A system of differential equations not having a analytical solution

In summary, chaotic differential systems do not have analytical solutions and can only be solved numerically. This is because not all mechanical problems can be solved analytically and there are only some special problems that can be solved in this way. When a differential equation/system gives a different response for different initial conditions, it implies that it does not have an analytical solution. This is because even with non-chaotic systems, the behavior can be drastically different depending on the initial conditions, making it difficult to predict the future behavior of the system. In the case of the differential equation x' = x(1-x), the solution for x(0) = 0 is simply x(t) = 0.
  • #1
marellasunny
255
3
I am going to quote an article below and there is a part I would like clarification.I do not understand why chaotic differential systems do not have an analytical solutions. In the simplest form,an equation such as this one [sin(x) + x - 0.5 = 0] does not have an analytical solution i.e it can be only solved numerically.
1.What would be its equivalent for a differential system not having a analytical solution?
2.When the differential equation/system gives a different response for different initial conditions,how does it imply "not having a analytical solution"? I'm a novice.


Not all mechanical problems can be solved analytically as many physicist thought before. They thought that if there were no analytical solution then it was only a matter of intelligence, it would need a more clever mathematical approach to solve these problems.
In fact there are only some special problems that can be solved analytically. This doesn’t mean that Newton was wrong, he predicted chaotic motion.
Henri Poincaré was the first to see this. In a contest, by King Oscar II of Sweden and Norway, a few problems was to be solved, one of them was to prove that the solar system was stable (i.e. analytical solution exist). Poincaré found that there could not be an analytical solution, even the simpler problem of a three-body system. He found that a small difference in the initial conditions could blow up to a totally different answer.
 
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  • #2
What equations do not have analytical solution depends on what functions are available.
usual examples would be
x''(t)+sin(x(t))=0
h x''(t)+t x'(t)-x(t)=0
h x''(t)+x'(t)-t x(t)=0

different response for different initial conditions is a different thing, but they often occur together
If you have the analytical solution it is (hopefully) easier to deal with the different responses
If we have an approximate solution we hope it is not to far from the actual solution
If we have very different solution we cannot distinguish we have a problem

We might want to use the differential equation to make a prediction. If tiny changes in the conditions make a huge difference in the response we cannot predict very well.
 
  • #3
marellasunny said:
I am going to quote an article below and there is a part I would like clarification.I do not understand why chaotic differential systems do not have an analytical solutions. In the simplest form,an equation such as this one [sin(x) + x - 0.5 = 0] does not have an analytical solution i.e it can be only solved numerically.
1.What would be its equivalent for a differential system not having a analytical solution?

Consider the one-dimensional system [tex]
\frac{dx}{dt} = f(x)
[/tex] with continuous [itex]f[/itex]. This has implicit solution [tex]
\int_{x(0)}^{x(t)} \frac{1}{f(u)}\,du = t.
[/tex] There is then a question as to whether the integral can actually be done analytically (which it almost always can't) and whether the resulting equation can be solved analytically for [itex]x(t)[/itex] (which it almost always can't). The only case we can always solve is the case where [itex]x(0)[/itex] is such that [itex]f(x(0)) = 0[/itex], and the solution is then [itex]x(t) = x(0)[/itex].

2.When the differential equation/system gives a different response for different initial conditions,how does it imply "not having a analytical solution"? I'm a novice.

"Sensitive dependence on initial conditions" is a phenomenon which marks a system out as being chaotic.

However even with non-chaotic systems the behaviour of a system may be drastically different depending on the initial conditions: consider [tex]
\frac{dx}{dt} = x(1-x),
[/tex] which falls into the vanishingly small category of systems for which a solution can be found analytically:
[tex]
x(t) = \begin{cases}
\frac{x_0 e^t}{1 + x_0(e^t - 1)} & 0 \leq t < \log \left(1 + \frac{1}{|x_0|}\right), x_0 < 0 \\
0 & 0 \leq t < \infty, x_0 = 0 \\
\frac{x_0 e^t}{1 + x_0(e^t - 1)} & 0 \leq t < \infty, 0 < x_0 < 1 \\
1 & 0 \leq t < \infty, x_0 = 1 \\
\frac{x_0 e^t}{1 + x_0(e^t - 1)} & 0 \leq t < \infty, x_0 > 1
\end{cases}
[/tex]
The salient points are firstly that if [itex]x_0 = x(0) < 0[/itex] then the trajectory diverges to [itex]-\infty[/itex] in finite time, secondly that if [itex]x_0 = 0[/itex] then [itex]x_0 = 0[/itex] for all time, and thirdly that if [itex]x_0 > 0[/itex] then [itex]x_0 \to 1[/itex] as [itex]t \to \infty[/itex]. Thus if all we know is that [itex]-\epsilon < x_0 < \epsilon[/itex] for some [itex]\epsilon > 0[/itex], then we cannot predict the future behaviour of the system.
 
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  • #4
[tex]
x(t) = \begin{cases}
\frac{x_0 e^t}{1 + x_0(e^t - 1)} & 0 \leq t < \log \left(1 + \frac{1}{|x_0|}\right), x_0 < 0 \\
0 & 0 \leq t < \infty, x_0 = 0 \\
\frac{x_0 e^t}{1 + x_0(e^t - 1)} & 0 \leq t < \infty, 0 < x_0 < 1 \\
1 & 0 \leq t < \infty, x_0 = 1 \\
\frac{x_0 e^t}{1 + x_0(e^t - 1)} & 0 \leq t < \infty, x_0 > 1
\end{cases}
[/tex]
The salient points are firstly that if [itex]x_0 = x(0) < 0[/itex] then the trajectory diverges to [itex]-\infty[/itex] in finite time, secondly that if [itex]x_0 = 0[/itex] then [itex]x_0 = 0[/itex] for all time, and thirdly that if [itex]x_0 > 0[/itex] then [itex]x_0 \to 1[/itex] as [itex]t \to \infty[/itex]. Thus if all we know is that [itex]-\epsilon < x_0 < \epsilon[/itex] for some [itex]\epsilon > 0[/itex], then we cannot predict the future behaviour of the system.

The above solution of x(t) does not work for x(0)=0. Is there an inherent assumption that I am missing? I wrote the solution I got for the D.E x'=x(1-x) below. After substituting for x(0)=0,I get a fraction that tends to 1.
[tex]
x(t)=\frac{1}{1+C_1*e^{-t}} \\
x(0)=\frac{1}{1+C_1} \\
C_1=x(0)-1 \\
\Rightarrow x(t)=\frac{1}{1+(x(0)-1)e^{-t}} \\
[/tex]
 
  • #5
marellasunny said:
The above solution of x(t) does not work for x(0)=0.

The solution in the case [itex]x(0) = 0[/itex] is [itex]x(t) = 0[/itex].

Is there an inherent assumption that I am missing? I wrote the solution I got for the D.E x'=x(1-x) below. After substituting for x(0)=0,I get a fraction that tends to 1.
[tex]
x(t)=\frac{1}{1+C_1*e^{-t}} \\
[/tex]

Separation of variables should give you [tex]
\log \left| \frac{x}{1-x}\right| = t + C[/tex] so that [tex]
\left| \frac{x}{1-x}\right| = Ae^{t}[/tex] from which it follows that [tex]
A = \left| \frac{x_0}{1 - x_0}\right|.[/tex] If at this point you set [itex]x_0 = 0[/itex] then you obtain [itex]A = 0[/itex], and hence [tex]
\frac{x}{1-x} = 0.[/tex]
 

1. What is a system of differential equations?

A system of differential equations is a set of equations that describe how a set of variables change over time, based on their rates of change.

2. What is an analytical solution?

An analytical solution is a solution to a problem that can be expressed using known mathematical functions. It can be calculated exactly and is considered the most accurate type of solution.

3. Why might a system of differential equations not have an analytical solution?

This can happen when the equations are too complex or involve variables that cannot be expressed using known mathematical functions.

4. What are some alternative methods for solving systems of differential equations without an analytical solution?

Some alternative methods include using numerical methods such as Euler's method or Runge-Kutta methods, or using computer simulations to approximate the solution.

5. Can a system of differential equations ever have both an analytical and a numerical solution?

Yes, in some cases, a system of differential equations may have an analytical solution for certain initial conditions, but a numerical solution may be needed for other initial conditions or to calculate the solution at different time points.

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