All Questions
Tagged with perturbation-theory nonlinear-system
29
questions
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Degenerate perturbation theory to nonlinear equation
I want to use perturbation theory to find the solution to the following nonlinear equation:
$$x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0,$$where $i=1\cdots N$ and $...
- 101
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1
answer
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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
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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 $x<1$ ...
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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 ...
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123
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Perturbation theory with coupled nonlinear differential equations
I’ve got a problem with a set of differential equations for which I’m trying to find fixed points (or rather restrictions for the parameters).
The equations have the form
(1) $\frac{d}{dt} R(t) = -B ...
- 1
5
votes
2
answers
244
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Solution to a nonlinear ODE
$$ a_1 y'''''+ a_2 y'''+\left(a_3 + y^2 \right) y' = 0 $$
where $a_1, a_2, a_3$ are constants with $a_1>0$ and $a_2,a_3 \in \mathbb{R}$.
Is there a general solution $y(x)$ to the above differential ...
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29
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Struggle understanding stability
I am a bit confused about the connection between linear stability analysis, bifurcation points and amplitude expansions. I have a non-linear system, given as, for $1\leq i \leq n$,
$$
\frac{d}{dt}\...
- 3,435
5
votes
2
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463
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Making the "Two-Timing" method rigorous (Perturbation theory)
I'm reading about the method of two-timing in section 7.6 of Nonlinear Dynamics and Chaos by Strogatz, and I have some questions about how to make this concept rigorous. In this section the book ...
- 1,054
1
vote
1
answer
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Melnikov's method, homoclinic orbits, and bifurcation values
In nonlinear dynamics, Melnikov's approach provides an intriguing way to detect homoclinic bifurcations and bifurcation values, i.e., the values of the parameter at which a dynamical system exhibits ...
user743313
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Perturbational approach to study a Non linear PDE
Lets suppose I have $\mathcal{\hat{H}}\psi=(\mathcal{\hat{H}}_0+\lambda \hat{V})\psi=0$ where $\hat{V}$ is some -non linear - perturbation to $\mathcal{\hat{H}}_0$ controlled by the parameter $\lambda$...
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The stability of a fixed point, given that the one of the eigenvalues of the linearised system is zero and the other it negative?
I have the following dynamical system
$$\frac{d x}{d \tau}=\gamma x(1-x)-\alpha x y$$
$$\frac{d y}{d \tau}=y\left(1-\frac{y}{x}\right),$$
where $\gamma$ and $\alpha$ are constant parameters. I am ...
- 7,337
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1
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62
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How to linearise $\dot \epsilon_y = (a+\epsilon_y)(1- \frac{a+\epsilon_y}{a+\epsilon_x})$?
I have the following dynamical system
$$\frac{d x}{d \tau}=\gamma x(1-x)-\alpha x y$$
$$\frac{d y}{d \tau}=y\left(1-\frac{y}{x}\right),$$
where $\gamma$ and $\alpha$ are constant parameters. I am ...
- 7,337
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42
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Solve the Following Non-Linear Differential Equation
The equation to be solved is
$r^2 u''(r)-A(r^2\ u(r)u'(r)+2\ r\ u(r)^2)+2\ r\ u'(r)-2\ u(r)=0$
with boundary conditions $u(0)=u_0$ and $u(\infty)=0$. Additionally it is assumed that $0<A<<...
- 11
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votes
1
answer
960
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Method of multiple scales on Mathieu's equation
I encountered a problem in Strogatz's "Nonlinear Dynamics and Chaos." Specifically 7.6.18. He takes the following equation : $$\ddot{x}+(a+\epsilon \cos t)x=0$$ where a is close to 1. There he asks us ...
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315
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Singular Perturbation in ODE (from the book Singular Perturbation Methods in Control Analysis and Design)
\begin{align}
\dot x&=\frac{x^2t}{z},&x(t_0)&=x^0=1,~~t_0=0,\\
ε\dot z&=-(z+xt)(z-2)(z-4),&z(t_0)&=z^0,
\end{align}
I am studying the book "Singular Perturbation Methods in ...
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