Dynamical Systems Bifurcation Quiz

What type of bifurcation occurs when a stable fixed point and an unstable fixed point collide and disappear?

What is the name given to the method for determining whether a limit cycle exists in a system of ordinary differential equations?

Which of these is not a type of bifurcation discussed in the text?

What type of bifurcation leads to the creation or destruction of a fixed point with a change in a parameter?

What is the main difference between a supercritical and a subcritical pitchfork bifurcation?

What type of bifurcation involves the creation or destruction of a periodic orbit?

Which of these is not a type of two-parameter bifurcation discussed in the text?

What is the main purpose of using phase planes for analyzing dynamical systems?

What is the integrating factor for the following first-order linear ODE: dU/dt + 2tU = t?

What is the general solution to the homogeneous first-order linear ODE: dU/dt + 3U = 0?

For the inhomogeneous first-order linear ODE dU/dt + p(t)U = q(t), what is the expression obtained after multiplying both sides by the integrating factor and applying the product rule?

What is the general solution to the following first-order linear ODE: dU/dt + 2tU = t?

What is the primary purpose of plotting a direction field for a first-order ODE?

In a direction field, which of the following represents the gradient of a solution curve at a given point (t, U)?

When solving the homogeneous first-order linear ODE dU/dt + p(t)U = 0, why does the solution involve an exponential term?

In a direction field, what does the density of the arrows indicate about the solutions to the ODE?

What is the value of the bifurcation point, $r_c$, for the equation $U̇ = r ln(U) + U - 1$?

How are saddle-node and transcritical bifurcations related to their 2D versions?

When a complex conjugate pair of eigenvalues moves across the imaginary axis as the bifurcation parameter is varied, what type of bifurcation occurs?

In the given context, why is the µ = 0 case of the pitchfork bifurcation analogous to the saddle-node bifurcation?

What is the main mathematical characteristic that distinguishes Hopf bifurcations from saddle-node, transcritical, and pitchfork bifurcations?

What does the equation V̇ = −V represent in the context of 2D bifurcations?

Heinz Hopf's contributions in mathematics often involved which geometric entity?

In the given context, why are supercritical and subcritical pitchfork bifurcations called "pitchfork" bifurcations?

What does the symbol 'U' represent in the equation 'rU − U^3 = 0' ?

Which of the following conditions ensures a supercritical pitchfork bifurcation?

In a supercritical pitchfork bifurcation, what happens to the stability of the steady state at U = 0 as r increases?

What does the phrase 'Normal Form' signify in the context of bifurcation theory?

What is the significance of the bifurcation point 'r = rc'?

What distinguishes a supercritical pitchfork bifurcation from a subcritical pitchfork bifurcation?

If we change variables in an equation exhibiting a bifurcation such that it takes the 'Normal Form' V̇ = ρV − V^3, what conclusions can we make about the system's behavior?

How is the bifurcation plot for a supercritical pitchfork bifurcation typically represented?

What is the defining characteristic of a limit cycle in a dynamical system?

In the system described by ṙ = r(1 − r^2), θ̇ = 1, what is the condition for a closed orbit to exist?

How is the stability of a closed orbit determined in this example?

What is the significance of the phase line in this scenario?

If the system were modified to ṙ = r(1 − r^2) − εr, where ε is a small positive constant, what would happen to the closed orbit?

Which of the following is NOT a necessary condition for a limit cycle?

What does it mean for an orbit to be 'isolated' in the context of a limit cycle?

Consider the system ṙ = r(1 − r^2), θ̇ = 1. If we change the system to θ̇ = ω, where ω is a positive constant, what effect will this have on the closed orbit at r = 1?