Questions tagged [stability-in-odes]

For questions concerning stability of equilibria and of other solutions of ordinary differential equations and their systems.

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Use the definition of stability of equilibrium to prove (1,0) is unstable. [duplicate]

Given the system $$x' = x(1-r)+y(x-r) \\ y' = y(1-r)+x(r-x)\\$$ where $r^2=x^2+y^2$. I don't know how to use the definition of stability of equilibrium to show that the equilibrium $(1,0)$ is ...
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Study instability of a system with Lyapunov functions in terms of a parameter

I am given the following system of odes \begin{gather} \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} - b(...
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How to check local stability of eigenvalues generated by a quintic characteristic polynomial?

Suppose we have a 5x5 matrix, how would we check the whether the eigenvalues are negative so that we can conclude they are locally stable? As requested in the comment by Woody3: here is the matrix: <...
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Trying to understand eigenvalues with respect to differential equations.

I am trying to understand how to find eigenvalues from a matrix consisting of exponential terms, considering a differential equation. The examples I've seen online are ODEs. Without using a vector ...
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Find an explicit example of a critical point $C$ said to be **Asymptotically stable** but not **stable**.

Is the critical point $C$ said to be stable and Asymptotically stable are same? I could find an example for a stable equilibrium point which is not asymptotically stable $$x'=-y$$ $$y'=x$$ I can ...
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$\dot{V}(x)=0 \iff \dot{x}=0$ imply the system always converges to an equilibrium point?

Say we have an system of first-order ODE $$\dot{x}=\Phi(x),$$ which has several equilibrium point. We have found a positive definite Lyapunov function $V(x)$, in particular $V(x)$ is radially unbound ...
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Can locally Lipschitz ODE system imply bounded solutions?

Consider a system of first-order ODE $$\dot{x}=\Phi(x).$$ $\Phi(x)$ is locally Lipschitz. By the theorem, given any initial condition $x(t_0)=x_0$, the solution $x(t)$ is unique and continuous on ...
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System which escapes infinitely often

Is there a system of ordinary differential equations (over the reals) of the form \begin{align} \dot{x}(t) &= f(x(t)) \\ x(0) &= x_0 \end{align} with the following properties: $f(0) = 0$...
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References for Patterning Speed

Imagine I have a system describing two species concentrations $x$ and $y$ defined on a ring of $N$ cells, given by \begin{align} \dot{x}_j&=F_x(x_j,\{x_{r}\},y_j,\{y_{r}\})\\ \dot{y}_j&=F_y(...
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Stability of Differential Equations on the Argand Plane

My book (Advanced Engineering Mathematics 2nd Edition - MD Greenberg) states in Theorem 3.4.3 that in order for a system of Differential Equations to be stable "it is necessary and sufficient ...