receipes. Looking for symmetries is the best advice. Having made this excursion to generalities, I looked back to Sajad's special example. What is the conserved quantity for the system

x' = y

y' = z

z' = w

w' = -aw + 2yz – y ??????

I have to confess that I was not able to find it.

Seems to be a challenge for bright people!

Here Julien has a more practicable approach with his observation that x no longer appears in the rhs of the equation. This lets the dynamics be determined by

y' = z

z' = w

w' = -aw + 2yz – y

and the let x follow this dynamics according to the kinematic law

x(t) = integral from t0 to t of y(t') dt'.

This is an exact and complete description of dimension lowering from 4 to 3 which is not induced (at least not directly) by a conservation law. (This is what fascinates me all the time with concrete examples: they hint to possibilities that general considerations tend to miss.)

Julien's notion of a network associated with a dynamical system is new for me and I would be interested to learn.

René's 'very simple way' is an example for a wide-spread strategy of scientific writers to hide difficulties (best case) or obscurities (usual case) of their concepts by qualifying them in this way. Also here I would be interested in seeing what's behind.

I have updated the question. Please see it.

To recognize that the dimensions of your problem are lower, you need to find a set of variables that only appear in a certain combination. In your examples, this combination is the expression you differentiate. Other examples would be expressions that actually depend only, say, xy (or x^2+y^3, or whatever) and its derivatives, instead of x and y themselves. Then you can replace this combination in the set of equations by a new variable, in this way reducing the dimension.

Dear Dr. Schleicher

Thank you. However I think this method is to some extent manual and I need a mechanism.

Holographic info theory suggests that almost any system can be thought of having almost any number of dimensions but certain numbers of dimensions seem more suited to certain systems.

Whicher ever model is easiest to use should be the obvious approach.

@Graham. Greetings to cloud-cuckoo-land!

There is a result in differntial geometry called the theorem of Frobenius which yields another approach. Suppose your ODE is defined by a vector X in 4 space and it can be written as a linear combinationn of 3 other vectors Y1,Y2,Y3. Then if the Y vectors constitute an involutive sysem (i.e. all commutators [Yi,Yj] are also linear combinaations of the Y's) the Y's are tangent to a set of three dimensional sub manifolds and the solution of your ODE is constrained to stay in whatever submanifold it starts on.

@Bernard.

In your inspiring contribution you should refer to the Y's as vector fields (not simply as vectors) since only then a commutator between such objects makes sense.

Heureka.

The solution is much simpler and more straightforward than we all thought:

The ODE system determines its Jacobian matrix. This, as any matrix, has a rank (the number of linearly independent row vectors). And this number is the effective number of equations. If the determinant of the Jacobian is different from zero, then we are sure that all equations are non-redundant, i.e. none of tham can be transformed away. Of course, the value of the Jacobian's rank may be different in different regions of the variable space. Then these regions need to be considered in isolation.

Computing the rank of a matrix is a routine task in linear algebra and is implemented in any decent linear algebra package. How things look in Mathematica can be seen in the appended file.

I had posted this result over the weekend and Monday morning I had to observe that it had disappeared. Strange things happen.

An interesting question with regard to Ulrichs answer seems to be: "What happens on or close to the boundary of regions of different rank?" Can a solution cross it under certain conditions? In the 3D quadratic jerk system as given in the question, the rank of the Jacobian normally is 3, but the rank is two on the hyperplane a1+2*a4*x+a7*y+a8*z=0. Directly on the hyperplane, one may eliminate x if a4 is nonzero, and treat it as a "spectator variable", i.e. solve the second and third equations for y and z as a function of t (interpreting '=d/dt), and obtain x either from the hyperplane equation directly or from a direct integration of the first equation. It might be a nice exercise to see whether both possibilities agree, if the initial conditions put (x,y,z) on the hyperplane and/or the initial velocity vector (x',y',z') is parallel to the hyperplane.

I expect that someone has general theorems for this case of different rank regions, probably along the lines of Bernard. But as noted above, an explicit example offers the chance for new insights.

I agree with @Ulrich answer with one comment only: the Jacobian or local linearization is a great tool, but sometimes the system is not well posed 'local linearizable'. Of course this needs further investigation, since almost everything in natural sciences are 'local linearization' approximations of the general nonlinear situation (even the simplest harmonic oscillator).

Ulrich Mutze comment regarding the Jacobian captures the essential features! For someone starting to learn about dynamical systems, the rank of the Jacobian is the first step to understanding complex dynamics. Bifurcations from a steady or periodic state that can be gleaned from the rank or eigenvalues are the next obvious step.

Ulrich Mutze has indeed provided a reliable means for determining that a nonlinear dynamical system does not have redundant equations by checking that the Jacobian matrix is of full rank. However, I believe the converse is not true. One can have a D-dimensional nonlinear dynamical system whose Jacobian has rank < D The simplest example I could construct with an oscillatory dynamic has D = 4 and is given by

dx/dt = y - x

dy/dt = -xz + u

dz/dt = xy - a

du/dt = -by

whose Jacobian has rank 3 for all (a, b) everywhere in space and whose dynamics are apparently 4-dimensional as evidenced, among other things, by its four distinct Lyapunov exponents of 0.0705, 0.0128, 0, -1.0832 for a = 2.6, b = 0.44. Can anyone provide an example with D < 4?

Dear Julien,

if the rank of the Jacobian is 3, then R^4 can be foliated into 3-dimensional submanifolds which are invariant under the dynamics. If one neverheless investigates the system on its whole state space R^4, one cannot recognize this kind of geometrically non-trivial restrictedness. Who defines dynamical systems on differential manifolds (instead of R^n) in the first place, will not be surprised to find geometrically nontrivial invariant submanifolds.

Similar to Dr. Low's example, consider x-dot=y, y-dot=-x ----> r-dot = 0, theta-dot = 1

Low's example is unbounded, and so I'm not sure what to say about that. Jafari's example is conservative with a Hamiltonian H = x^2/2 + y^2/2, and so the dynamics is indeed one-dimensional since it is constrained to the circle H = R^2/2 determined by the initial conditions. Perhaps Mutze will comment.

I think this could be an initial value problem.

For example, consider the 4-D system of equations proposed here:

x' = y

y' = z

z' = w

w' = -aw + 2yz – y

To solve this equation, we need four initial conditions, namely: x(0), y(0), z(0), and w(0). Then, let's consider two type of initial conditions:

1) x(0)=x0, y(0)=y0, z(0)=z0, w(0)=w0, where x0, y0, z0, w0 are different (or at least make w0 independent of the three first one).

2) x(0)=x0, y(0)=y0, z(0)=z0, and w(0)=-az0 + y0^2 – x0

Then, we ask the question again, for which of these initial conditions, the solution of the 4-D equations reduces to the solution of the 3-D equations, at least for the dynamic variables x, y, and z. It turned out to me (at least numerically) that the for the second type of initial conditions 4-D reduces to 3-D.

It seems that w here plays the role of an auxiliary variable, as also commented by Julien and Herbert, and (following the Julien's network model) w is driven by x, y, z, if initially they are also dependent 4-D reduces to 3-D.

Ulrich Mutze has a point there. I agree that the associated Jacobian matrix can give us some information, and his worked example is good. However, i'm not sure if the same condition applied into singular systems to EDO, or algebraic EDO.

Any way, I enjoyed the question. Interesting to think about.

@Robert

The 'Hamilton-Jacobi theory', the most advanced version of classical mechanics (with finitely many degreees of freedom), works by transforming (by a canonical transformation) to new coordinates (P_i, Q_i in my old books) the time derivative of which vanishes, so that the new coordinates are constants of motion.

You have transformed the system x-dot =x , y-dot=y to r-dot=r, phi-dot=0 so that half of the equations satisfy this canonical form . This does not mean that phi was 'transformed away' (phi = const, is non-trivial information). It is not the kind of functionally dependent quantity as w=f(x,y,z) in the example of the original question.

The nonvanishing Jacobian excludes that the variables under consideration are functionally dependent.

With your observation that all this is not a question of (un)boundedness you are perfectly right.

@Robert,

(r,theta) being a system of coordinates, how can theta be a function (of what kind ever) of r? You are right 'it's a somewhat different game'. We both seem not yet to see clearly which one. Hiqmet asked a good question. When the straightforward resolution of the original example was that the 'superfluous' quantity w was a function of x,y,z (actually -az + y^2 – x), what if we give initial values x0,y0,z0,w0 such that

w0 is different from -az0 + y0^2 -x0 ?

This question seems to be related to the notion of first integrals of the system. If we know k independent first integrals then, in general, we will be able to reduce by k the order of the system. And if k = n then the general solution of the system could be obtained directly without integration. A typical example is the predator-pray problem

x-dot = ax - by

y-dot = cx - dy

Here, a first integral is Q(x,y) = cx - d.ln(x) + by - a.ln(y)

The contour lines of Q offer a good insight about the evolution of x and y.

@Julien, and @Ulrich (with cc @Robert),

I read some comments here: (a) Julien has proposed the use of network approach, while (b) Ulrich suggested to apply the Jacobian matrix (but Robert Low showed a situation where the approach can fail). Personally, I not sure if the network approach is able to deal with all dynamical systems.

I believe that the problem posted by Sajad Jafari is linked to know if the system is "well structured" or not. The matroid theory can help us in such case (this is linked to the graph theory, therefore there is also a link with network approach).

Looking at the paper attached, Section 4 "Insvestigation of the problem structure", the Dulmage-Mendelsohn decomposition shows to us if we are dealing with a "well structured" problem or not. Such scheme can be applied to linear and non-linear system.

In addition to the answer by H.E. Lehtihet: the order of a dynamical system dx/dt=X(x,t) (where x is a vector) can also be reduced by k if this system admits a k-dimensional solvable symmetry algebra. This is a classical result dating back to Sophus Lie; the details can be found e.g. in Theorem 2.68 of the book P. J. Olver, Applications of Lie Groups to Differential Equations. 2nd ed. Springer-Verlag 1993.