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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6 views

How do I solving a system of three 1st order nonlinear ODE?

How would I go about solving the following system of non-linear ODEs for $x(t), y(t), z(t)$ $$x' = y $$ $$y'=sin(x)+z$$ $$z'=y+z$$ I have the following initial conditions; $$x(0) = 0$$ ...
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1answer
23 views

How to solve $ \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0$

How to find a solution to the following equation \begin{align*} \frac{1}{1+x}-\frac{c}{x}-2\log \left( \frac{1+x}{x}\right)+A=0 \end{align*} where $c$ and $A$ are some constants such that $c\ge 1$ ...
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1answer
27 views

Automatic way to have a good initial guess for the iterative methods ( newton method) and for high dimensional nonlinear problems

I am solving a variety class of nonlinear systems where I need to reduce in optimal manner the number of iterations of either the newton or modified newton method. For this end I am trying to figure ...
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0answers
17 views

Non-linear Systems, Impulse Responses, and Convolution

In linear system theory, it is easy to find the particular solution to differential equation by means of convolving the system's impulse response with the forcing function. My question is why can we ...
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2answers
22 views

How to represent non-linear operators computationally?

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$. I want to have a "general" form of this operator R. I think of the following series ...
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14 views

dominant balance for coupled differential equations

I have been trying to solve following set of nonlinear differential equations: $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - ...
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2answers
62 views

Non-linear system of equations

Solve following system of equations over real numbers: $$ x-y+z-u=2\\ x^2-y^2+z^2-u^2=6\\ x^3-y^3+z^3-u^3=20\\ x^4-y^4+z^4-u^4=66 $$ This does not seem as hard problem. I have tried what is obvious ...
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1answer
38 views

exact solution to lotka-volterra equations

I am looking for exact or perturbative solution realistic lotka-volterra (the one with logistic term in one of the equations) equations in population dynamics. Any reference where they have done it ...
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0answers
28 views

Well-posedness of nonlinear PDE system

The surface is parametrized by two variables $\sigma_1$ and $\sigma_2$. Moreover, this surface evolves in time. As a result, coordinates of the surface are: $\vec{F} =[x(\sigma_1,\sigma_2 , t), ...
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1answer
57 views

Averaging for nonlinear systems

I am trying to figure out how the following result has been obtained. Consider a function $J:\mathbb{R} \longrightarrow \mathbb{R}$ and a dynamical system: $$ \dot{ \hat{x} }(t) = k a \sin ( \omega ...
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4answers
36 views

How can you tell whether the equation of a non-linear relationship represents a parabola, a hyperbola or a circle?

How can you tell whether the equation of a non-linear relationship represents a parabola, a hyperbola or a circle?
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26 views

two variable perturbation analysis of nonlinear set of differential equations.

I have following set of equations, $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - 2 \epsilon_2 y(t) + 2 \epsilon_1 \epsilon_2 ...
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0answers
20 views

non-linear PDE finite difference approach

How to approach this equation using finite difference method ...
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0answers
15 views

Numerically solve: system of non-linear complex valued equations

I need to solve a system of equations numerically. Definitions: $$ -1\leq \epsilon_1,\epsilon_2\leq 1 $$ $$ E_1, E_2, \omega\geq0 $$ $$ E_0 < 0 $$ $$ n=1,2,3,... $$ $$ \frac{1}{t_f-t_i} \left( ...
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1answer
95 views

Non linear second order ODE

I really need help solving this : $$y_{xx}-\left(y^{3}-y\right)-\varepsilon\frac{1}{2}\left(1-y^{2}\right)=0 $$ With boundary conditions : $$ y(\pm \infty )=-1 $$ I need to find a solution that ...
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0answers
43 views

How to show that a system $dx/dt = f(x,y)$, $dy/dt = f(x,y)$ has a stable limit cycle which lies in the anular region?

I know I need to convert it to polar system but I don't know how to do this My system is $$\dfrac{dx}{dt}= x - y -x(x^2 + 2y^2)$$ and $$\dfrac{dy}{dt} = x + y -y(x^2 + y^2).$$ The annular ...
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12 views

Nonlinear System of Equations: Criteria for Existence of Solution

Let $\Omega \subset \mathbb{R}^n$ and $F: \Omega \rightarrow \mathbb{R}^n$ is at least once continuously differentiable (but not necessarily a polynomial). we want to find a point $x^* \in \Omega$ ...
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1answer
19 views

non linear curve fitting with newton-gauß

Shape the following problem as a non linear curve fitting problem and write the first iteration step with the gauß newton method. On a map are n radiostations $S_1,...,S_n$ which coordinates are ...
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4answers
223 views

Solving a system of non-linear equations with 10 equations and 10 unknowns

I'm working on a problem where I seem to have run into a system of non-linear equations. I have ten equations and ten unknowns. In the equations below, all of the $\phi_{ij}$'s are known, but all of ...
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4 views

Trapping region for a nonlinear system

I have the next system of equations and I want to find the optimal trapping region so: u'=-u+vu^2 v'=b-vu^2 I've tried different ways but I can't figure it out. ...
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1answer
18 views

Derivation for state equation linearization

In the following notes, how to linearize a state equation is described. The part I don't understand is why you can just remove the $\delta$ like that. I think the state equation should be: ...
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0answers
32 views

Solution to a state-space equation

In my notes on non-linear linearization there is the following example. It asks to verify the solution to the state-space equation. My understanding is that the solution is where the equilibrium point ...
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2answers
44 views

Convergence of a particular fixed point iteration scheme

Setup I have the following non-linear system of equations: $$ \mathbf{x} P(\mathbf{x}) = 0 $$ where $\mathbf{x} \in \mathbb{R}_{>0}^n$ is a probability distribution, i.e., $\sum_i x_i = 1$, and ...
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0answers
28 views

A set of nonlinear coupled ODE

I have reached a set of ODE as \begin{align} &\ddot{\vec{a}}(t)+\omega_0^2\frac{\cos{(b(t))}\sin{(a(t))}}{a(t)}\vec{a}(t)=0\\ &\ddot{b}(t)+\omega_0^2\cos{(a(t))}\sin{(b(t))}=0 \end{align} ...
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3answers
57 views

In control theory, why do we linearize around the equilibrium for a nonlinear system?

For example, in these notes: In the first example with the pendulum, they define the equilibrium as where the pendulum is at the vertical position (x=0), with a angular velocity of 0 (x'=0) and the ...
3
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4answers
93 views

Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$.

Consider the system $$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$ I have solved it by ...
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1answer
18 views

Nonlinear system, how to find Lyapunov

The nonlinear system: $\dot{x}_1 = \frac{5}{2}-2x_2-\frac{3}{2}x_1+x_2x_1$ and $\dot{x}_2 = x_1-1$. How do I find the Lyapunov function of this system? Or how to determine the existence of such a ...
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15 views

Finding the Lyapunov of the system

The nonlinear system is: $\dot{x}_1=\frac{5}{2}-2x_2-\frac{3}{2}x_1+x_2x_1$ and $\dot{x}_2=x_1-\frac{5}{2}$. My problem is trying to find the Lyapunov function for this system, if one exists. I've ...
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25 views

A set of nonlinear equations, their gradients, a nullspace and a uniqueness property

For an $n>m$ given $x \in \mathbb{R}^n$, $c(x) \in \mathbb{R}^m$, $c(x^*)=0$, $A := \left[ (\nabla_x c_i)^ T(x^*) \right]_{i=1,...,m}$ nonsingular and the columns of $Z \in \mathbb{R}^{n \times ...
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1answer
35 views

Solution of the equation… [closed]

How to find out the analytic solutions of the equation $1-4\sin^2\frac{\theta}{2}=\frac{\sin\theta(n-1)}{\sin\theta n}$ in the interval $(0, \pi)$? $n$ is an arbitrary integer constant.
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0answers
42 views

How To Solve This Non-Linear System (Ellipsoid-Plane-Cone Intersection)

Any help on how to solve this ellipsoid-plane-cone intersection problem or just even how to approach it will be greatly appreciated. All vectors are in $\mathbb{R}^3$ and I am trying to find ...
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0answers
14 views

Non-linearity on Random Sequence

I recently incorrectly assumed that applying a non-linear operation on a completely uncorrelated sequence would yield an uncorrelated sequence. Turns out that it is trivially easy to show that this ...
11
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0answers
100 views

How to solve non-linear differential equation [duplicate]

How to solve non-linear differential equation $$y'(x) = y(y(x)), \quad y\colon\mathbb{R}\to\mathbb{R}?$$ Of course, $y(x)\not\equiv 0$. If we substitute $y(x) = Ax^n$, we get complex $n$ and $A$. Any ...
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1answer
19 views

curiousity question on Lyapunov function of a nonlinear system

I've just begun studying nonlinear systems in my spare time. I'm using 'Nonlinear System Theory' by Rugh. My question is if there is a universal way to find the Lyapunov function of an arbitrary ...
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1answer
43 views

First order nonlinear ordinary differential equations

In my exercise I am stuck in a problem given below: $\ln\left(\frac{dy}{dx} \right) = x-y+1$ Although I could solve it if it was a linear equations. But ln() is a nightmare for me. Can anyone help me ...
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0answers
22 views

solving non-linear system of equation as optimization problem

I am trying to solve a system of non-linear algebraic systems through optimization. $f_i(x)=0$ for $i=1..n$ and $x \in R^n$. I saw few optimization versions: Minimize $\sum f_i^2(x)$, Minimize ...
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0answers
16 views

Help in identifying the one dimensional map

In the paper: http://inds08.uni-klu.ac.at/INDS2008/INDS08_System_Identification_using_Symbolic_Chaotic_Sequence.pdf there is a chaotic map in Eq(11) $$c_{n+1} = \frac{\gamma c_n(1-c_n^2)}{1+\rho ...
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1answer
95 views

Understanding stability of fixed points in 2D maps.

I'm trying to understand the stability analysis for a map of the form $$(x_{n+1}, y_{n+1}) = A(x_n,y_n)$$ Where A is a 2x2 matrix - assumed to be diagonalisable and with distinct eigenvalues. I ...
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0answers
15 views

solving non-linear systems inequalities

I am trying to solve a non-linear systems of 14 inequalities with 12 variables, involving exponential and polynomial functions. I have been searching over the web for leads,but without any success.I ...
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1answer
23 views

Finding solution of nonlinear differential system

$$x'=2x+y^2$$ $$y'=y$$ How to find a solution of above system if $x>0$? I found a solution of second equation, $y=y_0e^t$, but don't know how to use this to solve the system. Thanks.
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0answers
36 views

Theorem to show trajectories of differential equations are close after small change to initial condition

Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$ Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some ...
3
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2answers
100 views

Solve first order nonlinear differential equations

I want to solve this nonlinear 1-st order ODE, $$\frac{1}{1+x}=(\frac{1}{x-y}-\frac{1}{y})\frac{dy}{dx}$$ I find it non-separable, and Wolfram Alpha does not give me a closed form solution, but the ...
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2answers
82 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
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3answers
44 views

Analytically solving nonlinear second order ODE

I need help with providing an answer to this nonlinear ODE $a_1 + f_1(x) + f_2(x) y' - a_2\bigg((y')^2 - y''\bigg) = 0,$ where the $a_i$'s are constants and the $f_i$'s are arbitrary functions of ...
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1answer
50 views

Determining boundary of basins of attraction

Let's say that I have a dynamical system that displays multiple stable states with corresponding basins of attraction. The Lyapunov function for the system is not known. Is there an analytic or ...
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3answers
49 views

Linearizing an equation containing both $x$ and $\ln x$

The equation of interest is of the form: $$ k_1 \ln(y/x) = k_2 x $$ And I am wondering how can one linearize this equation for $x.$ Splitting the $\ln$ function would give something along: $$ k_1 \ln ...
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0answers
28 views

Linearize discretized nonlinear system model

For the following nonlinear system I want to find the linearization after a discretization: $$ \begin{pmatrix} \dot{x_{1}} \\ \dot{x_{1}} \\ \dot{x_{1}} \end{pmatrix} = 1/A \begin{pmatrix} ...
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1answer
36 views

Formula question (non-linear?) Y and X

I am trying to come up with a formula so I can get the value of $X$, with whatever $Y$ I put in. A few example values are listed down below \begin{matrix} Y & X \\ 1 & 0.9 \\ 10 & 0.5 ...
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0answers
18 views

Multi time scales analysis on nonlinear system of ODEs

So I have this coupled set of nonlinear ODEs that I want to do a multi time scales perturbation analysis on. $ u'(t)+\frac{C \epsilon u(t)^2}{Cl}-\frac{2 \epsilon p(t)}{Cl}-\frac{2 q_1'(t)}{Cl}=0 ...
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1answer
33 views

How to solve initial value problem $m\dot{v} = mg - kv^2$ for $v(t)$ when $v(0) = 0$

In this equation $v(t)$ represents the velocity of an object falling to the ground, $m$ is the mass of the object, $g$ is the gravitational acceleration and $k > 0$ is a constant related to the air ...