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 ...

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48 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+B \sin(2Cy)$

Analytical solution for a non-linear differential equation: $\frac{d^2y}{dt^2} = A \left(\frac{dy}{dt}\right)+ B \sin(2Cy)$ A,B are non-zero constants and y (position) is a scalar-value parameter ...
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1answer
41 views

Solving a system of non linear equations

I have got a system of non-linear equations of the form $$A x_1^B \exp \bigg(\frac{- C}{x_1} \bigg) = k_1$$ $$A x_2^B \exp \bigg(\frac{- C}{x_2} \bigg) = k_2$$ $$A x_3^B \exp \bigg(\frac{- C}{x_3} \...
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1answer
13 views

Separate translation from plane rotation

Given $$\cos(\omega)a_x-\sin(\omega)a_y+b_x=c_x$$ and $$\sin(\omega)a_x+\cos(\omega)a_y+b_y=c_y$$ we have that $$a_x^2 + a_y^2 - b_x^2 + 2b_xc_x - b_y^2 + 2b_yc_y - c_x^2 - c_y^2=0$$ Notice that $\...
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1answer
66 views

Analytical Solution to Coupled Nonlinear ODEs

I am looking to solve several coupled nonlinear ODEs like this one: $\hspace{20mm} \frac{d x(t)}{dt} = C_1 \cdot x(t) + C_2 \cdot y(t) + C_3\cdot (x(t)^2 + y(t)^2) x(t),$ $\hspace{20mm} \frac{d y(t)...
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57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
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1answer
34 views

System of two Nonlinear equations

I have a probably very simple problem here. A system of nonlinear equations. $$\left\{ \begin{align} & {{x}^{2}}+{{y}^{2}}=26 \\ & x+{{y}^{2}}=6 \\ \end{align} \right.$$ I started with ...
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48 views

classification of equilibrium points of 3d systems of ode's

I'm trying to find information about the classification of equilibrium points of 3d systems of differential equations, The qualitative analysis. I wonder if someone could refer me to some book or ...
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1answer
24 views

How to solve a nonlinear second order differential equation?

I have been trying to find ways to solve: $$J\frac{d²\theta(t)}{dt²}-K_m cos(\theta(t))=-\tau_f$$ With the initial conditions $$\theta(t=0)=0$$ $$\frac{d\theta}{dt}(t=0)=0$$ Without success. Is that ...
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1answer
16 views

Replacing variables in a nolinear system by their absolute values

I have to solve a nonlinear system (for some reactions) using newton's method to get molar fractions (positive values), sometimes I get negative values depending on the initial vector, to prevent this,...
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1answer
37 views

Elliptic limit cycle

The following equations, \begin{equation} \begin{split} \dot{F} &= -C + \frac{F}{\sqrt{F^2 + C^2}}\left(\alpha -\left(\frac{F}{a}\right)^2 - \left(\frac{C}{b}\right)^2\right),\\ \dot{C} &...
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1answer
62 views

Solve system of non linear equations

I am trying to solve the following system of non linear equations: $$\pi_1=\frac{k_2k_3(1-(1-x_2)(1-x_3))}{k_2k_3(1-(1-x_2)(1-x_3))+k_1k_3(1-(1-x_1)x_3)+k_1k_2(1-x_1x_2)}\\\pi_2=\frac{k_1k_3(1-(1-x_1)...
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0answers
21 views

Finding attractive and repelling part of the critical manifold.

We are given the following nonlinear system , $\frac{dI}{dt} = J \\ \frac{dJ}{dt} = -0.1\left(I^{3}(C - C_{0}\right)I - F - 0.2 J\\ \frac{dC}{dt} = \epsilon\left(F + \frac{C}{\sqrt{F^2 + C^2}}\left(1-...
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1answer
54 views

What is the difference between high dimensional and low dimensional chaos?

Often I read of high and low dimensional chaos. But, I don't know what is their difference. I have thought the following answer. Let us consider a time series $\{x_i\}_{i\in\mathbb N}$. According to ...
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1answer
30 views

Can I apply Hartman-Grobman when only one eigenvalue is zero?

Consider $\begin{bmatrix} \dot x\\ \dot y \end{bmatrix} =\begin{bmatrix} xy^2 -xy\\ x-y \end{bmatrix}$. If we take the Jacobian and evaluate it at $(0,0)$, one of the eigenvalues is $-1$ and the other ...
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21 views

Effect of dimension increase on the domain of attraction

Consider a nonlinear system with two locally stable fixed points $s_1$ and $s_2$ which have domains of attraction $D_1$ and $D_2$ respectively. Let $d_1$ and $d_2$ be domains of attraction of one ...
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1answer
71 views

Equation of the form $\mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)$.

Let $\mathbf{\Phi}(t)$ and $\mathbf A(t)$ be matrices satisfying the differential equation $$ \mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)\ . $$ If I am not mistaken, if $\mathbf A$ and its integral ...
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2answers
43 views

how to solve this non linear ode

$$ y y'(x) +y(x)^2(\sqrt {x^3}+{7\over4}\sqrt {x^5}+{1\over2}\sqrt {x^7})-{1\over2x}=0 $$ How to solve this equation?? I searched text book , and I only found bessel, legandre. But they are not same ...
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1answer
42 views

Is Lagrangian Multiplier Equivalent to Brute Force for binary decision variables

I have a set of variables $x_{i} \in \{1,k\} $ in a non linear optimization problem. As this variable has only two possibilities I have encoded this into a constraint. I assumed having equality ...
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1answer
92 views

Solving 3x3 Matrix Q using Nonlinear Least Squares or Cholesky Decomposition

I am trying to solve a system of equations using Cholesky decomposition. I would like to solve for the 3x3 matrix Q given: $\hat{i_f}^t Q Q^t \hat{i_f} = 1 $ $ \hat{j_f}^t Q Q^t \hat{j_f} = 1 $ $...
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19 views

Transcritical bifurcation for map function

Question: Determine the transcritical bifurcation for $x_{n+1}=\alpha x_{n}\left ( 1-x_{n} \right )^{2}$ I have determined the fixed point to be $x^{\ast}=0$ and $x^{\ast}=1$ Also, for values $...
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1answer
33 views

Proving that an equation has a unique stable limit cycle

I'm preparing for my exam and I stumbled upon a question and I am a bit lost on how to write the correct solution. The question goes as follows: Prove that the equation $\ddot{x} + \mu(x^{4}-1)\...
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20 views

Underdetermined, nonlinear system solvability criteria

The system under consideration is of the following form: $\ A(x)*x = \ B(x)$ In my case, this is a highly nonlinear underdetermined problem. Was wondering whether there is a way to determine ...
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1answer
38 views

Control of Nonlinear Cascaded systems

For control of cascaded linearized system, my objective is to design a stabilizing controller. For stability and performance analysis of such structures, I have been trying to find a book where such ...
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1answer
13 views

Get transfer function of a nonlinear diff. equation

I have this equation: $$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$ g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{...
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33 views

Solving system of nonlinear equations via iteration

I will give an example to illustrate the question: Assume I have the system: $$ xy + x + y = 7\\ x^2 + y^3 = 9 $$ and I want to solve for $x$ and $y$. It is a fairly common approach to rearrange ...
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26 views

Non Linear Systems : Broyden's Method

I am trying to implement Broyden's method for solving systems of non-linear equations following these documents http://heath.cs.illinois.edu/scicomp/notes/chap05.pdf http://web.mit.edu/jmartin3/...
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23 views

How to interpret multi-conditional piecewise functions.

I'm trying to simulate hysteresis and the its inverse for a control problem. This is a model found in [Tao & Kokotovic, Adaptive Control Systems with Actuator and Sensor Nonlinearities] to ...
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0answers
10 views

Is there an algebraic structure that corresponds to composition of $\max\{\vec{0}, A_2\max\{\vec{0}, A_1 \vec{x} + b_1\} +b_2\}$

I am not sure the question is well posed. I am wondering if there is an appropriate algebraic structure that represents a simple neural network (defined below): Let $\vec{x} \in \mathbb{R}^n$ and $...
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30 views

What it this theorem saying? - Regions of state space for which the flow eventually exists…

We have been given the following theorem to define regions in the state space for which the flow eventually exists. In questions, we use it to show that all trajectories eventually enter a bounded ...
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47 views

Poincare first return map, stability and bifurcations

Let $X= \mathbb R^3$ and consider the autonomous dynamical system $$\dot{x_1} = -x_2 + x_1 (1 - (x_1^2 +x_2^2)^2), \qquad{} \dot{x_2} = x_1 + x_2(1-(x_1^2 +x_2^2)^2), \qquad{} \dot{x_3}= \epsilon x_3$...
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1answer
34 views

Lyapunov number and Lyapunov exponent

Given two different points $x_{0}$ and$ x_{0}'$ where $x_{0}=x_{0}$ and $x'_{0}$=$x_{0}$+$\epsilon$ where $\epsilon_{n}$=$e^{n \lambda (x_{0})}$ is the n-th iteration of the separation distance ...
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41 views

Numerical method for solving equation with $u \frac{\mathrm{d}u}{\mathrm{d}x} + u$

I'm looking for a finite difference method to solve $$a(x) u \frac{\mathrm{d}u}{\mathrm{d}x} + u = b(x)$$ where $u(0) = c$. I tried to do a lagging convergence on the $u$ ie $$a(x) u^{(n)} \frac{\...
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1answer
51 views

How should I approach solving non-linear equations?

I need help creating a method for a program I'm making. I've worked on this countless hours and I can not seem to figure it out. what I need: A method that returns $x$. My variables ( initialized to a ...
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0answers
36 views

Finding a point coordinate given some distance restrictions relative to other points

I want to find the solution space of coordinates for point $p$ that satisfies the following system: $$ \begin{cases} [distance(p,a) - distance(p,b)] = k_1\\ [distance(p,c) - distance(p,d)] = k_2 \end{...
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0answers
27 views

Finding simple bifurcation point

$\dot{x}=x-rx\left ( 1-x \right )$ I know where the fixed point occurs but what about the bifurcation points. I thought I knew how to find the bifurcation points. At least, for simple ODE where I can ...
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0answers
19 views

Finding impact of multiple variables in a non linear equation

I am encountering a difficult(atleast to me) mathematical problem? I need to calculate the impact of a formulae on two sets of data for two different years Consider for example I have a simple ...
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33 views

Find the steady-states of the system of differential equations using sympy (in python) and determine their local stabilities.

The system is given by: $\frac{dx}{dt} = r x(1 - x) - \beta x y$, $\frac{dy}{dt} = \beta x y - \gamma y$. Analytically, I have found the Jacobian is given by: $J(x,y) = \begin{bmatrix} r(1 - ...
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1answer
26 views

How to Show that Lorenz equations are invariant?

I am struggling a little bit with this question. I know that that the Lorenz equations are: \begin{align} \dot{x} &= \sigma(y-x)\\ \dot{y} &= rx - y- xz\\ \dot{z} &= xy - bz \end{align} ...
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33 views

Does anyone have nice explanation about the theory? [closed]

I have hard time interpreting the Floquet theory. Does anyone have nice explanation about the theory?
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1answer
38 views

Homoclinic orbits of cubic potential

I found in Carles Simo's 'Hamiltonian Systems with Three or More Degrees of Freedom', among other references, that the homoclinic orbit for the cubic potential $\frac{y^{2}}{2}+\frac{x^{3}}{3}-\frac{...
3
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1answer
35 views

What is the meaning of “smeared limit cycle”?

I'm reading the paper Phase dynamics of coupled oscillators reconstructed from data by Kralemann et. al. (2008), which is about representing phenomena that exhibit a stable limit cycle (i.e. non-...
4
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0answers
101 views

Large system of nonlinear equations

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
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18 views

Solving system of equations involving ratios and exponents

I have two equations, $$\frac{x}{1-x}=\exp(A_{x}+Bx+Cy)$$ $$\frac{y}{1-y}=\exp(A_{y}+Cx+Dy)$$ I am interested in the number of solutions $(x,y)$ to these two equations as the parameters $A_{x},A_{y},...
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1answer
34 views

What does a negative time stepping mean? (Adaptive time stepping)

Summary behind the problem: The following code aims at solving a static elasto-plastic problem. Like a 2D square mesh based on an elasto-plastic constitutive model like Von-Mises or Drucker-Prager ...
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0answers
11 views

Nonlinear ODE Grimshaw solution manual

Anyone has a solution manual for Grimshaw's 'Nonlinear ODEs.' I had this class and really enjoyed it. Wanted to solve the rest of the problems in the book. If anyone possess a solution manual let me ...
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17 views

Complex Least Squares With Magnitude Equality Constraints

For $\mathbf{x} \in \mathbb{C}^N$, I'd like to solve the following problem: $$ \mathbf{x}^\ast = \arg \min_{\mathbf{x}} \Vert \mathbf{Ax-b} \Vert_2 \,\,\,\,\,\, \mathrm{s.t.} \,\,\,\,\, \Vert x_i \...
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0answers
6 views

Closed-form solution for a simple system of concave equations

I am trying to solve what looks like a simple system of equations: $$x_j = A_j\left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $$ for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, $0<\...
0
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1answer
30 views

Analytical solution of a non-linear equation with a 'min' function

I am building a mathematical model of a non-linear dynamical system and I have an expression of this form: $$x=\min\left(\frac{y}{a+y},\frac{y}{c+y(d+ex)}\right)$$ or let's consider any form like: $...
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0answers
18 views

Finding the exact real roots of a system of a sinusoidal and a line

Say I have a sinusoidal function $s(x)=\alpha \sin(\beta x - \gamma) + \delta$ and the linear function $f(x)=mx+b$. How can I find $x$ exactly such that $s(x)=f(x)$? I can't solve it like a normal ...
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4answers
27 views

How to show a zero of a function is a fixpoint of another function?

$g(x)=x \log(x+1)+x-1$ where the log has the base $e$. Set $G_1(x)=1/(\log(x+1)+1)$ and $G_2(x)=1-x\log(x+1)$. Show that the zero $x^*$ of $g(x)$ is a fixpoint of $G_1(x)$ as well as $G_2(x)$. My ...