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Questions tagged [non-linear-dynamics]

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Geometric significance of a bifurcation point with algebraic multiplicity $2$?

This is part of Strogatz exercise $3.2.3:$ This is the process by which I found the bifurcation point/points for $\dot x=x-rx(1-x)$: By the method of tangential intersection we have: $$x=rx(1-x)$$ $...
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36 views

Prove that $f$ has a periodic orbit of least period $n$ for each positive integer $n$.

$$f(x)=\begin{cases} 1/2+x&\text{ if }0 \leq x \leq 1/2,\text{ and }\\ 2-2x&\text{ otherwise. } \end{cases}$$ Moreover, let $I_0=[0,1/2]$, $I_1=[1/2,1]$. Prove that for each infinite ...
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1answer
24 views

Nondimensionalization of the logistic equation.

In the process of studying nonlinear dynamics by Strogatz, I saw how he did simplify the model for an insect outbreak with the use of nondimensionalization. So as an exercise I picked an equation and ...
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28 views

Classify the bifurcation that occurs at $\mu$ =0

$ dx/dt=\mu x+y+x^2+x^3 , dy/dt=-x+\mu y+x^2y$ What I have done so far is getting the matrix A with $A_{11}=\mu,A_{12}=1,A_{21}=-1,A_{22}=\mu$ at $(0,0)$.I can see the bifurcation is Hopf ...
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38 views

Lyapunov Stability for a Nonlinear, Time-varying system

I am currently trying to learn how to determine the stability of a solution using Lyapunov's Method for non-autonomous systems. Say we are given a nonlinear system: $$\dot{x_1}(t)=-x_1(t) + x_2(t)[...
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12 views

Nonlinear Gaussian State Space with Linear Observation Matrix Derivation

For the following State Space Model: $x_k = f(x_{k-1}) + v_k ~~ \sim ~ \Bbb N(0_{n_{v \times 1}}, \sum_v) \\ y_k = Cx_{k} + w_k ~~ \sim ~ \Bbb N(0_{n_{w \times 1}}, \sum_w)$ where $f: \Bbb R^{n_{...
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1answer
66 views

Describe discrete-time Langevin dynamics by its stationary distribution

Let's consider stochastic dynamics with discrete time step $t$ and states $x_t\in\mathbb{R}^d$ that evolve as $$ \mathrm{p}(x_{t+1}|x_t)=\mathcal{N}(x_t+f(x_t), \Sigma). $$ Moreover, we assume the ...
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36 views

System of nonlinear ODEs remains in positive cone

Given a system of three first order nonlinear ODEs $ \begin{align*} x_1' &= f(x_1,x_2,x_3) \\ x_2' &= g(x_1,x_2,x_3) \\ x_3' &= h(x_1,x_2,x_3), \end{align*} $ I'm hoping to ...
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1answer
67 views

Simple pendulum: How to solve the following second order nonlinear ODEs?

I'm trying to find equation of motion the following pendulum. To do this, let $$\vec r (t) = (l \sin (\theta (t)), -l \cos(\theta(t))).$$ then $$m D^2 \vec r(t) = ( -l \sin(\theta(t))\cdot \dot{\...
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28 views

Lotka-Volterra equation (predator-prey): given any initial condition, how can one know the steady-state behavior?

I am trying to find out if it is possible to determine the steady state behavior of the predator-prey system defined by the nonlinear equations: $$\begin{eqnarray} \frac{dx}{dt}&=&ax-bxy \\ ...
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1answer
68 views

How to deal with non-equilibrium operating point

Given the nonlinear system $$ \begin{align} \dot{x}_1 &= -4x_1 + 10x_2 + u \\ \dot{x}_2 &= -x_1 - 2x_2 - \log(1 + x_1^2) \\ y &= x_1 + x_2 \end{align} $$ Assume the system should be ...
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44 views

Phase portrait of a nonlinear system

I have an Lotka Volterra type of system as seen below; $$\begin{align} x' &= 10 x y -1/2 x - 1/10 x^2 \\ y' &= y - y^2 -x y \end{align}$$ I would like to plot the phase portrait of the ...
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linearizing dynamics about non fixed point for LQR implementation.

I am trying to implement LQR control for the cart pole system. I am curious if I can maintain a constant non-zero pole angle. So, I need to linearize my dynamics about my goal state. I know we can use ...
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2answers
58 views

Differential equations - why we do care so much about (non)linearity?

This is a very simple question, but I feel I'm missing the bigger picture. Authors will talk of the horrors of nonlinear differential equations and that they're very difficult to solve, but why are ...
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1answer
124 views

Does this dynamical system have another conserved quantity?

For the 3D system of ODEs: $$\begin{eqnarray}\dot{x} &=& -\beta x y \\ \dot{y} &=& \beta x y + \hat{\beta} z y - \delta y \\ \dot{z} &=& -\hat{\beta} z y + \delta y, \end{...
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1answer
56 views

A conserved quantity reduces the dimension of the system?

Suppose $\dot{x} = f(x)$ is a dynamical system on the state space $X$. My notes define a conservative system as one where there exists a (nontrivial) function $H: X \rightarrow \mathbb{R}$ such that $$...
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1answer
61 views

Solving the recurrence relation $y_{n+1}=y_n+a+\frac{b}{y_n}$

I am hoping to obtain a closed-form solution or an asymptotic result for the recurrence relation $$y_{n+1}=y_n+a+\frac{b}{y_n}$$ Any help would be very much appreciated!
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1answer
34 views

logistic map and stable cycle

Show that the logistic map $x_{n+1}=Ax_n(1-x_n)$ has stable $2$-cycle for all $A>3$. I am a newbie in non-linear dynamics and chaos theory. All I have been able to find out about the map is that ...
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1answer
36 views

Rewriting the Landau equation

I read an introduction to Landau equation: $$\frac{\mathrm{d} \vert A\rvert^2}{\mathrm{d}t}=2\sigma \lvert A \rvert^2 - \ell \lvert A \rvert^4.$$ And I encountered a problem on the solution of the ...
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28 views

Increase of free parameters in perturbed solution of an ODE

Lets take the ODE \begin{equation} ODE0: \qquad \frac{\partial y_0}{\partial x} = F(y_0,x) \end{equation} with known solution $y_0(c_0)$, where $c_0$ is (are) the free parameter(s) that I will fix ...
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40 views

To show that a 2D nonlinear ODE undergoes a pitchfork bifurcation

Show that a pitchfork bifurcation occurs when $\mu = 0$ for the system $$\dot{x} = \mu x + xy + 3y^2$$ $$\dot{y} = -2y + x^2 + 2xy^2$$. Attempt at a solution If $\dot{x} = 0$ then $$x = \frac{-3y}{\...
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1answer
34 views

To show that $\dot{x} = -y - x +y^2 - x^2,\; \dot{y} = xy$ has no periodic orbits.

I've tried using index theory, but there's a non-isolated fixed point at $(0, 0)$ (the remaining fixed points at $(0, 1)$ and $(-1, 0)$ are saddles). The terms in the equations have even indices and ...
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108 views

Proof check: Show that $x_{n + 1} = f(x_n) = 15 - x_n^2$ is a (Smale) horseshoe

A solution is given here: However, I believe it to be incorrect because $f(K_1) = f(K_2) \neq (-\sqrt{15}, \sqrt{15})$. Attempt at a solution: Referring to the diagram from the question... By the ...
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28 views

To classify bifurcations by calculating partial derivatives evaluated at the bifurcation point

Question: Given a one dimensional system $\dot{x} = f(x;\; \mu)$ for which a bifurcation of fixed points occurs at $(x^\star, \mu_c)$, where $x^\star$ is a fixed point and $\mu_c$ is the ...
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53 views

How to numerically solve this nonlinear differential equation with partial initial boundary condition and partial terminal boundary condition?

Let the two states be $\lambda:[0,T]\rightarrow \mathbb{R}^+$ and $p:[0,T]\rightarrow\mathbb{R}^{++}$. $b$ is a constant which is positive. We have $$ \begin{align} \dot{\lambda} &=-\frac{b}{\...
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Why are incremental dynamics written with force at current time instead of previous?

I am working on a question from the Stanford archive of EE 263: https://see.stanford.edu/Course/EE263/62. It is problem 6.2, shown here one page 31 https://see.stanford.edu/materials/lsoeldsee263/...
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16 views

Reynolds decomposition for nonlinear dynamics

Can we apply the Reynolds decomposition u(x,t)=U(x)+u'(x,t) to any strongly non-linear dynamics problem, where the final state is dependent on the initial condition ...
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144 views

Nonlinear state space model example

Is there any (preferably well-known) example of a nonlinear state space model of a system in the following form? $x(k+1)=Ax(k)+f(x(k))+u(k)\\ y(k)=Cx(k)$ where $f(\cdot)$ is a nonlinear function of ...
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77 views

Stability of Polynomial Planar Systems (differential equations)

Consider the general system of differential equations $\dot{x}=f(x,y)$ and $\dot{y}=g(x,y)$ where $f$ and $y$ are polynomial functions and $f(0,0)=g(0,0)=0$. What are the conditions on $f$ and $g$ ...
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84 views

Finding a Strict Lyapunov Function for $\ddot{x} + \dot{x} + x^3 = 0$

Consider the system $\dot{x}=y$ and $\dot{y}=-x^3-y$. We know that the function $V(x,y)=0.5y^2+0.25x^4$ is a Lyapunov function for the system and thus, the system is stable. However, we know that the ...
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2answers
130 views

Show that the origin of the following system is stable (non-linear vs linear stability)

$$\dot{x} = -2y -x^3 + x^2 y^2$$ $$\dot{y} = x - x^2 y$$ Jacobian evaluation at $(0, 0)$: $$\begin{pmatrix}0 & -2\\ 1 & 0\end{pmatrix}$$ Clearly the determinant is greater than zero but the ...
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34 views

How to express some coefficient of a generating function using only its derivatives

I have a series of terms, $a_{1},a_{2},...,a_{k-1}$ that depend on $t$. I describe them in the form of the generating function, $Q(x,t)=\sum_{i=1}^{k-1}a_{i}(t)x^{i}$. My goal is to end up with a ...
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26 views

determine resonance criterium by energy method of forced vibration in a mass spring sys

How to use energy method to find out that the applied frequency should be equal to the natural frequency when the sys is at resonance?
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46 views

Generalisation of Index of a curve to higher dimensions

Im studying Non Linear Dynamics and Chaos from Strogatz's textbook. In the sixth chapter, while talking about non linear flows in 2 dimensions he introduces the index of a curve in a vector field and ...
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2answers
136 views

Nonlinear dynamics book for self study

I have to learn basic nonlinear dynamics on my own for a project, and I'm trying to find a text that could serve the purpose. I'm in search of something not that cryptical, maybe easy to read, but at ...
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0answers
75 views

Solving Finite Difference Equation using Matlab

From the original Equation of Motion of an Inverted Pendulum: $$ 0 = \ddot\theta - \frac{g}{l} \sin\theta + \frac{1}{2ml}\cdot C_D\rho A (\dot\theta\cdot l)^2\cdot\operatorname{sign} (\dot\theta) + \...
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1answer
41 views

Finite Difference Equation From a Non-Linear Equation

Given a Non-Linear Equation that is: $$I\ddot\theta = mgl \cdot \sin \theta + F_D \cdot l + k\theta $$ Where, $$F_D$$ is representative of Drag Force and is equal to: $$-1/2C_D\rho Av^2\cdot \...
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Persistence of elliptic / parabolic fixed points of maps and flows

For an ode $\dot{x} = f(x)$, it is known that if an equilibrium $x^{*}$ is hyperbolic (none of eigenvalues of $Df(x^{*})$ have zero real part), then it persists for small $O(\epsilon)$ perturbations ...
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1answer
100 views

Nonlinear Dynamics: How to tell the stability of fixed points?

I understand how to find the stability of fixed points by using $f'(x)$ and inserting the values of the fixed points into that. However, I can't figure out how to tell the stability in systems such as ...
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1answer
43 views

Generalizing the estimate of norm of solution of an ODE to Banach spaces

I am reading Hale's Oscillations in Nonlinear Systems, and I want to generalize some results to Banach spaces. However, I cannot find a way to generalize the following part (which is in the ...
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1answer
46 views

Class K infinity function

A continuous function $\alpha:[0,\infty)\to [0,\infty)$, is said to belong to class $\mathcal{K}$ if it is strictly increasing $\alpha(0) = 0$ and $\alpha(t) \to \infty$ as $t\to\infty$. Let $s,r$ be ...
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1answer
49 views

Feedback linearizion for input-output linearizion - Lie Derivatives

Short introduction to feedback linearizion: If we got a nonlinear system: $$\dot x_1 = x_2$$ $$\dot x_2 = a x_1 ^2 + bx_1 + c x_2 + u$$ and we want to have state feedback by using feedback ...
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2answers
40 views

Methods for solving non-linear ode

I had the second order non-linear ODE $$f''\left(t\right)f'\left(t\right)=f\left(t\right)^{2}$$ And I managed to reduce it to a first order ODE $$f'\left(t\right)=\left(C^3+f\left(t\right)^{3}\right)^{...
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64 views

Integral of second moment

I am trying to solve the function $$\frac{\partial \left \langle f^{2} \right \rangle}{\partial t} = \frac{\left \langle f \right \rangle - \left \langle f^{2} \right \rangle} {N}$$ where $\left \...
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6 views

Solving Inverse Kinematics Equations into Approximate Expansion with Higher Order Terms

I have two sets of kinematics equations for a physical system, the first is: $a = \phi_1 - \phi_2 - tan^{-1}(\frac{R_{1}cos(\phi_1) - y_1}{R_{1}sin(\phi_1)-x_1}) + tan^{-1}(\frac{R_{1}cos(\phi_2) - ...
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1answer
93 views

Problems to understand Lyapunov stability - Nonlinear Control

I'm learning nonlinear control and I have already learn how to do phase plots. It was not a big deal. Just using ode45 in Octave/Matlab. But when I going to learn something, I only focus on practical ...
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0answers
39 views

normal form of 2 dimensional nonlinear maps with 2 parameter family

(My first time posting.) Consider a 2-d map: $x_{n+1} = y_n$ $y_{n+1} = \mu_1 y_n + \mu_2 - x_n^2$ I was asked to find what parameter will give saddle-node, period-doubling and Naimark-Sacker ...
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0answers
77 views

Quadratic dynamical system (Solution to the Initial Value Problem)

I'm interested in a solution to the initial value problem of the following quadratic dynamical system with 2 states $\bigl(x(t), y(t)\bigl)$: $$ \begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt}...
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1answer
76 views

Absolute stability for System with nonlinear output function?

Given a nonlinear system like $$ \begin{split} \dot{x} &= Ax - B\phi(y)\\ y &= C^T x. \end{split} \tag{1} $$ If the nonlinear function $\phi$ fulfills the well known sector conditons on ...
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0answers
28 views

Newhouse Ruelle Takens theorem and three frequency quasiperiodicty

NRT theorem suggests that there exist arbitrarily small perturbations of a vector field on an $m$-torus for $m\geq3$ leading to strange axiom A attractors. Although, three frequency quasi periodic ...