# 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  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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### 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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### 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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### 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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