# Questions tagged [nonlinear-dynamics]

This tag is for questions relating to nonlinear-dynamics, the branch of mathematical physics that studies systems governed by equations more complex than the linear, $~aX+b~$ form.

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### Does a spiral fixed point going from stable to unstable always indicate Hopf bifurcation?

Does a spiral fixed point going from stable to unstable (with change in some parameter) always indicate Hopf bifurcation? Could it be a homoclinic bifurcation instead? (Homoclinic bifurcation keeps ...
1 vote
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### Pontryagin Maximum Principle with terminal and initial conditions

Consider a control problem with Lagragian $L(t,x,u)$ (where $u$ is the control, $x \in \mathbb{R}^d$ the state) and dynamics $\dot{x}=f(x,u,t)$. I have mostly seen problems in which the dynamical ...
1 vote
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### Show the system has one equilibrium point

I was wondering how we would show that the system: dx/dt=-x^3+2x-4y dy/dt=-y^3+2y+4x has only one equilibrium point. I have seen cases where the system is, ...
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### Solving steady state diffusion with non-linear decay

I want to solve steady state diffusion with constant production term (in the source $[-L_s, 0]$) and a non-linear degradation term, where degradation takes place over the whole domain $[-L_s, L]$, but ...
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### How to find the correct upper and lower bound of a matrix

I am trying to find a correct upper and lower bound for an equilibrium point of the dynamics of matrix R. The equilibrium point is given as: $$R = ARA' + C$$ and I have been able to solve this ...
1 vote
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### How to render soliton solutions of nonlinear Schrödinger equation exponentially unstable?

The nonlinear Schrödinger equation in $\mathbb{R}^{3+1}$ is $$\phi_t + \nabla^2 \phi + f(|\phi|^2)\phi + V(x)\phi= 0, \quad \phi(x,0) = \phi_0(x).$$ (I require $\phi$ to be square-integrable, and also ...
1 vote
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### nonlinear differential equation for the hypocycloid

I've been trying to tackle a nonlinear first order differential equation that appears when trying to solve the brachistochrone problem through earth's gravitational field. Although being a long and ...
1 vote
36 views

### How Achieve This Specific Non-Linear Mapping

Let's say in a computer program, we have a Slider whereby a user selects a value. The slider widget itself produces a value, X, from 0.0 to 1.0. This value then maps to some other range. Most people ...
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I am interested in ideas for how to compute the maximal Lyapunov exponent of the nonlinear Schrödinger-Newton system given by $$\partial_t s(t, \vec{x}) = L(s(t,\vec{x}))$$ where L(s(t, \vec{x})) ... 0 votes 1 answer 39 views ### Calculate time from one state to another using 6DOF state space modeling of a quadcopter I'm new here. I have the 6DOF state space representation of a quadcopter given below, \begin{align} \dot x(t) &= Ax(t) + Bu(t) \end{align} Where, x(t) = State Vector, u(t) = Input (or control)... 0 votes 1 answer 44 views ### Do some nonlinear PDE steady state solutions depend on initial conditions (non unique)? I was told by a colleague that for some nonlinear PDEs the initial conditions can change the steady-state solution. So can a stable steady solution depend on the initial conditions for nonlinear PDEs? ... 0 votes 0 answers 25 views ### Fourier series for periodic and bounded solutions (very theoretical question - HELP) I have bounded periodic solutions to a non-linear differential equation. Drawing a phase portrait shows that they form closed curves (slightly irregular separate circles) with the center at the origin.... 5 votes 0 answers 192 views ### What are the solutions for y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)? What are the solutions for y(t)\cdot\left(y'(t) + a\right)=-b\sin(t)? It could be proben that there exists some solutions? Are these solutions unique? and obviously, which are these solutions? (... 0 votes 0 answers 22 views ### Approximation of coupled nonlinear system I have the following system and I wish to approximate it analytically:\dot{A}=\frac{P}{4A}(1-A^2)sin(2\chi) \dot{\chi}=\alpha(\frac{1}{A}-\frac{1}{A+1})-0.5\sigma-0.25Pcos(2\chi) And A(t),\... 0 votes 0 answers 44 views ### Non-linear initial value problem Problem Consider the following Initial Value Problem (IVP) \begin{equation*} \begin{aligned} \dot{\xi}(t) &= v(t)\,\cos[h(t)]\\ \dot{v}(t) &= a_{\parallel}\\ \dot{h}(t) &=\omega \end{... 0 votes 0 answers 69 views ### Non-linear differential equations where vector field does not change sign I am working on nonlinear dynamics wherein I have encountered a class of autonomous non-linear dynamical systems where the elements of the underlying Jacobian matrix of the vector field with respect ... 4 votes 0 answers 252 views ### It is possible to find a solution to y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4? It is possible to find an exact solution (hopefully in "close form") toy''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4$$? How?... There ... 0 votes 0 answers 14 views ### Nature of solutions to 2nd order ODE with 2 functions, initial and asymptotic conditions Given the ODE$$(\frac{a''}{a})^2 + 2(\frac{a'b'}{ab})^2 + 2(\frac{b''}{b})^2 + (\frac{1-b'^2}{b^2})^2 + 4\frac{a''a'b'}{a^2b} - 8\frac{a'b'b''}{ab^2} + 2\frac{a''(1-b'^2)}{ab^2} = 0$$with a even, ... 3 votes 1 answer 212 views ### Given (a_n) such that a_1 \in (0,1) and a_{n+1}=a_n+(\frac{a_n}{n})^2. Prove that (a_n) has a finite limit. [duplicate] Given (a_n) such that a_1 \in (0,1) and a_{n+1}=a_n+(\frac{a_n}{n})^2. Prove that (a_n) has a finite limit. Clearly a_n are increasing. Also,$$\frac{1}{a_{n+1}}=\frac{1}{a_n\left(1+\frac{... 