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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12 views

Separatrix and trajectories corresponding to solutions of an equation(pendulum).

The equation of the pendulum is: $\ddot{\theta}+\frac{g}{l}sin\theta$ After some manipulation, we get $H=\frac{\dot{\theta}^{2}}{2}-\frac{g}{l}Cos\theta$=positive constant Trajectories ...
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30 views

Analytic solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations. Mathematica can give me solutions up to certain sizes of the linear system, but I would like to have it for arbitrary size ...
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1answer
4 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
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23 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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1answer
24 views

Nonlinear equations systems

Can anybody help me to find a system with 3 equations and 3 unknowns and a bounded domain D = [a,b]x[c,d]x[e,f] such that the system has an unique solution in D? Also, i need nice equations, because ...
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13 views

What does the function $n(\gamma , z_{0})$ denote in this version of Cauchy Integral Formula?

In my lecture notes, the Cauchy Integral Formula for complex integrals is defined as $$ \int_{\gamma} \frac{f(z)}{z - z_{0}} dz = 2 \pi i \cdot n(\gamma , z_{0}) \cdot f(z_{0}) $$ What does the ...
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33 views

Solution of seemingly simple system of equation

I wand to find the solutions $p_H, p_L$ implied by the following two equations: (I) $\frac{(1-\lambda)(p_L-c_L)}{p_H \frac{q_L}{q_H}-c_L} = \frac{\lambda(p_L-c_L)}{p_H-(q_H-q_L)-c_L} - \lambda$ (II) ...
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17 views

Solve a system of simple (but not linear) equations

There are 24 variables and 24 equations in the system: $$ i=0,1,2,3.\\ variables: s_i, t_i,a_m \; (m=0,...,15)\\ constants: b_{ni} \; (n=0,...,5)\\ a_0t_i+a_1s_i=b_{0i} \\ a_2t_i+a_3s_i=b_{1i} \\ ...
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1answer
17 views

omega-alpha limit set and manifold

Definition: The $\omega$-limit set $L_{\omega}\left ( x \right )$ of $x \in \mathbb{M}$ >is the set of $y \in \mathbb{M}$ which for each y there exists a strictly increasing unbounded ...
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15 views

What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $p=1+\mathtt{i},q=1-\mathtt{i}$. Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & \color{blue}{q}\\ q & 0 ...
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1answer
18 views

Geometric intuition of an invariant set, positively invariant and negatively invariant

Definition: Invariant set A set $S \subseteq \mathbb{M}$ is invariant if $\Psi_{t}\left ( S \right )\subseteq S,\forall t$ -if $\forall x \in S$, $\Psi_{t}\left ( x \right )\subseteq ...
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2answers
66 views

Discrete one-dimensional 2-cycle system

Is it possible to classify all maps $x_{k+1} = f(x_k)$ that have the property that all orbits are period 2 cycles only? Also, how would I do it for period 3 system?
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14 views

linearization of a nonlinear modell

I am not sure about some definitions between Linearization around equilibrium point and reference trajektory. Because i am using linear Kalman filter and extend kalman filter with a nonlinear system. ...
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1answer
29 views

Eigenvector for a non-linear system

Using the reversibility arguments alone, show that the system $\dot{x}=y$ $\dot{y}=x-x^{2}$ has a homoclinic orbit in the half-plane $x\leq 0$ This is a non-linear system. A ...
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1answer
29 views

Eigenvector of unstable and stable manifold of a non-linear system with non-linear center

Show that the system $\dot{x}=y-y^{3}$ $\dot{y}=-x-y^{2}$ has a non-linear center and plot the phase potrait. My attempt: The system is non-linear so we linearise it: The ...
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26 views

Nonlinear optimization with eigenvalue problem as a constraint

I have an unknown matrix $\mathbf{A} \in \mathbb{R}^{2n \times 2n}$ which is a function of $n$ parameters $a_i, i=1,2,...,n$. The objective is to find these $a_i$'s and the objective function is as ...
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14 views

Finding equilibrium points of a system of nonlinear differential equations

I am currently working on a spatially explicit ODE model with dispersion to study the population dynamics of mosquitoes. I wish to compute the equilibrium values of the populations as functions of the ...
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1answer
54 views

Long term Behavior of Dynamical System

Given the following dynamical system: $ \dot x = -6x^2+yz+x-1 $ $ \dot y = 4xz-3y^2+y-2 $ $ \dot z = 9xy-2z^2+z-3 $ What can you say about its long term behavior? Attempt: First, finding the ...
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26 views

Prove any $n>2$ DE doesn't hold Poincaré-Bendixson theorem.

How can I build a differential equation to show that Poincaré-Bendixson theorem doesn't hold for $n≥3$ ? Is it easy to take any D.E with $n=3$ and prove it? More specifically, can you give me a ...
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21 views

Determine units in dimensional analysis

I am trying to do a dimensional analysis of a nonlinear system of differential equations. I have some problems with determine the dimensions of the different variables. The differential system is ...
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2answers
55 views

Dynamics of a three dimensional system

I have a dynamical system in three dimensions given by: $\dot x = (1-x^2-y^2-z^2)x+xz-y$ $\dot y = (1-x^2-y^2-z^2)y+yz+x$ $\dot z = (1-x^2-y^2-z^2)z-x^2-y^2$ I analyzed the system by first finding ...
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12 views

System of 2nd Order Non-linear ODE

I am trying to solve a system of 2 equations in Matlab. Both equations are 2nd order, nonlinear ODEs. I don't know which function(s) should I use? System of non-linear, 2nd order ODEs
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1answer
25 views

Constant difference between two percentage numbers (nonlinear)

I'm trying to figure out the correct way of detecting the difference of values based on two percentages, because my terminology is poor here is an example: The difference between two values where ...
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1answer
42 views

Constructing a Poincare Map

I need to construct a Poincare Map of the following dynamical system: $\dot x = x-(x+y)(x^2+y^2)$ and $\dot y = y + (x-y)(x^2+y^2)$ I changed the system to polar coordinates which gives me: $\dot ...
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2answers
39 views

Do four points lie on the circumference of a single circle? Can I solve this with matrices?

I think I managed to figure out a way to determine whether three points lie in a single line via matrix determinants (but correct me if there's a problem): Where $y - mx - b = 0$, I plug each of the ...
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1answer
44 views

Newton's method for a system of 7 equations

I want to use Newton's method to find two solutions to the system of equations using $L_\infty$ norm and an f-test with $tol_f = 10^{-6}.$ $$\frac{1}{2}x_1+x_2+\frac{1}{2}x_3 - \frac{x_6}{x_7}=0$$ ...
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56 views

Analytic solution for system of trigonometric equations

I have two equations as follows: $$ \left\{ \begin{array}{c} (\Delta_{11}\cos(\alpha) + \Delta_{12})\cos(\theta) + (\Delta_{21}\cos(\alpha) + \Delta_{22})\sin(\theta) = \Delta_{31}\sin(\alpha) + ...
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1answer
45 views

2D Bifurcation Classification

Given the system with m as a varying parameter: $\dot x = mx^2-y$ and $\dot y = m+y - x$ Determine any bifurcations that occur Attempt: x nullcline $y=mx^2$ y nullcline $y=x-m$ Fixed ...
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14 views

Stability of LTI systems under saturation

Consider the saturation function $$ \sigma(u)=\max(\min(u,1),-1) $$ for $u\in\mathbb{R}$. With slight abuse of notation, if $u\in\mathbb{R}^n$ let $\sigma(u)$ also denote the same function applied ...
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12 views

Dulac’s negative criteria - in a plane

I am trying to verify that a ODE system, $\dot x=f(x,y), \dot y=g(x,y)$, has no periodic orbits in the plane (presumed to be in $\mathbb R^2$). Thus I need to show for a function $h(x,y)$, that ...
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1answer
68 views

Non-Linear Regression for Parameter Estimation

I have a second order system, it's response to a step change can be expressed in the s-space as: $$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$ Which can be inverse ...
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27 views

Non-linear systems: Modelling the behavior of an oscillatory system when adding a perturbation.

I have a non-linear system of equations that describes an oscillatory system. the oscillations is generated by the interplay of two chemical species $A$ and $B$ Equation 1 \begin{align} ...
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19 views

Proving Exponential Convergence

Consider the function $\dot{x} = f(x,t)$. I want to show that if there exists a function $V(x,t)$ and some positive constants $h,\delta,k_1,k_2,$ and $k_3$ such that for all $x \in B(0,h)$ and for all ...
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38 views

Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
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14 views

What is an example of a stochastic nonlinear dynamic system with 2 separated stable orbits

I have some social science data to which I would like to fit a stochastic difference or differential equation in two variables. (I observe the system only at discrete intervals). This system that has ...
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1answer
43 views

Trapping Region for Dynamical System

Show that the dynamical system contains a closed orbit $\dot x = xf(x,y)+yg(x,y)$ and $\dot y = yf(x,y)-xg(x,y)$ Given Information: f(x,y) and g(x,y) are single valued functions and differentiable ...
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1answer
43 views

proving asymptotic stability dynamical system

I want to show the origin of the dynamical system \begin{align} \dot{x}_1 &= -2x_1+x_2+x_1^3x_2^2\\ \dot{x}_2 &= -x_1-2x_2+x_1^2x_2^3 \end{align} is asymptotically stable over an invariant ...
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1answer
35 views

H0w are second order nonlinear ordinary differential equations solved?

I conceived the following second order nonlinear ordinary differential equation: $$\frac{d^2y(x)}{dx^2}=\frac{k}{(y(x))^2}$$ I can tell it's nonlinear because of the $\frac{k}{(y(x))^2}$ term and ...
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16 views

Solving this system of equations of reciprocals?

Following on from How to solve this system of equations containing reciprocals?, my system of equations has become: $p_i = \frac{b_i}{b_i + l_i+mr_i}$ where $i\in\mathbb{Z}$ $b_i, l_i, r_i >0$ ...
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2answers
37 views

Nonlinear nature of fixed point

Given the system $\dot x = y-x^3$ and $\dot y = -x-y^3$ What can you say about the nonlinear nature of the fixed point? Attempt: I started out by employing linear analysis. The nullclines are ...
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22 views

ODEs, seting up a non linear system and Newton-Raphson-Method

at the moment I study for an examen regarding ODEs. I try to solve a task but I'm not sure how to do it. Task: We should use the trapozoidal Rule $y_1 = y_o * \frac{h}{2}[f(x_0,y_0) + f(x_1,y_1)] $ ...
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2answers
77 views

Solution of $y''+y^2=0$?

I'm trying to find solution for $$ \frac{d^2y}{dx^2}+y^2=0 $$ Can this differential equation(it is nonlinear, isn't it?) be solved analytically? Or do I have to resort to numerical method? The ...
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5 views

Homogeneus part Multivariate Quadratic Polynomial System

In my lecture say: In providing lower bounds on direct attack complexity, we rely on the empirical fact that the complexity of solving a non-linear polynomial system is determined by the ...
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1answer
14 views

formula for equation with exponent variable?

is there a closed formula for such an equation to find the value of $x$ in $ax = b^x$ if there isn't , are there any published attempts ?
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34 views

Help solving this 2nd order non-liner differential equation?

I am trying to compute the optimal path length for getting from Sydney to Hong Kong via a tunnel and using only the force of gravity. (See figure.) By using calculus of variations and the ...
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1answer
45 views

Index of circle containing fixed point

Given $\dot x = x^2-y^4 $ and $ \dot y = y^2 -x^4 $ Find the index of the circle $x^2 + y^2 = a^2$ with $a < 1$ Attempt: I employed linear analysis by finding all the fixed points. There are ...
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11 views

Magnetic dipole - sensing from 3 locations

The problem is to estimate the position $\vec{r}_0$ and direction $\widehat{𝑚}$ of a magnetic dipole (e.g. current loop) using electromagnetic field measurements by three sensors. A magnetic dipole ...
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2answers
61 views

How can I solve this 2nd-order, non-linear, stiff ODE numerically using Matlab? [closed]

I need to find a numerical solution for $T(x)$ to \begin{equation} aT'' - bT^4 + c = 0, \ \ \ T(0) = T_0, \ \ \ T'(L) = 0 \end{equation} where $a$, $b$, $c$, $T_0$ and $L$ are positive constants. ...
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1answer
17 views

How to transform a coupled differential equation into a system with diagonal linear part

Consider the system given by $$iu_t +u_{xx}+2|u|^2u = -v+iu$$ $$iv_t +u_{xx}+2|v|^2v = -u-iu$$ I am trying to transform the system into a system with diagonal linear part. I can solve a problem like ...
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17 views

Matrix Riccati Differential Equation

Is it possible to bring the system $$ \frac{d s_j(t)}{dt} = a_j s_j(t) + \sum_{i=1}^n a_i \prod_{l=1}^n (s_l(t))^{R_{li}} \quad(j \in 1,\dots,n; R_{li} \in \mathbb{N}_0; \sum_{l=1}^n R_{li} \leq 2)$$ ...