All Questions
Tagged with nonlinear-dynamics perturbation-theory
10
questions
5
votes
1
answer
120
views
Why is this approximate solution correct?
Consider the following differential equation
$$ y''=-y + \alpha y |y|^2, $$
where $y=y(x)$ is complex in general and $\alpha$ is a real constant such that the second term is small compared to $y$ ($||^...
- 331
1
vote
0
answers
47
views
Duffing equation with non-linearity factor greater than unity
I have been trying to solve the following non-linear equation taking help from the book regarding perturbative technique by H. Nayfeh (chapter -4)
$$\ddot{x}+\frac{x}{(1-x^2)^2}=0$$ where $x<1$ ...
- 344
1
vote
0
answers
79
views
Find the general solution of an ODE with a nonlinear perturbative term
Let's say I start with the linear differential equation
$$ y''=-y, $$
which has (for example) the two solutions $y=e^{\pm i x}$, therefore following the superposition principle the general solution is ...
- 331
0
votes
1
answer
111
views
Deriving a Hopf Bifurcation – Perturbation Method vs. Jacobian Matrix
When deriving a Hopf bifurcation of a dynamical system, the usual process is:
Find a fixed point $(x_0, y_0)$
Perturb the system about the fixed point $(x_0+\tilde{x}, y_0+\tilde{y})$
Linearize, ...
0
votes
0
answers
96
views
Approximating the solution to a system of 3 nonlinear ODEs with the KBM method?
Background
I have the following system of ODEs:
$\dfrac{\mathrm{d}x}{\mathrm{d}t}=x\dfrac{q}{Q}-x\dfrac{x+y}{M}\quad$ (Eq. 1)
$\dfrac{\mathrm{d}q}{\mathrm{d}t}=y(1-\dfrac{q}{Q})(1-c)(1-v)-aq-y\dfrac{q}...
- 261
0
votes
2
answers
123
views
Approximating the solution to a system of two ODEs with the KBM method?
Background
I have the following system of ODEs:
$$ \begin{aligned} \dot x (t) &= \alpha - \beta x(t) y(t) \\ \dot y (t) &= \delta x(t) y(t) - \gamma y(t) \end{aligned} $$
where all parameters ...
- 261
1
vote
0
answers
55
views
Applying multi-scale analysis directly on the exact solution
Usually, multiple scales analysis (e.g. Poincaré-Lindstedt method or other multi-scale expansions) is applied on an ODE. Suppose we start from the exact solution of the ODE, how do we obtain the ...
- 121
1
vote
1
answer
277
views
Expanding a PDE in powers of a small parameter?
I'm working on an assignment for my quantum mechanics class and I've arrived at a nonlinear inhomogeneous partial differential equation for a complex function $S:\mathbb{R}^2\to\mathbb{C}~;~S:(x,t)\...
- 12.5k
0
votes
1
answer
63
views
Distance of perturbed flow from the unperturbed stable manifold
Consider the following system
$ \dot x= v\\
\dot v= x - x^2(1+\varepsilon cost)$
Let $\phi_\varepsilon(x,v)=\psi_\varepsilon ^{2\pi}(x,v)$ the flow of system with initial condition $(x,v)$ at time $...
-1
votes
1
answer
139
views
The stability of a fixed point, given that the one of the eigenvalues of the linearised system is zero and the other it negative?
I have the following dynamical system
$$\frac{d x}{d \tau}=\gamma x(1-x)-\alpha x y$$
$$\frac{d y}{d \tau}=y\left(1-\frac{y}{x}\right),$$
where $\gamma$ and $\alpha$ are constant parameters. I am ...
- 7,337