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Questions tagged [nonlinear-system]

In mathematics, a nonlinear system of equations is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.

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System of linearly coupled ODE with quadratic term

I am looking forward to solve the system of equations $$\frac{\partial x_i}{\partial t}=a_ix_i^2+b_ix_i-\sum\limits_{j\neq i}^nb_jx_j-d_i,$$ with $x_i\geq 0$, $a_i,b_i,c_i,d_i>0$ and $i\in[1,n]$. ...
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23 views

Nonlinear Recursion Solution Process for $x_{n+1}=\Sigma_{i=1}^{n} x_{i}x_{n-i}$ (Known Solution)

I want to solve the equation $x_{n+1}=\Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
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Non-linear IVP with $f(u_1,u_2):=\sqrt{1+\sqrt{u_1^2+u_2^2}}\begin{pmatrix}u_1+u_2\\3 u_2-u_1\end{pmatrix}$

Does the following initial value problem \begin{equation*} \begin{cases} u'(t) = f(u(t)), \\ u(0) = u_0 \end{cases} \quad \text{with} \quad f(u_1,u_2):=\sqrt{1+\sqrt{u_1^2+u_2^2}}\begin{pmatrix}u_1+...
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1answer
17 views

Numerically solve Lotka-Volterra equations using Euler-Cromer

I am trying to solve this coupled pair of equations $$ {dx\over dt}=\alpha x - \beta xy,\\ {dy\over dt}=\delta xy - \gamma y $$ using the numerical method Euler-Cromer. It doesn't matter which method,...
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16 views

Initial guessing to bvp4c MATLAB

I am working on a 4th order non-linear variable coefficient homogeneous ODE bvp. I am having issues getting a solution using bvp4c. This could be one of many things. Not having a solution within the ...
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1answer
34 views

Mean value theorem to prove the existance of a solution

Consider the following nonlinear system P: $$ P = \begin{cases} \dot{x} = f(x,u)\\y = g(x)\end{cases} $$ where $f,g$ are of classes $C^1$, $x \in \mathbb{R}^n$ and $u,y \in \mathbb{R}$. I am ...
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1answer
28 views

solution to dynamical system with “xy” terms

$\frac{dx}{d\tau}=xy-x\;(1), \quad \frac{dy}{d\tau}=-xy-\alpha-\beta\;(2)$ I'm preparing for my final in non-linear dynamics and I want to find the fixed points for this system i.e. when both ...
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2answers
37 views

How to calculate the averaged equations for the weakly nonlinear oscillator $\ddot x+x+\varepsilon (x\dot x^2)=0$?

This is Strogatz exercise $7.6.5:$ For the system $\ddot x+x+\varepsilon h(x,\dot x)=0$, where $h(x,\dot x)=x\dot x^2$ with $0 < ε << 1$, calculate the averaged equations and if possible, ...
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38 views

Linear approximation of non-linear equation.

I am trying to understand what it means to make a linear approximation but will get more specific in a minutes. First, some context. The problem I am working on is more of a physics but the question ...
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1answer
24 views

Prove that no periodic orbits exist

Please help! I need to prove for the system below that no periodic orbits exist when $V_0=0$: $\frac{d^2x}{dt^2}+\zeta_1 \frac{dx}{dt}+(k_{11}+k_{12}x^2)x-2\gamma\frac{dy}{dt}=V_0 cos(\omega t),$ $\...
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36 views

How to minimize $\sum_i |a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|x_2|^2 - b_i|^2$ over $x_1, x_2 \in \mathbb C$?

Consider the following nonlinear minimization problem \begin{align} \tag{P1} \min_{x_1, x_2 \in \mathbb C} \sum_{i=1}^m \big|a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|...
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1answer
24 views

Exercise: Compute a distribution given the integral manifold

I am new in Stack Mathematics. I need your help in solving the follow exercise. "Compute a distribution $\Delta$ over $\mathbb{R}^3$ whose integral manifold is the surface of a sphere (i.e. the set ...
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1answer
25 views

Sufficient and necessary conditions for $y_1 = |x_1|^2$, $y_2 = x_1 \overline{x_2}$, $y_3 = \overline{x_1} x_2$, $y_4 = |x_2|^2$ in terms of $y$'s

Let $\mathbb C$ and $\mathbb R$ denote the fields of complex and real numbers, respectively. Suppose $x_1, x_2 \in \mathbb C$, and \begin{align} y_1 & = |x_1|^2 \tag 1 \\ y_2 & = x_1 \overline{...
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Using Newton Raphson's method to solve a non-linear boundary value problem?

(My specific question is at the end of the problem) Examine the boundary problem with a nonlinear right hand side $1+u^2(x)$ $$-u''(x) = 1+u^2(x) \quad \text{on} \quad 0 < x < 1 \quad \text{...
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1answer
25 views

Discretization of second order nonlinear ODE using finite difference approximation not correct

I have the differential equation $$y'' + x(y^2)' - 2y^2 = g(x) \Longleftrightarrow y'' + x2yy'-2y^2 = g(x).$$ Using finite-difference approximations $$y''(x_m) \approx \frac{Y_{m-1} - 2Y_m + Y_{m+1}}{...
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2answers
19 views

Problem with system of logarythmic equation.

So I have this problem with a system of logarytmic equations. Specifically how to get rid of the (ln(x))^2 in order to solve this one. I know you can get This as a solution for the first equation. My ...
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28 views

Singular points in nonlinear systems

Suppose, for example, that I have the following non-linear system of ODE's; $$(y^2-y'^2)g+y' y g=P$$ $$y'(y' g''+(y''+y)g')=S,$$ where $y,g,P,S$ are all functions of the dependent variable $x$. How ...
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1answer
54 views

How do I know if a second order nonlinear differential equation is analytically solvable?

How can I test if a non linear second order ODE is analytically solvable? for example: $$y''+\frac{3}{x}y'=ay^3+by$$ where $a,b$ are constants, I've been trying to solve such an equation for a ...
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26 views

Solve sine and exponential nonlinear differential equation?

Is it possible to solve this kind of differential equation with forward Euler? $$\ddot y^2 + sin(\ddot y ) + \dot y + y = u$$ I haven't even write this ODE on the first order form. If I would do ...
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1answer
10 views

Invariance of supercritical bifurcations under change of variables?

I am confused about one statement in Strogatz's textbook, Nonlinear Dynamics and Chaos: This equation ($\dot{x} = rx - x^3$) is invariant under the change of variables $x\to -x$. (56) How is this ...
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29 views

Geometric significance of a bifurcation point with algebraic multiplicity $2$?

This is part of Strogatz exercise $3.2.3:$ This is the process by which I found the bifurcation point/points for $\dot x=x-rx(1-x)$: By the method of tangential intersection we have: $$x=rx(1-x)$$ $...
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1answer
26 views

Nondimensionalization of the logistic equation.

In the process of studying nonlinear dynamics by Strogatz, I saw how he did simplify the model for an insect outbreak with the use of nondimensionalization. So as an exercise I picked an equation and ...
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27 views

Laplace transform - differential equation

I have the following differential equation: $$\ddotθ + \frac{b}{h}\dotθ + \frac{a}{h}\cos θ = \frac{1}{h}u$$ and I want to extract the transfer function (the equation actually describes a control ...
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1answer
426 views

How to solve $\dot{x}=f(x)/||f(x)||$?

How to solve the following ODE? \begin{equation} \frac{d}{dt}x=\frac{f(x)}{\|f(x)\|}, \end{equation} where $x: \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f: ...
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14 views

3D trilateration with 3 beacons of two different points with the same height

I know that 3 beacons are enough to find a point's position in a 2D region. By beacon, I mean a device that gives the distance from itself to the point (by calculating the intersections of all the ...
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1answer
65 views

Solving the nonlinear differential equation $ m \ddot x +\alpha x + \beta x^3 = 0$

As the header says: I want to solve the differential equation $ m \ddot x +\alpha x + \beta x^3 = 0$, with initial conditions $x(0) = -x_0$, $\dot x(0)=0$. It comes up in the solution to the equations ...
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2answers
52 views

Logistic map (discrete dynamical system) vs logistic differential equation

I have to roughly illustrate the logistic discrete dynamical system (as a model for population growth) to some non mathematics students. I'm not an analyst or an expert of dynamical systems. Looking ...
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1answer
31 views

Feasible way to find interpolating complex polynomial based on absolute value

Consider a complex degree-$(n-1)$ polynomial $p(z) = \sum\limits_{i=0}^{n-1} a_i z^i$. Given a number $0 < m < 2n$ of positions in the complex plane with absolute value requirements, i.e. $|p(...
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55 views

Lyapunov Stability for a Nonlinear, Time-varying system

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems. Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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1answer
39 views

Nonlinear equation analysis withe epsilon value [closed]

Consider the nonlinear equation $$\frac{d^2x}{dt^2}+\epsilon\sin(x)=0,~~\epsilon \ll 1\\ x(0)=0,~~\dot{x}(0)=1$$ and find... A. The value of $x_0$ as $\epsilon$ goes to $0$ B. The first order term $...
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21 views

Non-existence of a generic solution to system of nonlinear equations

I have the following system of nonlinear equations: $f_1(x_1,..,x_m,y) =0$ $...$ $f_n(x_1,..,x_m,y) =0$ where $f_i(\cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial),...
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2answers
18 views

Convergence of the Newton-Raphson method applied to a nonlinear system

I'm working on a nonlinear system of 2 equations and try to solve it with the Newton-Raphson method. I have a function $f(x,y)$. In order to use the method, I have to find a starting vector $(x,y)$. ...
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2answers
60 views

Mixed mathematical chemical problem

How to calculate the concentration of all species present in a solution with $\mathrm{0.3~M}$ $\mathrm{NaH_2PO_4}$? The whole truth is that any time that you add any phosphate ion into an aqueous ...
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0answers
42 views

On the existence and uniqueness of solutions of non-linear differential equation system

Are there theorems that could be applied to show the existence/uniqueness of solution of such system : $$ (S) \left \{ \begin{aligned} p''&= f_1(b) \\ q'' &= f_2(x,p,q,b)\\ q~~ &= f_3(...
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0answers
36 views

Solving non-linear system of equations $(u^2 + xv + e^y = 0, 2u+x^2 - xy = 5).$ with $(-1,0,2,5)$

In a book, this non-linear system of equations is given: $$ \begin{split} u^2 + xv + e^y &= 0, \\ 2u + x^2 - xy &= 5. \end{split} $$ It says that this system of equations can be solved by ...
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1answer
15 views

Understanding Rate/Speed of Convergence of a sequence.

Consider the following text taken from this link. What does it say? As far as I can get: If $p_n$ (i.e. $p_0, p_1, p_2, ...$) is a sequence, $p_n$'s point of convergence is $p$, $\lambda$ and ...
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34 views

Does a differential equation with no solution mean the function is just zero?

Does a differential equation with no solution mean the function is just zero? I am trying to solve a non-linear ODE and the solution will then be plugged into another equation, and having the ...
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1answer
42 views

Solve this system of trigonometric equations [closed]

How to solve this system of equations analytically? $$ \begin{cases} \:9\tan\alpha -\frac{4.9\cdot 9^2}{v^2\cos^2\alpha }=2.1\\ \:23\tan\alpha \:-\frac{4.9\cdot \:23^2}{v^2\cos^2\alpha \:}=2.44 \end{...
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0answers
17 views

Solving a system of ternary quadratic equation

I'm wandering if there are any appropriate ways to solve this system of ternary quadratic equations for $x,y,x$. \begin{equation} \left\{ \begin{aligned} x^2+xy+y^2 & = s(x+y) \\ y^2+yz+z^2 & ...
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0answers
59 views

Neural network as a nonlinear system?

I defined a very simple neural network of $2$ inputs, $1$ hidden layer with $2$ nodes, and one output node. For each input pattern $x⃗ ∈ ℝ×ℝ$ and associated output $o∈ℝ$, the resulting nonlinear ...
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3answers
36 views

Solve a system of trigonometric equations

How can I solve this system of trigonometric equations analytically? It is from physics class. $$ \begin{cases} 30t\cos{\alpha}=50\\ -30t\sin{\alpha}-4.9t^2=0 \end{cases} $$
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2answers
57 views

Non-linear partial differential equation with conditions

I have a pde with conditions, for which I'm looking for an analytical solution : $\partial_t f(t,x)+f(t,x)\partial_x f(t,x)=0$. $f(0,x)=0 \, , \, f(t,0)=0 $. $f(t,x)$ is defined over : $\mathbb{R}^+...
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2answers
91 views

Simple Neural Network: non linear system of equations? [closed]

I defined a very simple neural network of 2 inputs, 1 hidden layer with 2 nodes, and one output node. For each input pattern $\vec{x} \in \mathbb{R} \times \mathbb{R} $ and associated output $o \in \...
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1answer
29 views

finite element non linear boundary value problem

I have the following small finite element non linear boundary value problem: $$ -u''(x) = 1 + u^2(x) \quad \text{for} \quad 0<x<1 \quad \text{with} \quad u(0) = u(1) = 0 $$ with a grid with ...
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0answers
21 views

Solving nonlinear equation resulting from finite element method

Using the finite element method (for a uniform mesh in the spatial domain) I have the system with initial conditions $u_j(0)=\cos(x_j)$ for $j=1,\dots, N$ $$\frac{d\vec{u}}{dt}=A\vec{u}+B\vec{c},$$ ...
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0answers
23 views

Does Picard iteration affect the convergence order of a numerical scheme?

I have a common nonlinear differential equation, for example, the one in Stokes problem: Find $u$ (velocity) and $p$ (pressure) such that $$\nabla\cdot(-\mu(u)\nabla u+p\,I)=f\qquad\textrm{ in }\...
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0answers
18 views

Solving a non-linear system where x appears as matrix and vector

I have the following problem. I am trying to solve the following non-linear system of equations in MATLAB $\mathbf{b}=\mathrm{diag}\left(\mathbf{x}\right)\left(\mathbf{I}+\mathbf{A'A}\right)^{-1}\...
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1answer
48 views

How do I classify the equilibrium points of this dynamical system?

Consider the dynamical system $$\dot{r}=r(3-2r-s)$$ $$\dot{s}=s(2-r-s)$$ Find and classify the equilibrium points of the system. I have found the equilibrium points to be $(0,0), (0,2), (\frac{3}{2}...
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0answers
32 views

Solving nonlinear system of equations for variables

I would like to solve the following system of equations for $α_1$ and $α_2$: $$ \begin{bmatrix} \frac{\alpha_1 sin(\alpha_1 +\alpha_2)-sin(\alpha_1) \alpha_1 +sin(\alpha_1) \alpha_2}{\alpha_1 \...
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0answers
27 views

Design a parameter $k$ for a second order ODE $\ddot{x}(t) = f(t,k)$ subjected to a set of initial and terminal constraints.

I have a nonlinear differential equation as follows \begin{align} \ddot{x}(t) = f(\theta(t),k), \end{align} where $f$ is a function of the time variable $t \in \mathbb{R}^{+}$, $k \in \mathbb{R}$ is ...