Bifurcation Theory

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are nonnegative concentrations of a protein and messenger RNA, and \(a\) and \(\mu \) are positive constants.

\[ \dot {y}=a+\frac {x^2}{1+x^2}-y,\quad y(0)=y_0. \tag {2} \]

1. Show that the positive quadrant is positively invariant. Show that there is a rectangle in the positive quadrant that is positively invariant.
2. Show graphically that there is a region of the \(a-\mu \) parameter space for which there is a unique stable steady state solution, and a region of parameter space in which there are three steady state solutions. Identify the stability of the solutions. Give some identifying features of these regions.
3. What is the bifurcation structure of the solutions as a function of \(\mu \)?

Describe the bifurcations that occur in the equation as the parameter \(a\in \mathbb {R}\) increases.

Consider Rayleigh’s equation \[ \ddot {x}+\dot {x}^3-2\mu \dot {x}+x=0. \] What kind of bifurcations occur to the equilibrium point of system as the parameter \(\mu \) is varied? Determine the bifurcation values and describe how the solutions change near the bifurcation point.

Consider the system \[ \begin {aligned} \dot {x} &= \mu +x^2+y^2,\\ \dot {y} &= -y+x^2 \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation at the origin at \(\mu =0\).

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation from \((x,\dot {x},\epsilon )=(0,0,0)\) for the quadratic Duffing’s Equation \[ \ddot {x}+\dot {x}+\epsilon x+x^2=0. \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Using an approximate center manifold, determine the nature of the bifurcation at the origin when \(\epsilon =0\). \[ \begin {aligned} \dot {x} &= \epsilon x-x^3+y^2,\\ \dot {y} &= -y+x^2. \end {aligned} \]

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -xy,\\ \dot {y} &= -y+x^2-2y^2. \end {aligned} \]

Consider Griffith’s model for a genetic control system, where \(x\) and \(y\) are proportional to concentration of protein and the messenger RNA from which it is translated, respectively, and \(\mu >0\) is a rate constant \[ \begin {aligned} \dot {x} &= y-\mu x,\\ \dot {y} &= \frac {x^2}{1+x^2}-y. \end {aligned} \]

1. Show that the system has three fixed points when \(\mu <\mu _c\) and one when \(\mu >\mu _c\) where \(\mu _c\) is to be determined.
2. What is the nature of the bifurcation at \(\mu =\mu _c\)?

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -xy,\\ \dot {y} &= -y+x^2-2y^2. \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= -y-xz,\\ \dot {y} &= x-y^3,\\ \dot {z} &= -z-2xy-2x^4+x^2. \end {aligned} \]

1. State the Center Manifold Theorem for rest points. Briefly explain its importance in bifurcation theory.
2. Construct an approximation to the center manifold at the origin and use it to determine the local behavior of solutions. \[ \begin {aligned} \dot {x} &= xy+x^3,\\ \dot {y} &= -y-x^2y. \end {aligned} \]

State the center manifold theorem and explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-xy,\quad \dot {y}=-y-x^2. \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions.

State the center manifold theorem and briefly explain its importance in bifurcation theory. Consider the system \[ \dot {x}=-2x+y-x^2,\quad \dot {y}=x(y-x). \] Construct an approximation to the center manifold at the origin and hence determine the local behavior of solutions. [Hint: first perform a change of variables in order to diagonalize the linear part of the system].