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0
votes
1answer
19 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.
0
votes
0answers
17 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, ...
1
vote
0answers
21 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 ...
0
votes
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 * ...
0
votes
1answer
30 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 ...
2
votes
0answers
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 = ...
0
votes
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 ...
1
vote
0answers
27 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 ...
-2
votes
4answers
66 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: $(a+x)(b+y)=c$ $ab+ay+bx+xy = c$ ...
1
vote
0answers
10 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 ...
0
votes
0answers
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 ...
1
vote
1answer
58 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.
0
votes
1answer
21 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 ...
0
votes
0answers
31 views

Biological modelling math question?

I am trying to write a biological model that models protein interaction. I am having an issue with one aspect. Lets say protein A and protein B interact with eachother to form complex AB. Now every A ...
3
votes
0answers
35 views

Very stupid question about non-linear equations.. :D [closed]

This will get me lots of negatives because I won't be able to explain myself and it is a bit of a personal quandry, but!! :p - Why are non-linear system solutions searched for using a linear ...
0
votes
0answers
15 views

solve a set of linear + non linear equations in binary artihmetic

I have a set of linear and non linear equations and eventually i'd be able to find an algorithm (that i must write in C) that is able to find all possible solutions as fast as possible. These are the ...
1
vote
1answer
26 views

Numerical methods to solve nonlinear system of inequalities?

I know some methods to solve nonlinear system of equaltites? Relaxation Method, Newton method, nonlinear Jacobi method, nonlinear Seidel method. Is it exist some analogous method to solve nonlinear ...
0
votes
0answers
31 views

Blow-Up for Semi-Linear Wave Equation

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
3
votes
1answer
45 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
0
votes
0answers
23 views

Masters thesis topics in iterative learning control (ILC)?

Looking for ideas on thesis topics in iterative learning control or repetitive control. One topic of interest thus is far is ILC of discrete-time nonlinear systems. Any thoughts or recommendations ...
0
votes
1answer
86 views
+200

Method of solving no-homogeneous recurrence equation

I need to obtain a closed form of $M(t)$, satisfying the following recurrence equation: $$M(t+1)=a+bM(t)+\frac{c}{t+1}\sum_{t'=0}^tM(t')+df(t)$$ Where $f(t)$ is a kwonw function and $a$, $b$, $c$ ...
1
vote
0answers
29 views

Existence of periodic orbits (non-linear systems)

I'm trying to solve the following problem: Use the Poincaré-Bendixson's criterion to show that the system has a periodic orbit $$ \dot{x}_1 =x_2 \\ \dot{x}_2=-x_1+x_2-2(x_1+2x_2)x_2^2 $$ The unique ...
3
votes
0answers
78 views

Undergraduate research program (REU) with focus on nonlinear analysis (accessible to European students)

Is there an undergraduate summer research program (REU) -- to which also European students can apply -- which usually has projects (i.e., specialized faculty) on nonlinear analysis (i.e., ...
1
vote
2answers
64 views

Solving a non-linear, multivariable system of equations

I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations: $\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$ ...
1
vote
0answers
20 views

A Domain of Asymptotic Stability

For a system $$x'=-x^4+2x^5+y$$ $$y'=-x^5-y.$$ I found the Liapunov function as $$V(x,y)=(x+y)^2+2((x^5)/5-(x^6)/6)$$. A domain of asymptotic stability will be the interior of the largest level curve ...
0
votes
0answers
28 views

Control theory: Why doesn't the separation principle hold in nonlinear control theory?

It is widely known in control that separation principle is one of the best tool for pole placement and design of stabilizing controller in linear system. Many results also note the inability of ...
0
votes
0answers
17 views

Transfer Function of non-linear systems

I am trying to find an approximate transfer function of the following system using either Laplace or Fourier transform methods $$\frac{dy(t)}{dt} = k_1\times q_0\times x(t)-k1\times x(t)\times ...
0
votes
1answer
62 views

Closed form solution to $\frac{1}{a-1}= \log a$

I want to find a function that satisfies $$\Delta [f(x)]=f'[x]$$ Obviously the solution is the exponential function $f(x)=a^x$ with $a$ in between $2$ and $e$ because $\Delta[2^x]=2^x$ and ...
1
vote
1answer
36 views

What are properties of dynamical systems in non-integer dimension spaces?

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc. A 3D dynamical system ...
1
vote
2answers
43 views

Closed-form solution for 3D rotation angles given pre- and post-image

I'm working on some math involving a pinhole camera model, and I've run in to the following problem: given only $x$, $y$, $z$, $A$, $B$, and $C$, I need to solve for the angles $\theta$ and $\phi$ in ...
1
vote
1answer
26 views

Can an unstable limit cycle be contained directly within a stable one?

Can both the alpha and omega point sets of a trajectory be part of two different limit cycles? I.e. can trajectories being 'repelled' from one limit cycle be pulled into an 'attracting' (stable) limit ...
0
votes
1answer
27 views

Solve nonlinear equation by taking derivatives

I am seeking to solve a set of nonlinear equations analytically. One of them is 1-x/y = f(z) exp(x) Can I get further by applying derivative d/dx to both sides, which gives explicit expression for x ...
5
votes
3answers
62 views

Finding solution of non linear DE $x\ddot{x}-\dot{x}^2=1$

I am looking for help on how to find the solution of the following differential equation, $$x\ddot{x}-\dot{x}^2=1,$$ which comes from solving the Euler-Lagrange equations for the lagrangian ...
1
vote
0answers
105 views

(Still open) Convergence of a certain matrix product representing a class of piecewise linear dynamical systems.

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline ...
2
votes
1answer
55 views

Is it possible to find an analytical solution for “x” in this equation?

In my research I have come across the equation $$\prod_i^n \left( \frac{a_i}{x} \right)^\frac{b_i}{x} = \prod_i^n (1-d_i)^{(1-b_i)c}$$ Is it possible to obtain $x$ from this analytically, or do I ...
0
votes
2answers
55 views

Solve recurrence equation $T(n)=2T(n-1)-4$

I got such recurrence equation which I cannot solve, I tried with mathematical induction, but I've got information, that this one is not linear and cannot be solve like that. And really have no idea ...
0
votes
0answers
21 views

Van der Pol oscillator method of multiple scales boundary conditions

Could someone explain to me where the boundary conditions right at the bottom of page 42 come from? ...
0
votes
1answer
23 views

Linearising thermal radiation

I'm trying to linearise in real time (part of a simulink simulation) radiative resistances. What I've done is the following: Radiative law: $$P=\frac{\Delta T^4}{R_r}$$ At any given timestep, ...
2
votes
0answers
21 views

Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
1
vote
0answers
52 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
1
vote
1answer
44 views

Difference: Newton's method, Newton-Rhapson method, Gauss Newton-method.

I would appreciate some clarification w.r.t. algorithms for solving nonlinear systems of equations. 1 - I don't understand the difference between Newton's method and Newton-Rhapson method. In [1], ...
0
votes
1answer
21 views

funny money exchange

Say a friend agrees to square my pocket change, I have 10 x 10p coins = £1. He gives me 100 x 10p coins, I have £10 Next day I have 100 x 1p coins also = £1, he gives me 100 x 100 x 1p = £100 Why ? ...
2
votes
0answers
48 views

Solving first order non linear ODE

I am trying to solve the following first order non-linear differential equation: $$ \frac{\partial y}{\partial x} = -\sqrt{\frac{2(\sigma + 1)}{\sigma}} \sqrt{-\frac{1}{2y^{2}} + ...
4
votes
3answers
103 views

Why Rotations (in CG) are not linear?

One of my teachers told me that a fundamental problem in Computer Graphics is that the rotations are not linear. The transations and scalar are linear, but not the rotations. He tried to explain to me ...
5
votes
1answer
147 views

Mixed Boolean Arithmetic Identity

I'm trying to prove/derive the equivalence of the following formula: $$ x*y = (x \land y) * (x \lor y) + (x \land \neg y) * (\neg x \land y) $$ whereas $(\land, \lor, \neg)$ correspond to bitwise ...
2
votes
2answers
51 views

Property of the solution for a specific system of non-linear equations.

We are stuck on a proof, and would appreciate any help: Let $\gamma >1$ be a known scalar and let $g,h=1,...,G$ and $s=1,...,S$. Let $\pi _{gs}$, $\beta _{s}$ and $y_{g}$ be known variables with ...
0
votes
0answers
14 views

Non-linear perturbation definition

What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric? I have seen so called "linear perturbations" which look like $$ds^2 = -(1+2\Phi)dt^2 ...
0
votes
0answers
18 views

Algorithm to scan Lyapunov candidate functions

Let's say you are trying to establish the stability of an equilibrium point in a nonlinear system, but you are having trouble coming up with an appropriate Lyapunov function. (Or, you're lazy and ...
1
vote
0answers
36 views

Solving nonlinear differential equation using boundary value at infinity

I want to solve the following differential equation subject to the condition that $f(0)=0$ and $\lim_{x\rightarrow\infty}f(x)=1$. Also $|f| < 1 $ always. Can anybody suggest me a concrete way ...
0
votes
1answer
19 views

Calculating decreased cost with increasing quantity

I have a hand made table I've been using to give customers price per unit on my items, which gives a better price for the more items that they buy. My sample table right now I need to keep the ...