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18 views

Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...
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0answers
16 views

nonlinear coupled partial differential equations

Is there any order for solving a set of nonlinear coupled partial differential equations analytically i.e. without a numerical algorithm. I cant solve the following set of equations $$ ...
2
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1answer
32 views

Solve a second order nonlinear equation

I have a second order nonlinear equation: $$-u''+ \frac{1}{4}(u')^2+au=x^2.$$ I am only interested in the solutions in $[0, \frac{x^2}{a}+\frac{1}{a^2}]$. One paper claims without proof that the ...
1
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0answers
24 views

Nonlinear Schrodinger Equation

Consider the equation $$i u_{t} = u_{xx} + au - bu|u|^{2},$$ where $a, b > 0$ are real constants and $|u|^{2} = uu^{*}$. (a) Find the dispersion relation for the equation and discuss the behavior ...
1
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1answer
56 views

ordinary differential equation solving

I have a diffeq: I have a nonlinear Diffeq: $$\frac{d^2x}{dt^2}+\beta \frac{dx}{dt}+\varepsilon e^{- \lambda x} = f(t) $$ where $f(t)$ is a function that is known, and $\beta$ and $\lambda$ are ...
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0answers
16 views

Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): Here is $\alpha, \beta, \gamma, \delta$ - some constants, $u$ - control variable. ...
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0answers
31 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
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0answers
3 views

the notion of Wazewski set in shooting method

I come across the notion of Wazewski set when studying shooting method for proving existence of a boundary value problem. Some authors who use shooting method to prove existence start with ...
0
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1answer
26 views

System of Non-Linear ODE

Does anyone have any clue of how to find an analytical solution for the following system: $$ \frac{dF_1}{dt}=(p+qF_1-rF_2)(1-F_1) $$ $$ \frac{dF_2}{dt}=vF_1(1-F_2) $$ $p$, $q$, $r$ and $v$ are ...
1
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2answers
23 views

$yy'=\sin(t),y(0)=1$ phase portrait

I need to draw a phase portrait for the equation $y(t)y'(t)=\sin(t)$ with the initial condition $y(0)=1$. So far i've found that $y(t)= \sqrt{3-2\cos(t)}$ and ...
0
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0answers
15 views

Discrete time adaption rule

Is it possible to find an update rule for $d(k)$ that satisfy following equation $$\log\frac{d^2(k+1)+1}{d^2(k)+1}=-c\log\left(|f(d(k))|+10\right)$$ where $c>1$ . I appreciate the time you'll take ...
0
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0answers
18 views

Nonlinear discrete time systems

Is it possible for discrete-time parameter $a(k)$ with an update rule like $a(k+1)=f(a(k))$ & always $|f(a(k))|<= c|a(k)|$ where $0<c<0.5$ to converge from the initial value $c_1$ ...
0
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2answers
46 views

How to get the perfect square for the following equation

The problem is defined as follows: $$ \min_X tr(X^T A X)-\alpha tr(X^T B) $$ I want to get the equal perfect square equation as that above, that is $$ \min_X \| X-C\|_F^2 $$ where $C$ is related to ...
1
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2answers
61 views

Non linear effect in differential equation

Suppose I want to study the non linear ODE $\frac{d^2}{dx^2}f(x) + a f^2(x) + b f(x) + c=0$ I know the solution for $a=0$ and I know $a$ is a small parameter. How can I study the effect of the non ...
0
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0answers
14 views

Fastest way to solve specified system of nonlinear equations

I have a following system of equations \begin{equation} \begin{aligned} \sum\limits_{i = 1}^3 g_i V_{i, x} & = (\sum\limits_{i = 1}^3 g_i n_{i, x})t + P_x \\ \sum\limits_{i = 1}^3 g_i ...
1
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1answer
27 views

Solving a Non-linear Multivariable System of equations

How would I go about solving a system of nonlinear equations where the highest degree is two? For example: $$f_1(x) = f_1(x_1, x_2,\dots, x_n) = 0,$$ $$f_2(x) = f_2(x_1, x_2,\dots, x_n) = 0,$$ ...
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0answers
17 views

Non-linear system of exponential equations with 2 boundary conditions: $p(y_m)=p_m$ and $\frac{dp(y)}{dy}$

I have this equation: $$ p(y) = -\left(e^{-\tfrac{K_{py}zy}{p_u+c e}}-1\right)\left(p_u+c e^1\right)-c\left(1-e^{-y}\right)^d\left(e^1-e^{1-y}\right) $$ The two unknowns are $c$ and $d$ and the system ...
0
votes
1answer
35 views

Solve a viscous Burgers' equation with a Newton-GMRes method

I implemented a preconditioner for a GMRes method. To test this preconditioner I want to solve this one dimensional viscous Burgers' equation $$\partial_t u(x,t) + u(x,t) \partial_x ...
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0answers
24 views

How to find the root of this non-linear equation?

I am trying to solve this non-near equation using Matlab but it doesn't give me the correct answer (as shown in the document that I am doing it from). The Matlab code gives me imaginary root. Could ...
1
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1answer
27 views

Need to draw phase portrait near the equilibrium points of differential equation

So, this equation $$\ddot{x}+3\dot{x}-4x+2x^2 = 0.$$ I can write like a system \begin{equation} \left\{ \begin{array}{ll} \dot{x} = v, \\ \dot{v} = 2x^2 - 4x - 3v. \end{array} \right. \end{equation} ...
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0answers
36 views

Properties of periodic solutions of nonlinear ODE system

Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function ...
1
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0answers
14 views

state space linearization

I am trying to linearize the non-linear state space model of: $\dot{x_1} = \frac{1}{C_p}(i_{pv} - x_2u)$ $\dot{x_2} = \frac{1}{L}(R_o(i_o - x_2) - R_Lx_2 - x_3 + (V_D + x_1 - R_mx_2)u) - ...
0
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1answer
42 views
1
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1answer
21 views

Non-linear system with all trajectories converging on the line $x=0$, rather than $(2,0)$?

I have the following nonlinear system: $$\begin{pmatrix}\dot{y}_1\\\dot{y}_2\end{pmatrix}=\begin{pmatrix}2y_1\\y_1^2\end{pmatrix}$$ Which I set up to $F=\dot{y}$ Giving the jacobian of ...
0
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1answer
51 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
2
votes
1answer
28 views

How to find initial estimate of roots from graphs?

I have f1(x,y) = x^2 + 3y^2 - 1 = 0 and f2(x,y) = (x-2)^2 + (y-1)^2 - 4 = 0 I am suppossed to find the roots of these nonlinear ...
19
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9answers
6k views

System of nonlinear equations that leads to cubic equation

The system of equations are: $$\begin{align}2x + 3y &= 6 + 5x\\x^2 - 2y^2 - (3x/4y) + 6xy &= 60\end{align}$$ I can solve it through substitution but it is an arduous process to reach this ...
1
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1answer
40 views

How to set a residual bound for fsolve() in Matlab?

Peace be upon you, I have the following system of equations \begin{align*} \begin{cases} \psi(x_1)-\psi(x_1+x_2)+0.6931471805599456\\ \psi(x_2)-\psi(x_1+x_2)+0.6931471805599456 \end{cases} ...
0
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1answer
58 views

How to prove symmetry of the following Maxwell-Bloch equations?

I have the following Maxwell-Bloch equations: $\dot{E}=-\alpha_{1} E+ k_{1}P$ $\dot{P}=-\alpha_{2}P+ k_{2}ED$ $\dot{D}=-\alpha_{3}(D-\lambda) -k_{3}EP$ In this system ...
0
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0answers
25 views

Help solving a non-linear differential equation with an integral.

I am having trouble solving the following type of differential equations. Mathematical methods by Boas or other intro books don't seem to have any hints on solving the equation. Any help would be ...
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0answers
17 views

Non linear ordinary differential system

Is there an analytical solution (in the general case) to the following differential system (Cauchy Problem) : $\dot{f}=\frac{Af}{(f^2+g^2)^{1/2}}$ $\dot{g}=\frac{Bg}{(f^2+g^2)^{1/2}}$ with the ...
2
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0answers
83 views

Translational invariance and zero eigenvalue

Page 2 (506), line 18 of http://www-personal.umich.edu/~orosz/articles/NonlinScipublished.pdf says that "The presence of translational symmetry in the nonlinear equations gives rise to a relevant ...
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2answers
49 views

Long term behavior of the solution of $u'=e^{-u}-u$

Consider the autonomous differential equation $$\left\{\begin{matrix}u'&=&e^{-u}-u&=:&f(u)\\u(0)&=&u_0&\in&\mathbb{R}\end{matrix}\right.$$ How can we analyze the *long ...
1
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1answer
37 views

Linearisation of a system of equations - answer check

Find all of the critical points for the following nonlinear system. $$\begin{pmatrix}\dot{y}_1 \\ \dot{y}_2\end{pmatrix}=\begin{pmatrix}-y_1+ y_2 - 2\\ y_1 -y_1y_2^2\end{pmatrix}$$ and then use ...
1
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1answer
36 views

Method of characteristics (quasilinear pde- nonlinear transport )

I want to solve the following pde ( nonlinear transport - I guess ?) $\phantom{}$ $ u_t - a(x) u_x = 0 $ with $a(x) > 0 $ and $u(x,0) = u_0(x)$ $\phantom{}$ I tried the method of ...
0
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0answers
41 views

What are the most advanced mathematics that apply to architecture?

I am doing an investigation about the mathematics in architecture, yet I still haven't quite discovered a higher-level mathematical concept that is actually used in the design and construction of ...
0
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0answers
22 views

Solving large non-linear polynomial equation system

I have a 2 order equation system of 7 unknowns. It is constructed as this: F1=0,F2=0,F3=0...F7=0 of which F1=f1*f2,F2=f3*f4... And f1=a1*p1+a2*p2+a3*p3+a4*p4+a5*p5+a6*p6+a7*p7 a1~a7 are known ...
0
votes
1answer
33 views

Plotting the phase portrait of $\dot x = x(x-y)$ and $\dot y = y(2x-y)$

I am trying to plot the phase portrait of $\dot x = x(x-y)$ and $\dot y = y(2x-y)$ Now I have already found the fixed points of the system, (0,0). I have also found the Jacobian of (x,y) and when ...
1
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1answer
45 views

State space and linearization

I have a question about state space representation. How can I represent an equation in which I have only the second and first derivatives? For example where $u$ is the control input. If I put ...
0
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1answer
21 views

How do I solve $0 = x\times114 - x\times\log_3(x) - 20.28\times y$ in matlab for different values of $y$?

I have $y = 10^3, 10^6, 10^9, 10^{12}, 10^{15}, ...$ and above mentioned equation. How do I solve (i.e. getting values of x for different y) and plot this equation in MATLAB ?
0
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0answers
30 views

Solving equation in maxima not placing variable on one side

I'm trying to solve an equation but the variable ($\varphi$ PHI) will not factor out to one side. Is there any other way to do this? I'm using maxima version 5.32.1 Here's the equation in latex as ...
3
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1answer
29 views

Can someone explain linearisation on nonlinear systems to me?

I want to find all critical points of the following nonlinear system: $$\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}$$ $$\b y_1' \\ y_2'\e = \b 5y_2 -15 \\y_2^2 - y_1 ^2\e$$ Then use linearisation ...
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0answers
41 views

solve complicated system of non-linear equations numerically

I have two algebraic equations I am trying to solve in MAPLE. They are: $14\,{a}^{26}{b}^{2}-91\,{a}^{24}{b}^{4}-364\,{a}^{22}{b}^{6}-1001\,{a} ...
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0answers
18 views

bibliography for weak solutions of ODE's

Some one could recommend to me some bibliography about weak solutions of ODE's, and solutions of ODE's that are not lipschitz??
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0answers
50 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
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0answers
41 views

Invariants of a nonlinear ODE

Given a nonlinear ODE and a simple constraint $x \leq c$ for some constant $c$, how can we describe the largest set (or an approximation thereof) such that if the initial value of the solution of the ...
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0answers
52 views

Non-linear least squares solver to solve a system of non-linear equations?

Can I use a non-linear least squares solver to find the solutions of a system of non-linear equations? From Wikipedia: "The method of least squares is a standard approach to the approximate ...
2
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0answers
54 views

Examples of nonlinear systems that cannot be modeled by multilinear systems?

In university I have learnt about the concept of tensors, which are multilinear maps in that it is a map such that it is linear with respect to all its arguments ...
4
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1answer
135 views

Is chaos theory really a theory? Why not just call it non-linear dynamics?

This may just be semantics, but it's always confused me. What is the thesis of Chaos Theory? I have read an entire book about it, and as far as I can tell, its just a bunch of analytical techniques, ...
1
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1answer
51 views

Lyapunov linearized stability analysis

I have this system: $\dot x=-(x-1)(x-2)^2$ I'm asked to find the equilibria and to study the stability using: i) linearization ii) appropriate Lyapunov function How should I linearize the system? ...