Question

Problem 1. (robustness of saddle-node bifurcations and imperfect transcritical bifurcation) (a) What happens if you add a small imperfection to a system that has a saddle-node bifurcation? (b) Consider the system \frac{dx}{dt} = h + rx - x^2 ... (1b). Note that, when $h = 0$, this system undergoes a transcritical bifurcation at $r = 0$. Show how the bifurcation diagram of $x^*$ vs $r$ is affected by the \"imperfection\" of including the parameter $h$. a) Plot the bifurcation diagram for equation (1b), for $h < 0$, $h = 0$, and $h > 0$. Be sure to justify your answer using a graphical and/or algebraic analysis of the system. b) Sketch the regions in the $(r, h)$ plane that correspond to qualitatively different vector fields, and identify the bifurcations that occur on the boundaries of those regions. That is, derive algebraic equation(s) for the curves of bifurcations in $(r, h)$ parameter space, and plot the bifurcation curves, plot the \"stability diagram\" or \"two-parameter bifurcation diagram\". Label the diagram thoroughly.

          Problem 1. (robustness of saddle-node bifurcations and imperfect transcritical bifurcation)
(a) What happens if you add a small imperfection to a system that has a saddle-node bifurcation?
(b) Consider the system
\frac{dx}{dt} = h + rx - x^2 ... (1b).
Note that, when $h = 0$, this system undergoes a transcritical bifurcation at $r = 0$. Show how the
bifurcation diagram of $x^*$ vs $r$ is affected by the \"imperfection\" of including the parameter $h$.
a) Plot the bifurcation diagram for equation (1b), for $h < 0$, $h = 0$, and $h > 0$. Be sure to justify your
answer using a graphical and/or algebraic analysis of the system.
b) Sketch the regions in the $(r, h)$ plane that correspond to qualitatively different vector fields, and
identify the bifurcations that occur on the boundaries of those regions. That is, derive algebraic
equation(s) for the curves of bifurcations in $(r, h)$ parameter space, and plot the bifurcation curves, plot
the \"stability diagram\" or \"two-parameter bifurcation diagram\". Label the diagram thoroughly.

Problem 1. (robustness of saddle-node bifurcations and imperfect transcritical bifurcation)
(a) What happens if you add a small imperfection to a system that has a saddle-node bifurcation?
(b) Consider the system
(dx)/(dt) = h + rx - x^2 ... (1b).
Note that, when h = 0, this system undergoes a transcritical bifurcation at r = 0. Show how the
bifurcation diagram of x^* vs r is affected by the ïmperfectionöf including the parameter h.
a) Plot the bifurcation diagram for equation (1b), for h < 0, h = 0, and h > 0. Be sure to justify your
answer using a graphical and/or algebraic analysis of the system.
b) Sketch the regions in the (r, h) plane that correspond to qualitatively different vector fields, and
identify the bifurcations that occur on the boundaries of those regions. That is, derive algebraic
equation(s) for the curves of bifurcations in (r, h) parameter space, and plot the bifurcation curves, plot
the s̈tability diagramör ẗwo-parameter bifurcation diagram.̈ Label the diagram thoroughly.

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