Differential Equations: Autonomous Equations and Stability

Consider the differential equation: $\frac{dx}{dt} = x(1 - x)$. Identify the equilibrium points and determine their stability.

Come up with an example of an equation that exhibits an unstable equilibrium where one arrow is going into the equilibrium and one arrow is going out.

Is there any configuration of stable and unstable equilibria which could not occur as the phase-line for a differential equation?

Given an autonomous differential equation $\frac{dy}{dt} = f(y)$ where $f(y) = 1 + y (1 - y)$, identify the equilibrium solutions and determine their stability.

dy/dt = 3y(y - 2)

Given an autonomous differential equation $F(y) = 10 + 3y - y^2$, find the critical points and determine their stability as attractors or repellers by analyzing the sign of $F(y)$ in various regions divided by these critical points.

What is the maximum sustainable fishing-rate? At what rate do I know that I can fish without killing off the fish population entirely?

Analyze the behavior of the differential equation $\frac{dz}{dt} = 4z - z^3$.

For different values of parameter $c$, how does the behavior of the dynamical system described by $\frac{dz}{dt} = 4z - z^3 + c$ change? Summarize these changes in a bifurcation diagram.

Given a differential equation representing logistic population growth with harvesting: $\frac{dy}{dt} = y(K - y) - \alpha$, analyze how the changing harvesting rate $\alpha$ affects the population by using bifurcation diagrams. Identify bifurcation points and describe the stability of equilibrium solutions.

Consider the differential equation $\frac{dX}{dt} = xe^{-ax} - x^2$. Here, $a$ is the bifurcation parameter.

Sketch a bifurcation diagram and indicate the stability of each section created by the bifurcation diagram. Additionally, identify any thresholds where a change in long-term behavior occurs.

Find the second derivative in terms of $x$ and $y$. Describe the region in the $xy$-plane in which all solution curves to the differential equation are concave up.

Let $y = f(x)$ be a particular solution to the differential equation with the initial condition $f(0) = 1$. Does $f$ have a relative minimum, a relative maximum or neither at $x = 0$? Justify your answer.

Given a differential equation $\frac{dx}{dt} = F(x)$, determine the stability of its equilibria by graphical analysis.

For a differential equation $\frac{dy}{dt} = f(y)$, determine the stability of the equilibrium solutions using the phase line and the graph of $f(y)$. Define if each equilibrium solution is asymptotically stable, unstable, or semi-stable.

Consider a system where there is no non-conservative work and potential energy is associated with forces in the system. The potential energy is modeled by the function $V(x) = bx^3$. The force is given by $F(x) = -\frac{dV}{dx}$. Calculate the zero points of the force and analyze the stability of these equilibrium points.

Consider the autonomous differential equation $\frac{dx}{dt} = f(x)$, and suppose $x(t) = x^*$ is an equilibrium point; that is, $f(x^*) = 0$. What can we say about the stability of the equilibrium point $x^*$?

For a nonlinear function $f(x)$, explore the stability of equilibria using the derivative of the function at the equilibrium point. Assume $f'(x_e)$ is non-zero.