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Questions tagged [non-linear-dynamics]

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Bounce distance off wall

A ball launched at speed V and angle alpha to the horizontal bounces off a vertical wall at a horizontal distance d. The coefficient of restitution is r. How can I calculate how far horizontally from ...
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Limit Map of Discrete Dynamical System

Let $f:H\rightarrow H$ be a countinuous map from the separable hilbert space into itself, for every $x\in H$ define the discrete dynamical system $$ \xi_x^{n+1}\triangleq f(\xi^n_x);\qquad \xi^0_x\...
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1answer
29 views

Variation of the Diffusion Equation

Let $\mathcal{L}=D\dfrac{\partial^{2}}{\partial x^{2}}-v\dfrac{\partial}{\partial x}+\beta$ be a differential operator describing diffusion ($D$) with drift ($v$) and a source ($\beta$). As part of a ...
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31 views

tracking error state space, non-linear control example

I am trying to understand an example from [1]. In detail I do not understand how the equation for the dynamic of the tracking error is chosen. I am not a mathematician so please forgive me if I may ...
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Lyapunov Indirect Method

I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium ...
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36 views

Partial differential equations that involve an infinite “continuum” of variables: “Each point in space contributes additional degrees of freedom”?

Page 11, Nonlinear Dynamics and Chaos, by Strogatz, says the following: This is the domain of classical applied mathematics and mathematical physics where the linear partial differential equations ...
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1answer
41 views

Relating the Nonlinear and Linear operators in the Homotopy Analysis Method

The question refers to chapter two of the book Liao, Shijun. Homotopy analysis method in nonlinear differential equations. Beijing: Higher Education Press, 2012. Link to book pdf from Chinese .edu ...
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38 views

How to calculate the averaged equations for the weakly nonlinear oscillator $\ddot x+x+\varepsilon (x\dot x^2)=0$?

This is Strogatz exercise $7.6.5:$ For the system $\ddot x+x+\varepsilon h(x,\dot x)=0$, where $h(x,\dot x)=x\dot x^2$ with $0 < ε << 1$, calculate the averaged equations and if possible, ...
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50 views

How to solve this non-linear second order difference equation

Is there a direct method to analytically solve this non linear second order difference equation problem ? \begin{equation} \begin{split} \left \{ \begin{array}{ll} \frac{\beta\nu}{2\sqrt{\Phi_{t+1}}...
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40 views

Nonlinear system of equation in real life

Good night (from my country) for all Mathematicians here. Last week my teacher gave me an indivudal assignment to do during the holidays. He asked us to provide either nonlinear or nonlinear system of ...
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64 views

Improved model of a fishery: $\dot N=rN(1-\frac{N}{K})-H\frac{N}{A+N}$

Strogatz exercise $3.7.4.a:$ An improved model of a fishery is: $$\dot N=rN\left(1-\frac{N}{K}\right)-H\frac{N}{A+N}.$$ a) Give a biological interpretation of the parameter $A$; what does it ...
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Fundamental matrix of ODE system $\dot{x} = (At+B)x$

Suppose I have a system of ODEs $\dot{x} = (At+B)x$, where $x(t)$ is a, say, $n \times 1$ vector, and $A$ and $B$ are constant $n \times n$ matrices. What is the fundamental matrix of this system? I ...
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33 views

Solve sine and exponential nonlinear differential equation?

Is it possible to solve this kind of differential equation with forward Euler? $$\ddot y^2 + sin(\ddot y ) + \dot y + y = u$$ I haven't even write this ODE on the first order form. If I would do ...
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29 views

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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1answer
35 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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37 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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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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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
74 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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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
75 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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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
75 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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45 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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22 views

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
62 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
125 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
62 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
63 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
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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
38 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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33 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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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
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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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121 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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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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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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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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160 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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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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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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133 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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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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1answer
27 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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1answer
51 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
193 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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85 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
43 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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0answers
47 views

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 ...