Find the fixed point for the equation :

A first-order linear inhomogeneous equation:

Plot the unstable solution for :

Plot the stable solution for :

Second-order linear equation:

Third-order linear equation:

Higher-order inhomogeneous ODE:

Solve the ODE using coordinates of the fixed point as initial values:

Stability analysis of a nonlinear differential equation:

Use StreamPlot to demonstrate the stability:

The stability of a first-order nonlinear equation:

Plot the solution for the initial value :

Use StreamPlot to demonstrate the stability at point :

Consider a second-order nonlinear ODE:

DSolve is unable to solve this equation:

Find the fixed points for the equation using DFixedPoints:

Transform the equation into a system of first-order ODEs:

Plot the trajectories of the system in the plane:

A stable linear system of uncoupled equations:

Trajectories of the system:

An unstable linear system of uncoupled equations:

Trajectories of the system:

The stability of a linear system with constant coefficients:

Use StreamPlot to visualize the stability:

Unstable system with constant coefficients:

Trajectories of the system:

Stable system with constant coefficients:

Trajectories of the system:

Inhomogeneous unstable system:

Trajectories of the system:

Inhomogeneous stable system:

Plot the solutions:

Linear system with symbolic coefficients:

System of four linear ODEs:

Plot the solutions:

10×10 linear system with random constant coefficients:

A nonlinear first-order system:

Analyze the stability of the points:

Use StreamPlot to visualize the stability:

A nonlinear system with periodic fixed points:

Analyze the stability of the points:

A nonlinear system with unstable fixed point at origin:

A system of two nonlinear equations has an infinite number of periodic fixed points:

Use Assumptions to specify the range of a dependent variable:

Stability analysis for the spring-mass system with damping:

Use assumptions to simplify the stability conditions:

Solve the spring-mass system equation:

Plot the solution for given values of parameters:

Do stability analysis for the electric circuit equation:

Solve the electric circuit equation:

Plot the solution for given values of parameters:

Stability analysis for the damped pendulum equation:

Plot the phase portrait of the system:

Plot the solution for the initial conditions , :

Stable system of Lorenz equations:

Use StreamPlot3D to visualize the Lorenz attractors:

Unstable system of Lorenz equations:

Solve the system and plot the solution:

Stability analysis for the predator-prey model (Lotka–Volterra equations):

Plot the phase portrait of the system:

Solve the system for the initial conditions , :

Plot the solutions:

The Rosenzweig–MacArthur predator-prey model:

The chemostat model represents biological systems in which microorganisms grow on abiotic resources:

Find the fixed points:

Analyze the stability of the model for and :

The Brusselator is a theoretical model for a type of autocatalytic reaction. The rate equations of the Brusselator model:

Find the fixed point of the system:

The point is stable if b<1+a²:

The point is unstable if b>1+a²:

Analyze a satellite's attitude dynamics starting from Euler's equations of motion:

Euler’s equations with principal moments of inertia , , :

Find the fixed points of the equation for fixed values of , , :

Choose the fixed point as an operating point:

Construct a state-space model:

The satellite’s attitude is unregulated if disturbed:

Verify the controllability of the model:

Study an inverted pendulum using the Lagrangian:

The position of :

Its velocity:

The kinetic energy of the cart and pendulum:

The potential energy of the pendulum:

The Lagrangian:

The generalized forces:

The equations of motion:

A state-space model:

The non-positive eigenvalues make it an unstable system: