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Questions tagged [nonlinear-system]

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 one.

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Seeking a closed-form solution of a system of nonlinear trigonometric equations (converting Euler angles to spherical)

Following this and this questions, I want to solve the system of nonlinear trigonometrical equations (as part of an inverse kinematics calculation): $$\begin{align} \sin{\phi} \sin{\gamma} &= \...
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Decomposing a symmetric matrix as a sum of nilpotent matrices

Assume that a real-valued symmetric matrix $M$ with trace zero can be written as $$ M = A + A^T, $$ with $A^2=0$. Given that $M$ is known, how (if possible) can $A$ be found? The diagonal elements ...
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Finding roots of a non linear equation

Can the roots of the following equation can be found analytically? $$f(x)=-x+x^2+5sin(x)$$ I'm of course referring to the non trivial one (besides $x=0$).
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Writing coupled system of non-linear ODE's as a coupled system of nonlinear first order ODEs

Below I am given a coupled system of nonlinear ODEs: I first considered equation $(1)$, and I defined some new functions. $x_1 = h \implies x_1' = h' = x_2$ $x_2 = h' \implies x_2' = h'' = x_3$ $...
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45 views

Existence of periodic solutions.

I have the dynamical system $$\left\{ \begin{array}{c l} \dot{x} &= y\\ \dot{y} &= -x-y-y^3 \end{array}\right.$$ And I want to show that there exists no periodic solutions. ...
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Solution of the Lotka Volterra system of differential equations in $\mathbb{R^2}$ is cyclic

I'm currently reading an abstract about the Lotka Volterra equations and I've some questions. Please be aware I'm an amateur. First I will give you the system of differential equations: $$ x^{'} = ...
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50 views

An first integral of nonlinear differential equation as like forced pendulum nonlinear diff. eq.

I'm trying to face this nonlinear differential equation: $$ y''(x)+\omega^2\sin\,y(x)=a\,x \,\;(1)$$ and I'm interested to found the solution of $ y'(x)$ (an first integral) The homogeneous part of ...
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Proving that a pair of non-linear equations has a unique solution

Assuming that we have a pair of equations with two unknowns $a$ and $b$: \begin{cases} f(a,b) = 0 \\ g(a,b) = 0 \end{cases} What kind of conditions are needed to show that the solution to this pair is ...
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70 views

Numerical solution to a non-linear PDE

I have this Non-linear PDE $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ Where C is a function of (x,t) It comes from the ...
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Show whether the following function is radially bounded or radially unbounded: [closed]

my function is V(x) = (Sin(x))^2 Please give me an appropriate answer
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30 views

Non linear system

Anyone can help me with this system? $$(I)\ 2x\sin(\theta)\cos(\theta) - 4x\sin(\theta) + 10\sin(\theta) = 0$$ $$(II)\ 10x\cos(\theta) - 2x^2\cos(\theta) + x^2(\cos^2(\theta) - \sin^2(\theta)) = 0$$ ...
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nonlinear differential equations

I am interested in solving coupled differential equations with the following shape: \begin{equation} \ddot y_p(x) =\sum_{lmn=1}^N C(l,m,n,p)y_l (x)y_m (x)y_n(x) \end{equation} where $p=1\cdots N$. ...
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Phase trajectory must always enclose a fixed point

I found this problem in strogatz nonlinear dynamics . The theorem says, A closed phase space trahectory must enclose a fixed point . The question is asked as, is this true for phase surfaces other ...
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Jacobian of a Fixed Point Method

I am trying to determine the Jacobian of a multi-dimensional fixed point method with the following form $\begin{equation} x = -c(x)Ax + Bx, \end{equation}$ where $A$ and $B$ are square matrices (not ...
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Four equations with five unknown and the Wolframalpha can solve them

$$a+b+c=1\tag 1$$ $$a*k_1+b*k_2+c=1\tag2$$ $$2a*k_1^2+2b*k_2^2+2c=1\tag3$$ $$3a*k_1^3+3b*k_2^3+3c=1\tag4$$ the Wolframalpha gives the solution as follow: How did this happen?????
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Linearization of Partial Differential Equation

I have this beautiful Non-linear PDE $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ Where C is a function of (x,t) It comes from ...
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Conditions for the solution of the ODE $\dot{x}(t) = f(x(t))$ continuous on $t \in [0, +\infty)$?

Given the ODE: \begin{align} \dot{x}(t) = f(x(t)) \end{align} It is known that the solution $x(t)$ is continuously differentiable on a compact interval $t \in [0, t_{1}]$ when $f(x(t))$ is ...
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Is there any way to solve the following system of non linear equations

$$a\,(k_1-1)+b\,(k_2-1)+c\,(k_3-1)=-1/2$$ $$a\,(k_1-k_1^2)+b\,(k_2-k_2^2)+c\,(k_3-k_3^2)=1/6$$ $$a\,(k_1-k_1^3)+b\,(k_2-k_2^3)+c\,(k_3-k_3^3)=2/8$$ $$a\,(k_1-k_1^4)+b\,(k_2-k_2^4)+c\,(k_3-k_3^4)=3/10$$...
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Solve non-linear boundary-value-problem with finite difference method (FDM)

Given a boundary-value-problem $u''(x) = (10^{-7}u(x) + 10^{-8}x(x-2))\cdot (1 + u'(x)^2)^\frac{3}{2}$ with conditions $u(0) = 0$ and $u(2) = 0$ and formulars for first and second derivation: $u''(...
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84 views

How to solve a system of nonlinear equations containing logarithm

I need to solve the following system of equations for $x_1,x_2,x_3,x_4,x_5,x_6$ $$x_1= 0.64\times x_2 + 0.64\times x_3$$ $$x_5\times (x_2)^2= 0.392 - 1.25\times x_4 \times (x_1)^2$$ $$x_5\times(x_2)^2=...
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What is the solution of the equation $xyp^2 + (3x^2 - 2y^2)p - 6xy=0$, where $p = \frac{dx}{dy}$

What is the solution of the equation $xyp^2 + (3x^2 - 2y^2)p - 6xy=0$, where $p = \frac{dx}{dy}$ I was trying to solve it by dividing the whole equation by $xy$ and then integrate it $\frac{dy}{dx}...
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How to linearize this nonlinear system?

Take a look at this nonlinear system $$ \dddot{x} +4\ddot{x}+24|\dot{x}| + 5\cos(x)|\dot{x}| + 50x = u $$ The objective is to linearize the system about the equilibrium point. First, we compute the ...
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Chaos and utilization of complex systems

In reliability engineering we deal with complex systems we do not fully understand. Usually we understand the linear functioning of the system at low utilization, but as utilization increases some non-...
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45 views

Is chaos a topological property for continuous dynamical systems?

Following the definition of chaos given by Devaney, a continuous map $f$ on $(X,d)$ separable metric space with no isolated point is said chaotic if it is topologically transitive, that is for any ...
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1answer
45 views

Stability of equilibrium of a nonlinear system of ODE's

Suppose we have the nonlinear system of ODE's $$\begin{cases} \dot{x_1} = -\beta x_1 x_2 \\ \dot{x_2} = \beta x_1 x_2 - \gamma x_2 \end{cases} $$ Where we take $\beta, \gamma > 0$ arbitrary for ...
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What is the necessary and sufficient condition for this nonlinear system of equations to have nonzero solution?

Let $f : \mathbb C^n \to \mathbb C^n$ be a function defined by \begin{align} f(z) :=\phantom{{}}& A^H \mathrm{Diag}((Az) \circ (\overline{Bz})) Bz - A^H \mathrm{Diag}(b) Bz + \\ &\phantom{{}} ...
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What is the source of symmetry in this recurrence plot of the Circle map?

I was reading about the Circle map in this Wikipedia entry. This is a dynamical map with a single variable $\theta$, and dynamics defined by $$\theta_{n+1}=\theta_n + \Omega -\frac{K}{2\pi} \sin (2\...
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1answer
25 views

Existence of solution of nonlinear system

I have a system of non-linear equation such that $$f(x,y,z; \alpha, \beta) = 0,$$ $$g(x,y,z; \alpha, \beta) = 0,$$ $$h(x,y,z; \alpha, \beta) = 0,$$ where $\alpha \in [0,1]$, and $\beta \in (0,\infty)$...
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Solve $\ln(1-x)-\ln(x)+\frac{a}{x}=c$ for $x$

Is there an analytical (maybe involving special functions) solution of an equation of the form: $$\ln(1-x)-\ln(x)+\frac{a}{x}=c$$ Here I want to solve for $x$, which should satisfy $0\le x\le1$, and ...
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Derive transfer function of an unstable plant [duplicate]

Is there a method to mathematically estimate/model the transfer function of a plant around an unstable equilibrium point? A constant step input causes the plant output to increase exponentially with ...
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Algorithm to solve large number of transcendental equations as quickly as possible?

I'm solving a large number of (multivariate) non-linear equations with the following properties: The roots of the variables are real. The equations are transcendental, but are otherwise linear. (All ...
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Transforming nonlinear equations into linear equations

I recall a guest lecture in school in which my class was shown that a large class of nonlinear differential equations can be transformed into linear equations by defining new and more variables. I am ...
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26 views

Need help solving a system of nonlinear equations - SIRS model

I am working on a research paper and developing a modified SIRS model with the following equations. S˙ = −αS − β(1 − ξ)SA − βξSP + εP + δR + µ(P + R) + µ*A P˙ = αS − (ε + γ + µ)P A˙ = γP + σR + β(...
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1answer
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Discrete maps: Stability of periodic orbits

Suppose we have a discrete map of the form $$x_{n+1} = f(x_n) \qquad x \in \Bbb R$$ where $f:\Bbb R\rightarrow \Bbb R$. Then by definition, a $2$-cycle is a periodic orbit $\{y_1,y_2\}$ with $y_1 \...
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Finding equilibrium solutions of a nonlinear system of differential equations

I am practicing for exams by going over problems in the book, but I stumbled upon a problem that I can't seem to solve: $$\frac{dx}{dt}=gz-hx\qquad \frac{dy}{dt}=\frac{c}{a+bx}-ky\qquad \frac{dz}{dt}=...
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1answer
39 views

Nonlinear map having conserved quantity

I am reading the following discussion:                           Does this simple 2D dynamical system have a conserved quantity?    Does this dynamical system have another conserved quantity? ...
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Finding the stable, unstable, center manifolds of a nonlinear system

Question: Consider the system \begin{align} \frac{dx}{dt} & = x-xy \\ \frac{dy}{dt} & = 2x-3y+y^2 \end{align} Find the stable, unstable, center manifolds about origin $(0,0)$ up to and ...
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Does $\square u \pm |u|^pu = 0$ always depend on sign for all $p$?

I remember reading somewhere that the general behavior of nonlinear equations $\square u \pm |u|^pu = 0$ always depend on the sign, that a $+$ sign makes the operator scatter initial data, and the $-$ ...
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SI model analysis

We have an SI model with susceptible individuals $$\frac {dS}{dt} = bM\left(1 - \frac M K\right) - \mu S - \beta SI$$ and infected individuals $$ \frac {dI}{dt} = \beta SI - (\alpha+\mu)I.$$ In this ...
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Variable degree polynomials - Is this a valid solution?

$$ \begin{matrix} a_1 \left(-b^{m-1}+b^{m-3}-b^{m-4}+b^3-b^2+1\right) \\ +a_2 \left(b^{m-1}-2b^{m-2}+2b-1\right) \end{matrix} = -2b^{m-1} + 2b^{m-2} + b^{m-4} + b^4 -b^3 -2b +1. $$ is my equation to ...
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The level sets of integral are invariant sets (Wiggins' textbook)

I am reading the following book: Introduction to applied nonlinear dynamical systems and chaos, Stephen Wiggins On p. 77, for a general vector field $$\dot{x} =...
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1answer
48 views

Solving non-linear equations of four variables.

I am not sure if I can find explicit expressions for a, b, c and d (all of these positive)from the following system of equations: $ a \log(2b) -\frac{a}{2} \log((1 +2b)^2+4)- (2c)^d-\log(5) =0$ $ a \...
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Bifurcation analysis, limit cycle collapses on two symmetric fixed points

Coming back on the system I already mentioned in another post, this time I am working on some bifurcation analysis of a 2D System. The system is defined by the following equations. I am assuming $\...
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How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
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Proving a property of a solution to a set of nonlinear polynomial equations

Consider the following system of equations for $R_{i}(\lambda)$ \begin{align} R_1 &= \frac{\lambda}{4}(1 + R_3 + 2 R_2 R_1) \tag{1.1}\\ R_2 &= \lambda \left[q + \left(\frac{1}{2} - q\right)...
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1answer
29 views

Plotting an iterative system of nonlinear equations using MATLAB

Consider the following coupled system: Let $x(n) = \left[ \begin{array}{c} x_1(n)\\ . \\ .\\ .\\ x_{32}(n) \end{array} \right]$, and the system of $32$ first order nonlinear ...
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1answer
16 views

Test if a function given as a non-integrable ode set is Bijective

Given that state space trajectories of an autonomous system do not cross, can I deduce that a mapping function f:(x,y)→(x',y') given by a solution of an ODE of an autonomous system is bijective? ...
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21 views

Revisit “example of an unstable fixed point for which the linearized dynamics are stable”

I am reading the following discussion: example of an unstable fixed point for which the linearized dynamics are stable The above discussion is for the vector field (continuous time). Is there an ...
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1answer
22 views

stability of $(0,0)$ for $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$

Given the system $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$. For the fixed point $(0,0)$ I can see through linearisation that the flow corresponds to a centre which moves anticlockwise Why is ...
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how to find form of $h(x)$ in reduction to centre manifold

given the system $\dot{x} = y - x - x^2$ and $\dot{y} = x - y - y^2$ I can find that the centre subspace is spanned by $E^c = [1,1]^T$ It says the centre manifold can be expressed as $y=h(x) = x + ...