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1
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1answer
35 views

Prove there are exactly three solution

In this question, the OP asked to find the solutions of: $$a^{−x}+{\log x \over \log a}=0$$ In my answer, I showed that when $a<e^e$ there can be at most one solution. It is also clear that any ...
0
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1answer
42 views

Nonlinear system of equations / factoring two-variable cubic over $\mathbb{R}$

About halfway through a homework problem, I end up with a three-way identity: $$\frac{uw}{v+w} = \frac{uv}{u+w} = \frac{vw}{u+v}$$ (I say I end up with..., but this is the method suggested by my ...
2
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0answers
32 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
1
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1answer
18 views

Approximation of non-linearities

I have the following differential equation $$\ddot q=M^{-1}(q) (C(q,\dot q)\dot q+G(q))+M^{-1}(q)u+M^{-1}(q)D(t) $$ The author of the textbook refers to the terms $M^{-1}(q) (C(q,\dot q)\dot q+G(q))$ ...
4
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0answers
65 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. I am especially interested in the case $$S(x)=\tfrac{1}{2}x^2+\tfrac{1}{2}x.$$ I have no ...
1
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2answers
74 views

Lyapunov function for non-autonomous non-linear differential equations

I have read some lecture notes about Lyapunov’s Second Method for autonomous system. Now, I want to deal with the stability of a non-autonomous system. Suppose there is a non-autonomous non-linear ...
2
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0answers
41 views

perturbation theory solution of forced Duffing's equation

Question: Find the leading order of the asymptotic expansion for large t: $\frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=Fcos(\frac{1}{3}\big(1+\varepsilon\omega)t\big)$ I have ...
3
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0answers
47 views

Problems in Nonlinear Matrix Equation [Question1]

Let $\theta, \beta$ be 3 × 3 skew-hermitian matrices, and $\sigma$ be a 3 × 3 matrix. Find hermitian matrices S and T such that $$(S − \theta) · (T − \beta) = \sigma$$ The answer must provide a clear ...
3
votes
1answer
77 views

How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
0
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0answers
12 views

Describing function of a non linearity with memory

Can anyone help me on finding the correct methodology to compute the describing function of the following NL function? ...
0
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0answers
39 views

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
1
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1answer
81 views

linearize a nonlinear ode

Could anyone suggest me how to linearize the following system of nonlinear odes (special attention to (2) \begin{align} -cU'&=-U''+UV\tag{1}\\ -cV'&=-k(k+1)V^{k-1}(V')^2+(k+1)V^k ...
0
votes
0answers
26 views

Parameter estimation of Lorenz system (nonlinear dynamical system)

My problem is as follows. I have to estimate parameters of Lorenz system using given data. Lorenz system is described by following system of ODEs: $$ \frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = ...
2
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0answers
96 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
0
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0answers
21 views

How to derive a Gaussian mean's functional dependence on a variable

I have three variables: $E, I,$ and $X$. $I$ is a function of $E$: $I(E)=gauss(\mu(X), \sigma=1)$ I sampled this distribution with a very simple $\mu(x): 3\times x$. I want to fit that data to ...
2
votes
1answer
51 views

Nonlinear first order ODE with quadratic in the derivative

This equation shouldn't be so hard, and yet I'm stymied. $$ \left( \frac{dw}{dz} \right )^2 + \alpha \frac{dw}{dz} + w \beta = 0 $$ with $w(0) = w_0>0$ $w(L) = 0$ for some known L and ...
2
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0answers
39 views

nonlinear system exercise and eigenvalues

I've been struggling with this exercise from Strogatz', I would appreciate any correction of what I've done so far since I've been self studying all of these topics and some (if not all) of it might ...
0
votes
1answer
97 views

Newton-Raphson Method for Non-linear System of 3 variables in Matlab

I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. Here are the 3 non-linear equations: \begin{equation} c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0 ...
0
votes
0answers
17 views

Transform and numerically solve an ODE with heaviside of form $F'(z) = g(F(z)) + d + c \mathbf{H}(\bar{z} - z)$

I have an ODE in $F(z)$ (really a system of equations, but assume the vastly simplified form here) $$ F'(z) = g(F(z)) + d + c \mathbf{H}(\bar{z} - z) $$ Where $g(\cdot)$ is some non-linear operator ...
4
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2answers
68 views

Solution of system of non-linear equations

Is there a general condition for the existence and uniqueness of solution of a system of simultaneous non-linear equations similar to the determinant test for a system of linear equations. What are ...
0
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0answers
8 views

Critical damping for system with nonlinear damping and stiffness term

I have two questions about an unforced, critically damped nonlinear spring which has a damping coefficient $\beta $ scaling $\dot x^3$ and a Duffing stiffness coefficient $\kappa $ scaling $x^3$. The ...
0
votes
1answer
76 views

how to find point where two exponential type functions intersect

I have two functions who intersect each other and i want to find time at which they intersect. The two functions are, $\left(1-\frac{1}{\text{X2}}\right)-\frac{(\text{X1}-1) (\text{X2}-1)}{e^{4 t ...
0
votes
2answers
29 views

Solving for points in a plane based on line lengths and geometry

I have the following points and lines in a plane: The problem is this: Given that we know the lengths of lines A, B and C, how can we calculate the coordinates of each point a, b and c? The ...
2
votes
2answers
39 views

where did I go wrong in solving this sytem of nonlinear first-order ODEs?

To communicate my experience level and intent: I'm an undergraduate, this is not homework, I'm trying to write a physical simulation for fun and xp and am stuck just before (what looks to me like) the ...
1
vote
1answer
63 views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
5
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0answers
171 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
3
votes
1answer
27 views

Does Hyperbolic + Not Asymptotically Linearly Stable imply Not Asymptotically Stable?

Topic: Stability of Autonomous Non-linear ODEs I'm wondering whether having a hyperbolic critical point that's not asymptotically linearly stable (ALS) in the linearisation of a system implies that ...
0
votes
1answer
28 views

Finding solution to a unidirectional nonlinear wave equation

I can do the parts a), b) and c) and find that in part c) that the condition in which the solution will break down is when $1+tf'(x-tu)=0$ However I am unable to part d) I tried ...
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0answers
25 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
0
votes
1answer
37 views

Are there standard approaches, to solving a system of nonlinear PDE?

If we have a system of PDE's, where each PDE is different i.e. for $u:U\subset \Bbb R^2\to\Bbb R^3$, $u(x,y)=(a(x,y),b(x,y),c(x,y))$, which needs to satisfy $ \left\{ \begin{array}{ll} ...
0
votes
0answers
42 views

Transformation of quadratic formula with fractions

I'm trying to implement a nonlinear least squares based trilateration algorithm into a code project. The algorithm I'm using I found in a paper by Yu Zhou (An Efficient Least-Squares Trilateration ...
1
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0answers
19 views

How to measure the nonlinearity of a real-valued multivariate function?

In general, how to measure the nonlinearity of a real-valued multivariate function? The particular case I'm interested in is multivariate sigmoid function.
4
votes
2answers
54 views

Solve numerical system of nonlinear equations?

I need to solve a nonlinear system of equations that looks like this ...
0
votes
1answer
46 views

Optimization problem with a minimization sub-problem as a constraint

I have a problem, for predefined $x_0,z\in\mathbb{R}$, which looks like $$\min_{\alpha,x} \sum_{i=1}^n \alpha_i f_i(x_i,z) $$ subject to \begin{align} \sum_{i=1}^n \alpha_i &= 1 \\ \sum_{i=1}^n ...
1
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0answers
31 views

Hopf bifurcation phase portrait orientation

Say for a system $$\dot{x}=y$$$$\dot{y}=-x+\mu y -y^3$$ I have confirmed a hopf bifurcation occurs at the origin and that the branches are stable i.e., a stable limit cycle and the origin being stable ...
1
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1answer
87 views

Nonlinear first order system of ODEs

While solving some physical problem, I have obtained the following system of differential equations with boundary conditions: $$\left\{\begin{matrix} \frac{d\phi_1}{dz}=\frac{m^2}{\lambda}- ...
2
votes
1answer
37 views

$4$ variable system of equations

Find all the solutions to this system of equations: $$\begin{cases} -2d^3+3a^2d+d+2c^3-3a^2c-c=0\\ 2d^3-3db^2-d-2c^3+3b^2c+c=0\\ -6c^2b+b^3+b+6c^2a-a^3-a=0\\ 6d^2b-b^3-b-6d^2a+a^3+a=0 \end{cases}$$ ...
4
votes
4answers
93 views

Solve system of nonlinear equations using non-numerical method

Is there any non-numerical method to solve this kind of system of nonlinear equations for $c_1, c_2, x_1, x_2$: $$c_1+c_2 = 1$$ $$c_1x_1+c_2x_2 = 1$$ $$c_1x_1^2+c_2x_2^2 = 2$$ $$c_1x_1^3+c_2x_2^3 = ...
1
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1answer
45 views

solving a non-linear (trigonometric) system of equations with two equations and two variables

I'm trying to solve the following system of equations: $$l_1*sin(\alpha)=l_2*cos(\gamma)+l_3*sin(\beta)$$ $$l_2*sin(\gamma)+l_1*cos(\alpha)=l_3*cos(\beta)+l_4$$ with the unknowns $\beta$, $\gamma$ ...
0
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0answers
57 views

How to represent Non-linear equation in State Space form? (To solve in MATLAB)

I have a set of differential equations. I am using state space representation to convert it 1st order form and then am solving via RK method using ode45 function. I know how to do this when the ...
1
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1answer
39 views

How to solve system of nonlinear differeintial equations

System follows: $$ y'=\frac{y^2}{z-x}; z'=y+1$$ I was found the 2 ways. The both are wrong 1) $$z = x + \frac{y^2}{y'}; z'=1+\frac{2yy'^2-y^2y''}{y'^2}=y+1; => (p(y) = y')=> yp(yp'+p)=0;$$ ...
1
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2answers
38 views

Finding solution of first order non linear equation

I have $$m\frac{dv}{dt}=mg-kv^2$$ and I want to find v(t). I tried to separate the derivative over both sides but I am getting no where. At the moment I have $$v+\frac{v^2}{gt}=\frac{kt}{m}$$ Can ...
1
vote
1answer
43 views

Jacobian linearization, does it need to be around a hyperbolic fixed-point?

Everything that I read about Jacobian linearization of systems of nonlinear equations is about approximations near hyperbolic fixed-points (cf. the Hartman-Grobman theorem). It seems to me that even ...
4
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0answers
66 views

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What's an approach to solve this kind of equation?
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0answers
18 views

Parameter Iteration Method for Nonlinear ODE

I recently came across the parameter iteration method to solve the Blasius equation. It is very simple and gives surprisingly accurate results. This paper discusses applying the method to the Blasius ...
0
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2answers
52 views

Is the reference point (x, y) above or below the non-linear equation?

BACKGROUND In short, I have a series of 3 to 10 data points that will be used to represent a curve. For example: $X=0, Y=10$ $X=4, Y=7$ $X=9, Y=12$ $X=16, Y=10$ What I am trying to do is ...
1
vote
1answer
31 views

How to go about this non-linear eigenvector problem

I have these non-linear coupled differential equations: $$ \frac{\mathrm{d}A_1}{\mathrm{d}z} = C_1 A_1^* A_2 \\ \frac{\mathrm{d}A_2}{\mathrm{d}z} = C_2 A_1^2 $$ Where $C_1$ and $C_2$ are constants and ...
0
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0answers
36 views

Linearization of multiple normal functions

I have noticed that it takes a very long time to perform non-linear least squares fitting on datasets similar to this: where there are multiple Gaussian distributions to be fit to experimental ...
1
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1answer
22 views

If at a specific value of x in a non-linear ODE a term is cancelled so that it looks just like the linear ODE, should the result be the same?

If at a specific value of x in a non-linear ODE a term is cancelled so that it looks just like the linear ODE that I am comparing to, should the result be the same (graphic at bottom)? I am ...