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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10 views

Nonlinear Schrodinger equation - modified

I have a nonlinear Schrodinger equation: $ia_1\dfrac{\partial A}{\partial x}-a_2\dfrac{\partial^2 A}{\partial t^2}+|A|^2A=0$, $A$ is the amplitude and the above equation governs the slow modulation of ...
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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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33 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 ...
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2answers
63 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
60 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
40 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
31 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
48 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
21 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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16 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
28 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 ...
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3answers
25 views

Nonlinear system with complex numbers

Solve the following system under the complex numbers (without eulerian form or polar form) $$z^3 + w^5 = 0 \\ z^2 \bar w^4 = 1$$ I have found that $(\pm 1, \mp 1)$ satisfy the equations as well as ...
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0answers
23 views

solubility of nonlinear overdetermined PDE for holonomic scaling of a frame

This question is essentially a tweak of under what conditions can orthogonal vector fields make curvilinear coordinate system? . Suppose we have a frame of vector fields $\nu_i$ on $\mathbb{R}^n$. As ...
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0answers
28 views

is Lyapunov direct method applicable for infinite dimensional nonlinear system?

after linearize the infinite dimensional system, I have an A matrix in which each element is in terms of the dimension index k. And as k goes to infinity, A matrix has some positive eigenvalue. But if ...
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0answers
32 views

solutions that don't converge to equilibriums

suppose a nonlinear autonomous system has more than two asymptotically stable equilibrium, how do I find a solution that doesn't converge to any of these equilibrium?
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71 views

Does uniformly attractive imply lyapunov stable in autonomous system?

If a autonomous system is uniformly attractive, can we say that the system is stable? If not, do we have counter example? Thank you!
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0answers
11 views

Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
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1answer
16 views

Find correleation between values and degrees

I have an arc that starts at $252$ degrees and ends at $288$ degrees, I would like to assign non - linear values on it with this ratio: $1 - 180$ degrees. $5 - 135$ degrees. $10 - 90$ degrees. $30 - ...
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1answer
38 views

Lyapunov function

My lecturer gave us the definition of a Strong Lyapunov function. She then said that if V is positive definite but $dV/dt$ is also positive definite (instead of negative definite) in a region ...
2
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1answer
34 views

Second Order Non-Linear Ordinary Differential Equation

I have the equation $$x_{tt}+cx_t+x=x^2$$ where $c$ is constant and $x=x(t)$. If the $x^2$ wasn't on the right hand side of the equation then I could solve this easily by the method of ...
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1answer
23 views

Finding the characteristic timescale of a first-order nonlinear ODE

I know that to find the timescale of a first order linear equation $$\frac{dX(t)}{dt} + aX(t) = b$$ you just take the inverse of the integrating factor, so $$t_x = \frac{1}{a}$$ Henning and ...
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0answers
13 views

Is there a connection between topological mixing and squashing functions used in neural networks?

Sigmoid, ReLU, tanh, logistic -type "squashing" functions are popular in neural networks to introduce nonlinearity into the transformations of the input vector, allowing the network to fit complex ...
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0answers
17 views

Extraction of quadratic terms with state-space representation

I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: $T_i + f_i(\omega, \alpha) = 0;\;i:1-4$ . ...
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1answer
36 views

Simplifying the method of solving a problem

One of my peers from high school asked me how can this problem be solved: $$\begin{cases} x^2+y^2=z \\ x+y+z=m \end{cases}$$ Considering the mentioned equations, find $m$ such that the system has ...
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2answers
41 views

How to solve the equations of linear combination of sigmoid functions?

Let $\sigma(x)=\frac{1}{1+e^{-x}}$ be the sigmoid function. How to solve such kind of equations? \begin{align*} \sigma(x+y)+\sigma(x-y)=a\\ \sigma(2x+y)+3\sigma(3x-y)=b\\ \end{align*} I guess this ...
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0answers
8 views

Schwartzian derivative of a vector function

I am not a mathematician and hence I really do not know where to look for answer. I have a system which is governed by the three equations. say $x_{n+1} = f(x_n,y_n,z_n)$ ,$y_{n+1} = ...
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1answer
29 views

First order non-linear ordinary differential equation

The following ODE is given: $a\pm\sqrt{b+c*(x(t)+d))}=e*x'(t)+f*x(t) $ from Matlab I'm able to get a solution for the differential equation (actually two solutions, one for the + and one for the - ...
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0answers
19 views

How to apply the laplace transform to this second order ODE?

Can I apply the Laplace transform on a the following second order nonlinear PDE? $$ \frac{\partial y}{\partial t}=\frac{\partial^2 y^n}{\partial x^2}$$ where $n$ is a natural number? I mean apply the ...
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1answer
54 views

Can the following nonlinear first order ODE be solved?

I have tried solving this equation from several manners but no luck. Can it be solved? $$\frac{d f}{d t} = A f^2 +g(t)$$ The solution for the homogeneous is (I think; somebody should confirm) ...
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1answer
29 views

How to obtain an exact solution to nonlinear second order ODE

I need help in analytically solving this nonlinear second order ODE, $A y(x) + y'(x) \Bigg( B + \frac{C y'(x)}{D y'(x) - y''(x)} \Bigg) = 0$. Any help is appreciated.
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2answers
63 views

Techniques to solve nonlinear first-order ODEs

I am trying to solve the following nonlinear ODE: $$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$ where $a, b$ are constants and $a>0$. Moreover, you may assume that $b \neq 0$ if necessary. This ...
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0answers
21 views

Oscillations of a feedback interconnection

I have a feedback interconnection described via the following transfer function $G(s)=\frac{1}{s^3+5s^2+6s+1}$ and the nonlinearity $\psi(e)=\text{sgn}(e)$. I have used the describing function method ...
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0answers
20 views

Methods of solving nonlinear systems of equations derived from combinatorial problem

I'm trying to find a way to generalize the expression of polynomials of degree $n-1$ such that $$ k_1+k_2x+k_3x^2+k_4x^3+\dots+k_nx^{n-1}=\frac ...
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0answers
21 views

Find out the optimization type

I am formulating a problem and intend to solve it by optimization. Here is the current result: *Objective:*$\quad\min\quad c + f_1(x)x_1 + f_2(x)x_2$ Constraint: $\quad ax_1 + bx_2 <= d$ where ...
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0answers
26 views

Which is the best method/algorithm to predict rank in an exam based on historical data?

I have the 10 year historical data(mark and rank) of all students who took an exam (exam happens once in an year and the patter of the exam is always same(number of questions and marks per question) . ...
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0answers
29 views

Can't replicate solution to a non-linear PDE ($- \square \varphi + \lambda \varphi^3 = 0$)

I am trying to replicate the solution of this paper : http://arxiv.org/pdf/0807.2179.pdf Which is, roughly, for the quartic scalar field theory, $- \square \varphi + \lambda \varphi^3 = 0$ a set ...
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1answer
32 views

What is the largest invariant set?

I think the largest invariant set is on other than $\{x:\dot{x}=0\}$, is this correct, is there other way to establish the largest invariant set? Please give an example.
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1answer
34 views

Is this system stable?

I got this control system with such dynamics: \begin{equation} \dot{x}(t)=-\frac{\partial{H(x)}}{\partial{x}},~H\geq 0,~H(x)=0\Rightarrow x=0 \end{equation} $x(t)$ is a $n$-dimension vector, ...
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0answers
25 views

Solving $du/dt=a \Delta u + b (\Delta u)^2$

Consider the function $u(\boldsymbol{x},t)$, where $u:\mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$ ($\mathbb{R}_+$ denotes nonnegative reals). My question is related to the PDE ...
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0answers
9 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
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1answer
53 views

How do we plot nonlinear differential equations

If this is not nonlinear I apologize, I'm still learning differential equations. I am attempting to make a stream plot of a predator-prey model of eccentric closed curves by using the following ...
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27 views

How to solve the equation $Au+Bv=C$

How do I solve $Au+Bv=C$ Where $A$ and $B$ are constant known matrices that are nxn, $C$ is a constant known nx1 vector while $u$ and $v$ are unknown nx1 vectors with the condition given that $u_i = ...
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0answers
12 views

Conditions for existence and uniqueness of systems with linear and nonlinear equations

Given a system that contains a mixture of linear and nonlinear equations, under what conditions can we guarantee that a solution will exist and that it will be unique? For example, the system ...
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32 views

Hopf bifurcation in linear delay differential equations?

Can equilibria of a system of linear delay differential equations undergo a Hopf bifurcation? I am able to generate oscillatory solutions by tuning a time delay in the system, but would it make sense ...
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4answers
69 views

(a+x)(b+y)=c for x,y over integer [closed]

$(8192+x)(2789+y)=22855543$ How to get x and y for integer? WolframAlpha can get the integers, I have worked on this so long, here is what I have so far: $(a+x)(b+y)=c$ $ab+ay+bx+xy = c$ ...
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0answers
15 views

Sufficient conditions for the existence of the solution to the system of non-linear equations

The question of existence of the solution for an arbitrary system of non-linear equations $F(x)=0$ where $F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is undecidable (following Existence and ...
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9 views

transformation of variables in Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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1answer
60 views

Solving a nonlinear 1st order ODE

I need help solving this nonlinear first order ODE. Any help will be appreciated. $y'(x) + ay(x) + \frac{bx^2}{y(x)} + cx + d = 0$ Thank you.
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1answer
28 views

Solving three-variable nonlinear equation systems

A physical problem which I've been studying leads to the following nonlinear equation system to be solved: $$\alpha\cdot79\cdot A_1 +(1-\alpha)\cdot 1025 \cdot B_1 = C_{11}$$ $$\alpha\cdot145\cdot A_1 ...