We know that a higher-order ode can be converted to a dynamical system by replacing each higher-order derivative by a new variable. What about the inverse problem? Does a dynamical system convert to a system of odes?
A dynamical system can be defined by a difference equation or a recurrence relation and, in general, they can not always be transformed to a system of odes.
For example, you can check in "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields" from Guckenheimer and Holmes. As far as I remember it should be explained there.
In general, dynamical systems are "defined" as a group action of the reals or integers into a set (usualy a manifold) . There are many kinds of dynamical system, namely: differential dynamical systems (induced by ode's), discret dynamical systems (induced by diffeormorphims), symbolical dynamical systems, etc.
For instance, if you take a difeomorphis f:M->M, you can cosider the dynamical system in M given by the action a:ZxM->M such that a(n,x)=f^n(x). It defines a discret dynamical systems which is not a ode.
As a reference you can see the book of Arrowsmith and Place, "Introduction to Dynamical Systems".
I think that dynamic systems could be converted to the ode's on a time scale since the last ones contain the discrete equations.
I think you should be more precise: What is a dyn sys? What is the meaning of "to convert", etc.
Dynamic system is a set of odes and its equivalence to a higher-order ode iff if the solution of one is known, so does the other through a proper mapping
I am still not sure what you mean: Now that you have defined a dynamical system as a set of odes, what is meant by your question "Does a dynamical system convert to a system of odes?"?
I mean the equivalence between a higger-order ode and a set of first-order odes
Let me try to understand
Given a higher order differential equation x^(n)=F(t,x,x',x'',...,x^(n-1)), taking x=x_0, x^(i)=x_i, you can transform it in a system of differential equations as follows:
x_0'=x_1
x_1'= x_2
x_2'=x_3
:
.
x_i'=x_{i+1}
:
.
x_{n-1}'=F(t,x_0,x_1,...,x_{n-1})
Ok, it is always possible to do.
Now, given a system of differential equations
x_1'=F1(t,x_1,x_2,...,x_n)
x_2'=F2(t,x_1,x_2,...,x_n)
:
.
x_n'=Fn(t,x_1,x_2,...,x_n)
you want to know if there exists an equivalent higher order differential system y^(n)=F(t,y,y',y'',...,y^(n-1)).
Right?
Here is an engineer's perspective. Consider the following system of autonomous ODEs:
dx1/dt=H1(x1,x2), dx2/dt=H2(x1,x2), subject to a set of initial conditions for x1 and x2.
One would certainly call this a dynamical system expressed in this case as set of ODEs. And there are plenty of examples of such systems in standard texts on dynamical systems.
Now we can also write this system of ODEs as a higher ODE:
d^2x1/dt^2=grad H1.(dx1/dt, dx2/dt)=grad H1.{H1,H2)
This second order ODE is subject to the same ICS as before. So this is another way of writing our dynamical system. But we have one additional requirement: grad H1 must exist. This requirement was not required for the original set of ODEs.
If the original system of ODEs described a physical problem, then by solving the second order system we have imposed the additional requirement that H1 and H2 must be differentiable. But the physics of the problem does not require it!
Often this is not a problem. In fluid mechanics this is done routinely when we express the equations of motion in terms of a stream function, which results ina fourth order system.But one has to be cautious in the interpretation of the results if the fourth oder system predicts singularities in the flow domain, such as cusps. But that is another topic....
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9th Sep, 2013
Shuchang Zhang
Shanghai Jiao Tong University
Dear Novaes,
That's what I mean. Do you have any idea?
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9th Sep, 2013
Douglas Duarte Novaes
University of Campinas
I think the answer is no.
Consider the planar autonomous differential system:
(1) x'=0
(2) y'=0
The system (1,2) has all points of the plane as singularities.
On the other hand a second order differential equation u''=F(u,u') can be written as the differential system
(3) u'=v
(4) v'=F(u,v)
Clearly any point (u,v) such that v\neq 0 is a regular point of the system (3,4).
Hence system (1,2) can not be equivalent (in any sense) to a second order differential equation.
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9th Sep, 2013
Manuel Mazo Jr
Delft University of Technology
The way you originally formulated the question the answer is trivially yes, as a dynamical system (under your definition of the term) is actually a system of 1st order ODE's.
Now, if the question is if you can transform the system of 1st order ODE's, with some of its "state variables" (in control systems terms) as the output, as a single high-order ODE the answer is in general NO.
There are particular cases in which this is possible but I believe (and this is just a, possibly poorly educated, "claim") that such set of dynamical systems is a measure zero set of all possible systems.
11th Nov, 2013
Andrea Giacobbe
University of Catania
The answer is no, and there is a simple linearization argument. At an equilibrium a system of first-order ODEs can have a linearization with Jordan blocks of any dimension, on the other hand the linearization at an equilibrium of a higher order ODE has only Jordan blocks of maximal dimension (i.e. the geometric multiplicity of any eigenvalue is always 1, regardless of what their algebraic multiplicity is).
This is nicely explained in Arnol'd book "Ordinary Differential Equations".
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