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Questions tagged [stability-theory]

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

Stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle.

Many parts of the qualitative theory of differential equations and dynamical systems deal with asymptotic properties of solutions and the trajectories — what happens with the system after a long period of time. The simplest kind of behavior is exhibited by equilibrium points, or fixed points, and by periodic orbits. If a particular orbit is well understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior.

Stability theory addresses the following questions:

Will a nearby orbit indefinitely stay close to a given orbit?

Will it converge to the given orbit? (The latter is a stronger property.)

In the former case, the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting.

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If an LTI system is dissipative, does this mean that it is stable?

Suppose that we have the LTI system $$\dot{x}=Ax+Bw\\z=C x$$ and we know it is dissipative with respect to the function $s(x,w)=z^\top w=x^\top C^\top w$, i.e. $\exists X=X^\top >0$ and a quadratic storage function $V(x)=x^\top X x$ such…
cholo14
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What is steady state of differential equation?

I am a beginner in Dynamical Systems and Stability Analysis. The theory starts from the definition of steady-state solution (equilibrium) of differential equations, but I cannot understand how the following two explanations of steady-state solution…
chloe
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Number of Equilibrium Points in a system

Suppose we know that the number of asymptotically stable equilibrium points is two for the system \begin{equation} \dot{x} = f(x) \end{equation} where $x \in R^1$ and $f$ is continuous. Is the statement "the number of all equilibria must be at…
eet
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how to obtain the stability boundary? (numerically)

Let suppose we have system $$ \dot x=f1(x,y,\alpha,\beta),$$ $$ \dot y=f2(x,y,\alpha,\beta),$$ For determining the stability of the system, I use a small pertub from fixed point like $$ x = x0+x1 $$ $$ y = y0+y1 $$ where $ x0, y0 $ is the fixed…
Nguyen
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Stability of a complex valued system using Jacobian

I want to find the conditions for stability of a system by constructing the Jacobian matrix and finding what values of my parameters will give me negative eigenvalues. My system is of the form $ \dot{x} = f(x) $ , and $f(x)$ is complex. I have seen…
user38729
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Stability of the equilibria

Could somebody help me to prove that the equilibrium is stable? \begin{equation}\begin{cases} u'=u(a_1-b_1u+c_1v+r_1w),\\ v'=v[(1-k)a_2+b_2u-c_2v],\\ w'=kb_3v-(r_2u+q)w.\end{cases}\end{equation} I have constructed two species mutualistic cooperation…
Kevin Maple
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Intersection Between Stable and Saddle-Point Solutions -Stability Analysis

I have a free energy function: $$G(N_b, l_b)= -N_b E_b + \frac{1}{2} N_b \kappa_b (l_b - 1)^2 - F(l_b - 1) + \frac{A}{2} k_g (u_g - l_b)^2 + (N_t - N_b) \text{Log}\big(\frac{N_t - N_b}{A}\big) + N_b \text{Log}\big(\frac{N_b}{A}\big)$$ Physically,…
Klaas-Jan
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Stability (solution?) of nonlinear difference equation

I'm trying to solve the following difference equation: $$x_n = x_{n - 1} + 2\pi \lambda \sin(x_{n-1})$$ I'm an electrical engineer, so I only briefly learned about difference equations. I'm looking for a solution, but more important, I need to show…
pschulz
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Shifted passivity of LTI system

I am given a linear system $\dot{x} = Ax + Bu, y = Cx$, which is given to be passive with storage function $S(t) = \frac{1}{2}x^T Q x$, where $Q=Q^T\geq 0$. I am now looking for a way that I can show that the system is also shifted passive with…
Thom
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Stability Proof is Wrong?

I read the book "A Linear Systems Primer" [1] and confused about a proof for a theorem: In this section we first study the stability properties of the equilibrium $x = 0$ of linear autonomous homogeneous systems: \begin{equation} \dot{x}=A x,~…
winston
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For which matrix $Q$ does the lyapunov equation have a unique solution?

Suppose I have the equation $$A^\top K +KA+Q=0$$ I know that $A$ is stable and $Q$ is symmetric. I want to prove that the solution $K$ to this equation is unique. What constraints should I put on $Q$? How can I prove the uniqueness of the solution?
cholo14
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