Questions tagged [stability-theory]

Stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.

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Burger's equation: explicit finite difference method withtout using the Hopf-Cole transformation

Consider the one-dimensional viscous Burger's equation, $$ u_t+uu_x-\nu u_{xx}=0 \tag{1} $$ In order to solve it numerically using the explicit method of finite differences, one can transform the ...
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Assumption of continuously differentiable function in the Lyapunov Stability Criterion

According to the proof of Lyapnuov's theorem given in [1] the assumption of continuity of partial derivatives is necessary to prove asymptotic stability while for simple stability it is not. I wonder ...
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What is the relationship between zero dynamic stability, closed loop stability, and open loop stability? [closed]

What is the relationship between the size of the three stable boundaries?
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Structural stability of Arnold's Cat Map

In Wikipedia it says that "hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable", so I am trying to understand why this is true. What $C^1$-small ...
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Examples of dynamical systems that have structural stability

I am looking for simple examples of structural stability, I read the definition of structural stability but couldn't figure out a concrete example of a system, its perturbated version and its ...
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Deduce stability from strict convexity of gradient systems

Given a twice differentiable function $f(x):\mathbb R^n\rightarrow \mathbb R$, and its corresponding gradient system $$\dot x=-\nabla f(x)$$ My question is: If $f(x)$ is strictly convex, can we ...
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Why an LTI system with some zero eigenvalues still stable?

The textbook says an LTI system $\dot x=Ax$ is stable if and only if the eigenvalues of $A$ have the strictly negative real part. However, I found a counterexample. If $$A= \begin{bmatrix}-3 & -1 ...
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LQR definitions

I have to define the choice of parameters I have chosen to create an LQR controller for a drone, and I have written the following: High penalties in the Q matrix mean that the state will try to ...
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Acoustic wave equation will be numerically unstable anyway?

Since acoustic wave is longitudinal, its equation is exhibited nonlinear. In particular, I derived an acoustic wave equation within uneven, varying pipe. It goes: $$ m_{tt} (m_x)^2 -2 m_{tx}m_tm_x + ...
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Preserving Hurwitz stability of block matrices with same eigenvalues of each block

Let us assume that $$A=-\text{diag}(d_1,\ldots,d_m) +Q \text{ is Hurwitz},$$ where $d_i>0, \forall i\in \{1,\ldots,m\}$ and $Q\in \mathbb R^{m \times m}$ is a possibly full matrix. Can we deduce ...
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Stability of a system of two second order, linear difference equations

I have a system of two linear difference equations in the form: $$ x_{1,t+2}=-Ax_{1t}-Bx_{1,t+1}-Cx_{2t}-Dx_{2,t+1}+Ex_{2,t+2}+F $$ $$ x_{2,t+2}=Gx_{2t}-Hx_{2,t+1}+Ix_{1t}+Jx_{1,t+1}-Kx_{1,t+2}+L $$ ...
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Stabilizing a transfer function With a PD controller

I have the following Plant function: $$ P(s) = \frac{50(s-1)}{(s-5)(s+5)} $$ And a controller $ C(s) = K_p + sK_d$ I want to Check if I can stabilize the closed loop system with this controller. My ...
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If steady states of a dynamic system exist only as limits, are they actually steady states?

I have a nonlinear dynamic model in discrete time. A simplified version of my dynamic system is: \begin{equation} x_{t+1} = \frac{1}{1 + \exp(f(x_t))} \end{equation} where $$f(x_t) = −\beta \left(2d \...
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Are these steady states of non linear dynamic system actually steady states?

I have the following non linear dynamic system in discrete time: \begin{equation} x_{t+1} = \frac{1}{1 + \exp\left(- \beta \left( 2 d \left(c + \frac{(1 - c)}{1 + a (1 - x_{t}) d}\right) - b - d \...
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Is this formula stable? $\frac{|x|-|y|}{x-y}$ as $x$ approaches $y$.

I want to analyze the stability of this formula $\frac{|x|-|y|}{x-y}$ as $x$ approaches $y$. But this formula is not a recursion! I used to analyze the stability of recursion by computing the first n ...
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Proof for stability of stationary point

Given a twice continuously differentiable function $f$ used for a difference equation $x_{n+1} = f(x_n)$, we can show that a stationary point $f(s) = s$ is asymptotically stable (see e.g. here for ...
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Question about the proof of zubov equation

I am reading section 34 (Zubov's method) in the book named stability of motion written by Wolfgang Hahn. The proof of theorem 34.1 seems only proves one side: $A$ is a subset of domain of attraction. ...
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Finding the CFL condition of second order $u_j^{n+1} =u_j^n +aH(u_{j-1}^n)+bH(u_j^n)+cH(u_{j+1}^n)$ with $u_t=H(u)_{xx}$ and $0\le H'(u)\le d$.

We have the following partial differential equation $$u_t =H(u)_{xx},~~~ 0\le x<1$$ with an initial condition $u(x,0) = f(x)$ and periodic boundary condition. Here $0 \le H'(u) \le d$. Consider the ...
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Homoclinic and heteroclinic orbit of a system of first order ODEs

I want to check if there exists any homoclinic or heteroclinic orbits of a system. I have the following system of first order ODEs $$u'=w\\v'=z\\w'=\frac{\beta}{{\delta}}uv-\frac{c}{{\delta}}w\\z'=-\...
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construct different ode systems but with the same lyapunov function

I am thinking of whether there are some ode systems that are different with each other, suppose all of them have zero as an equilibrium point. Moreover, they have a common lyapunov function that can ...
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Asymptotic stability of a unrelated equation

I am studying stability of dynamical systems and Lyapunov theory, and I am trying to solving the following exercise: Provide sufficient conditions for the asymptotic stability of the dynamical systems ...
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a question regarding subsets of basin of attraction

I am reading the review paper named review on computational methods for lyapunov functions which can be seen here My question regarding the lower part of page 4 in the paper, which specifies certain ...
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Conditions on $g$ so that $x=0$ is stable

Consider the following recurrence: $$x_{n+1} = x_n g(x_n), \qquad n \geq 0$$ where $g : [0, \infty) \rightarrow [0, \infty)$ is strictly decreasing. I want to set some necessary and sufficient ...
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Show uniform stability of the system

the LTV system $\dot x(t)=A(t)x(t)$ is called uniformly stable if $\exists \gamma>0$ such that $\left\| {\Phi \left( {t,{t_0}} \right)} \right\| \leqslant \gamma $ for all $t\ge t_0$ where ${\Phi \...
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Understanding Fourier stability analysis of leap-frog scheme in Morton Book

I try to understand the Fourier stability analysis for Leap-Frog scheme to solve linear advection equation, $$\dfrac{\partial u}{\partial t}+a\dfrac{\partial u}{\partial x}=0.$$ Above screenshot is ...
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Lyapunov function to prove globally asymptotically stable

I have the system $x'=-x^3+2y^3$ and $y'=-2xy^2$. I need to prove that the point $(0,0)$ is asymptotically globally stable. Here's what I did: if we have a Lyapunov function $v(x,y)=ax^2+bxy+cy^2$, ...
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Numerical stability of Mehod of lines for Wave equation

Hello mathematicians, I focus on the numerical solution of partial differential equations (PDEs). Recently, I was studying the numerical stability of method of lines (MOL). Method of lines transforms ...
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Why are repeated poles at the origin regarded as unstable?

I thought you needed the poles and zeroes to be at the right hand side to make the system unstable. Therefore here is the question in two parts: Why are repeated poles unstable at the origin? What ...
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Stuck with a stability analysis and order 4 polynomial

Long story short, I'm simulating the behavior of a ropes for fun. I have a discrete model of a mass/spring/damper network that I've made and runs discrete simulations on it, no solver involved. ...
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Is it possible to exist stable subset of equilibrium in a continuum of equilibria?

If there exists a continuum of equilibria (see black curves in the following figure) of a dynamical system, is it possible that in this set of equilibria, there exists isolated asymptotic stable ...
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stbility if a pair of matrices

We know that a matrix $A$ is positive stable if the real part of its eigenvalues is positive. Also, by Lyapunov Theorem, A is positive stable iff there exists a positive definite matrix $P$ such that ...
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stability and fixed points of $y_{i+1} = 0.5(a_{i+1}/b_{i+1})y_{i} + 0.5(y_{i} + (b_{i+1}-a_{i+1})c)$?

I have a data set of a time series, and determined that the data fits this equation, where $y_{n}$ is the dependent variable, $a$ and $b$ are independent variables and $c$ is a constant $y_{i+1} = 0.5(...
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Lyapunov function for a second order system involving trigonometric functions

I am studying the stability of the following system: \begin{aligned} \dot{x}_{1} &= -x_{1}^{2} - \sin x_{2}\\ \dot{x}_{2} &= x_{1} - \frac{\cos x_{2}}{x_{1}}\\ \end{aligned} The system ...
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Unstable forward difference scheme with cross products

Let $u(t) = (u_1(t), u_2(t), u_3(t))$ be a solution of the ODE $$\frac{d\mathbf{u}}{dt}=\mathbf{a}\times\mathbf{u}.$$ where $\times$ denotes the cross product and $\mathbf{a} = (a_1, a_2, a_3) \neq 0$....
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How to pick a Lyapunov function and estimate PID gains? [closed]

I am currently trying to estimate the range of PID gains by developing a Lyapunov function for a nonlinear 6-Dof quadrotor system. The system is of the following form: $$M(q)\ddot{q}+C(q,\dot q)\dot q+...
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Is this 3-equation nonlinear system stable?

I have a system of equations such that $F:\mathbb{R}^{3} \to \mathbb{R}^{3}$. It can be described as follows: $e_{r,t}=\frac{Ae_{r,t-1}-B}{[A(\lambda_{t}e_{r,t-1}+(1-\lambda_{t})e_{p,t-1})-B]^{1-\...
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Counterexample of system stability based on theorem: $\rho<1$

A well-known theorem states the following: For any bounded set of matrices $\mathbb{K}$ such that $\hat{\rho}(\mathbb{K})\neq 0$, the joint spectral radius can be defined as \begin{align} \hat{\rho}(\...
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SDE error and stability

I'm interested in understanding the properties of the long term behaviour of a system of stochastic differential equations. For example, given a coupled system \begin{align} d x&=f(x,y)dt+\phi(x,y)...
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show that the matrix $A$ has stable eigenvalues?

Assume that for the system $\dot x=Ax$ there exist $P,Q>0$ and suppose $\mu>0$ such that $A^TP+PA+2\mu P=-Q$. I want to prove all eigenvalues of $A$ have real part less than $-\mu$
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C^1 unstable attractor

I am aware of the existence of unstable attractors of an iteration of a map $x^{k+1}=H(x^k)$ in the continuous but non-smooth case but I am not aware of an example in the smooth case and I am ...
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Proving stability of non-linear fixed-point iteration

I am trying to prove to myself that if the eigenvalues of the Jacobian of a non-linear fixed-point iteration are strictly within the unit circle, then the iteration converges (see the theorem on ...
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Using Jury Conditions to Show Instability

If I want to force an equilibrium point of a discrete dynamical system to be unstable can I just violate one of the conditions for stability stated in the jury conditions $$|\mbox{Trace} (J)| < 1 + ...
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If M holds DCC then Th(M) is not $\omega$-stable

I have some questions regarding the proof that if M is a group and Th(M) is $\omega$-stable then there is no infinite, strictly decreasing sequence of definable subgroups, $R_0\subsetneq R_1\subsetneq ...
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1 answer
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Instability of a parameter varying system whose parameters belong to a compact set

Suppose, there is a system $$\dot{x}=f(t, \gamma_p(t), x)$$ with $x\in\mathbb{R}^2$. For my specific case, parameter vector $\gamma_p$ is a scalar and known monotonic function with a compact image set ...
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Lyapunov function for an arbitrary equilibrium point

Typically, Lyapunov function assumes $0$ as an equilibrium and require $V(0)=0$. If we wanted to analyze the stability of a nonzero equilibrium point $x_0$, most references asks to do a state ...
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Stability properties of a GNC controller with decomposed control laws

Crossposted at Engineering SE. I am interested in designing a GNC controller for a 3DOF underactuated vehicle that follows a path in 2D space. The two available control inputs are the thrust force in ...
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Stability and Boundedness of Solutions of LTV system

I have the differential equation: $$\dot{x} = \Phi(x,t)[Ax+f(t)] $$ where: $$A \in R^{n\times n}- \ Hurwitz $$ $$ f(t) \in R^n \ and \ \Vert{f(t)\Vert < \infty}; $$ and $$ \Phi(x,t) = diag\{\frac{(...
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Stability of discrete non-linear dynamical system with dominant eigenvalue equal to 1

Suppose that $F :\mathbb{R}^3 \to \mathbb{R}^3$ is a smooth map with a fixed point $e \in \mathbb{R}^3$ and that the Jacobian $J_e(F)$ of $F$ at $e$ has three distinct real eigenvalues, two of which ...
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Non-gravitational example of unstable equilibria

A bit of a soft question: When talking to beginning students in dynamics, I often use some intuitive examples to illustrate the idea of stable vs unstable equilibria. The classics are a pendulum ...
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when is this inequality feasible?

I have the following inequality that has to hold for all frquencies $w$ $${\left[ {\begin{array}{*{20}{c}} {H\left( {jw} \right)}\\ 1 \end{array}} \right]^*}\left[ {\begin{array}{*{20}{c}} { - 2mn}&...
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