# 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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### 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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