# Questions tagged [stability-in-odes]

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

505 questions
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### How to adjust the diagonal so that a matrix is on the stability threshold?

I am working on the stability of food webs, which can be represented by a Jacobian matrix showing the interaction strengths between species. I know that a matrix is locally stable if all real parts of ...
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### Stability of the following ODE

Consider the following nonautonomous, nonlinear ODE: $$y'(t)=\rho(W(t)y(t)+b(t)),$$ where $y(t),b(t)\in\mathbb{R}^{n},W(t)\in\mathbb{R}^{n,n}$ and $\rho:\mathbb{R}\to\mathbb{R}$ is some nonlinear ...
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### Bifurcation points of system of non linear differential equations

I had a ODE of second order, then I built the following system of differential equations: $$x' = y$$ $$y' = -\lambda sin(x) -2sin(x) -sin(x)cos(x)$$ With $\lambda = \frac{w^2}{\Omega^2}$ . I want ...
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### On the physical sample of “ discontinuous differential equations ”

The following source contains several physical examples: https://arxiv.org/pdf/0901.3583.pdf. How can I get the differential equation in "Example 2: Brick on a frictional ramp"? How can I grasp this ...
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### Bifurcation points of differential equation (example)

Assume the differential equation: $$x'=\lambda^2-8a\lambda x+2x^2, \quad a\in \mathbb{R}.$$ The critical points are the solutions to the equation: $$x'=0 \iff 2x^2-8a\lambda x +\lambda^2=0\tag{1}$$...
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### Bifurcation in a linear system with 2 equations and 1parameter

I have the following system $$\frac{dx}{dt} = ax+y$$ and $$\frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for ...
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### How to find stability of a third order non-linear system

Suppose we have a third order system, reduced to three first orders in the form $\dot x_1 = x_2 \\ \dot x_2 = x_1 + x_3F(x_1) \\ \dot x_3 = x_3F(x_1)$ Suppose we know $F(0) = 0$ How do we find the ...
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### Finding a Poincaré map for the following ODE and discuss Stability.

I am trying to construct a Poincaré map for: $x’=p(t)x+q(t)$ Where $p(t) \ \text{and} \ q(t)$ are 1 periodic. I’m then asked to discuss the stability if, $\bar{p}=\int^{1}_{0}p(s)ds$. My first ...
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### Implication of stability of Van der Pol oscillator.

Consider the homogeneous Van der Pol equation, $\ddot{x} + \mu (x^2-1)\dot{x} + x = 0$, with $\mu>0$. We convert it into a dynamical system, \dot{\bf x} = (y, -(x+\mu(x^2-1)y), \ \mathbf{x} \...
I was trying to prove that Let $x^*$ be a fixed point of a continuous map f. Show that $x^*$ is asymptotically stable with respect to the map $g=f^2$, then it is asymptotically stable with respect to ...