Stability and Phase Plane Analysis Calculator

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Example

Created on 2024-06-20Asked by Sofia Thomas (Solvelet student)

Determine the stability of the equilibrium point for the system:

\begin{cases} \dot{x} = x - y \\ \dot{y} = x + y \end{cases}

Solution

To determine the stability of the equilibrium point for the system:

\begin{cases} \dot{x} = x - y \\ \dot{y} = x + y \end{cases}

1. **Find the Jacobian matrix:**

J = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}.

2. **Find the eigenvalues of

J

:**

\det(J - \lambda I) = \begin{vmatrix} 1 - \lambda & -1 \\ 1 & 1 - \lambda \end{vmatrix} = (1 - \lambda)^2 + 1 = 0.

\lambda^2 - 2\lambda + 2 = 0.

\lambda = 1 \pm i.

3. **Determine the stability:** Since the real part of the eigenvalues is positive (

\Re(\lambda) = 1

), the equilibrium point

(0,0)

is unstable. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Jack King on Solvelet

1. Sketch the phase portrait for a nonlinear oscillator exhibiting limit cycle behavior.2. Analyze the stability of a mass-spring-damper system described by the second-order differential equation

d^2x/dt^2 + 2\zeta\omega_n dx/dt + \omega_n^2 x = 0

, where

\omega_n

is the natural frequency and

\zeta

is the damping ratio.,

DefinitionStability/phase plane analysis is concerned with studying the behavior of dynamical systems by graphical means. Stability analysis studies behaviour of the system given small perturbation and phase plane analysis represents actual path followed by the system in state space. Example: Phase Plane of a Nonlinear System dtdx=y,dtdy=−x+x2−y to Determine System Stability and Behavior

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