Nonlinear system of differential equations

In summary, the PhD student is trying to calculate the variance of x as a result of the stochastic variation of z, but is having difficulty doing so because of the complexity of the solution. He has found that the system can be solved analytically, but that the result requires a parametric form to be expressed.
  • #1
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Hi! I'm working with my PhD thesis at the moment, and I've stumbled upon a pretty involved problem. What I have is a system of equations like this:

[itex]\frac{dx}{dt} = A \cos(z)[/itex]
[itex]\frac{dy}{dt} = B x \frac{dx}{dt}[/itex]
[itex]\frac{dz}{dt} = y[/itex]

where [itex]A[/itex] and [itex]B[/itex] are constants. I also have a stochastic term to [itex]z[/itex] according to:

[itex]\delta z(t) = \lim_{N \rightarrow \infty}\pi\sqrt{\frac{t}{N\tau}}\sum_{i = 1}^{N}\zeta_i[/itex]

where [itex]\zeta_i[/itex] are random numbers of unit variance (normal distributed probability), and [itex]\tau[/itex] is the time scale for the decorrelation of [itex]z[/itex]. I wish to calculate the variance of [itex]x[/itex] as a result of the stochastic variation of [itex]z[/itex], i.e.,

[itex]\langle(\Delta x - \langle\Delta x\rangle)^2\rangle[/itex]

where [itex]\Delta x = x(\tau) - x(0)[/itex] and [itex]\langle ... \rangle[/itex] is the average of the expression within the brackets with respect to a variation of the values of [itex]\zeta_i[/itex], weighted according to their probability. I've already calculated the variance of [itex]x[/itex] for [itex]B = 0[/itex] for which [itex]z = y t + z_0[/itex] and [itex]dx/dt[/itex] can simply be integrated in time to obtain an analytical expression for [itex]x(t)[/itex]. How can I continue to get a more general solution to the problem? Can I e.g. use some perturbation theory for small values of [itex]B[/itex] to begin with?
 
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  • #2
If you want a numerical solution, then I would go for Newton's method, send me a message if you want further help.
 
  • #3
Perhaps the following will be a bit helpful?

[tex]\dot{y}/\dot{x} = \frac{dy}{dx} = Bx.[/tex]

You can then solve for [itex]y(x)[/itex]. Similarly,

[tex]\dot{z} = \frac{dz}{dt} \frac{1}{\dot{x}} \Rightarrow \frac{dz}{dt} = \dot{x} y(x) = A \cos (z) y(x).[/tex]

This is a separable equation that you can use to solve for z(x). Once you solve for z(x), you can plug that into your equation for x to get

[tex]\frac{dx}{dt} = A\cos z(x),[/tex]

which is again separable (but you may not be able to express the integral in terms of elementary functions. I haven't tried but I'm guessing the integral to do won't be nice).

I'm not entirely sure how to add in the stochastic term, but hopefully this can get you started with something.
 
  • #4
The system of 3 EDOs can be analytically solved, but the result has to be expressed on a parametric form, because the last integral cannot be expessed in terms of a finite number of elementary functions. Even in the simplest cases of constants C1 and C2 (nul for example), the integral involves some elliptic functions on a very complicated form.
 

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  • #5
Oh, this is really great! Thanks everyone for your help :D
 

1. What is a nonlinear system of differential equations?

A nonlinear system of differential equations is a set of equations that describe the relationship between multiple variables, where the rate of change of each variable is dependent on the values of the other variables. Unlike linear systems, the equations in a nonlinear system may involve powers, products, or other nonlinear functions of the variables.

2. How is a nonlinear system of differential equations different from a linear system?

The main difference between a nonlinear and linear system of differential equations is that in a linear system, the equations are only allowed to involve the variables and their first derivatives, while in a nonlinear system, higher order derivatives and nonlinear functions of the variables may also be present. This makes nonlinear systems more complex and difficult to solve analytically.

3. What are some real-world applications of nonlinear systems of differential equations?

Nonlinear systems of differential equations are commonly used to model complex physical and biological phenomena, such as population growth, chemical reactions, and electrical circuits. They are also used in fields such as engineering, economics, and epidemiology to study and predict the behavior of systems with multiple interacting variables.

4. How do you solve a nonlinear system of differential equations?

Unlike linear systems, there is no general method for solving nonlinear systems of differential equations. Often, numerical methods such as Euler's method or Runge-Kutta methods are used to approximate solutions. In some cases, analytical solutions can be found for specific types of nonlinear systems using techniques such as separation of variables or substitution.

5. What are some challenges in studying and solving nonlinear systems of differential equations?

Nonlinear systems of differential equations can exhibit complex and unpredictable behavior, making it difficult to accurately model and solve them. In addition, analytical solutions are often not possible, and numerical methods can be computationally intensive. Furthermore, the behavior of a nonlinear system can change drastically with small changes in the initial conditions or parameters, making it challenging to make accurate predictions.

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