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Chapter 8: Problem 6

Consider the system \(\dot{x}=y-2 x, \dot{y}=\mu+x^{2}-y\). a) Sketch the nullclines. b) Find and classify the bifurcations that occur as \(\mu\) varies. c) Sketch the phase portrait as a function of \(\mu\).

Short Answer

Expert verified
a) The nullclines are \(y = 2x\) and \(y = \mu + x^2\). b) Bifurcations: - Saddle-node bifurcation when \(D > 0\) (4 - 4\(\mu > 0\)). - Transcritical bifurcation when \(D = 0\) (4 - 4\(\mu = 0\)). - No bifurcation when \(D < 0\) (4 - 4\(\mu < 0\)). c) Sketch the phase portrait for each case based on the nullclines, equilibrium points, and trajectories in the phase plane.

Step by step solution

01

a) Sketch the Nullclines

First, we will find the nullclines, which are the lines where the derivatives of the variables are zero. The nullclines for this system are given by: 1. \(\dot{x} = y - 2x = 0\) 2. \(\dot{y} = \mu + x^2 - y = 0\) Solve each equation for y: 1. \(y = 2x\) 2. \(y = \mu + x^2\) The first nullcline is a straight line with a slope of 2. The second nullcline is a parabola opening upwards, having its vertex at (-\(\mu\), 0). Now, we can sketch both nullclines (1) and (2) on the same graph.
02

b) Find and Classify Bifurcations

First, find the equilibrium points where both nullclines intersect, i.e., find the values of \(x\) and \(y\) that satisfy both equations: 1. \(y = 2x\) 2. \(y = \mu + x^2\) Setting (1) equal to (2), we have: \(2x = \mu + x^2\) Now, rearrange to have a quadratic equation in terms of x: \(x^2 - 2x + \mu = 0\) This quadratic equation will determine the bifurcations of the system. We can find the discriminant of the quadratic equation, which informs us on the nature of the equilibrium points: \(D = b^2 - 4ac = (-2)^2 - 4(1)(\mu) = 4 - 4\mu\) Now, classify the bifurcations based on the discriminant: 1. If \(D > 0\), there are two distinct real equilibrium points, indicating a saddle-node bifurcation. 2. If \(D = 0\), there is one real equilibrium point, indicating a transcritical bifurcation. 3. If \(D < 0\), there are no real equilibrium points, indicating no bifurcation.
03

c) Sketch the Phase Portrait

To sketch the phase portrait, we will study the system's behavior as \(\mu\) changes. Based on the classification done in step b), we will have different cases: 1. If \(D > 0\), there are two distinct real equilibrium points. To visualize the phase portrait, draw the nullclines and equilibrium points on the same graph. We can observe trajectories flowing towards or away from the equilibrium points, which shows the dynamics of the system. 2. If \(D = 0\), there is one real equilibrium point. In this case, draw the nullclines and the single equilibrium point on the same graph, then observe the trajectories in the phase plane. 3. If \(D < 0\), there are no real equilibrium points. Sketch the nullclines solely. The trajectories will flow along nullclines without equilibrium points. By sketching each case's nullclines and equilibrium points on separate graphs, we have the complete phase portrait for the given system as a function of \(\mu\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bifurcation
Bifurcation in dynamical systems refers to a change in the number or stability of equilibrium points as system parameters are varied. It's a critical concept that helps us understand how a system transitions from one behavior to another as conditions change. For instance, altering the parameter \(\mu\) in the given exercise, \(\dot{x}=y-2 x, \dot{y}=\mu+x^{2}-y\), can lead to different types of bifurcations.

Identifying Bifurcations

We determine bifurcations by analyzing changes in the discriminant \(D = 4 - 4\mu\) of the corresponding quadratic equation for the equilibrium points. Three scenarios are essential:
  • If \(D > 0\), we have a saddle-node bifurcation, indicating the creation or merging of two equilibrium points.
  • When \(D = 0\), it signals a transcritical bifurcation, suggesting that the stability of the equilibrium point is changing.
  • If \(D < 0\), no real equilibrium points exist for those values of \(\mu\), meaning no bifurcation occurs in this range.
By assessing these scenarios, we can predict how the system's behavior evolves as \(\mu\) changes, which is paramount for understanding complex systems in fields like ecology, economics, and engineering.
Phase Portrait
The phase portrait of a dynamical system is a visual representation of all possible trajectories of the system's state variables. Often used in conjunction with nullclines, it provides a global overview of the system's dynamics.

Creating Phase Portraits

To create a phase portrait for the exercise's system, we would plot the nullclines, identify the equilibrium points from the intersections, and characterize the flow of trajectories around these points. The phase portrait changes with the parameter \(\mu\), which affects how equilibrium points appear and disappear.
  • For \(D > 0\), we illustrate two distinct equilibrium points and observe the system's behavior around them.
  • At \(D = 0\), precisely one equilibrium point is highlighted, showcasing different dynamics.
  • With \(D < 0\), no equilibrium points are depicted, so we focus solely on the nullclines' directions.
Phase portraits are an essential tool for understanding how a system might behave over time without solving the differential equations analytically.
Equilibrium Points
Equilibrium points, also called fixed points, are where the state variables of a dynamical system cease to evolve, leading to a constant state. By determining where the system's derivatives equal zero, we find the equilibrium points.

Classification of Equilibrium Points

The exercise provides the foundation to classify equilibrium points:
  • Set the derivatives \(\dot{x}\) and \(\dot{y}\) to zero to obtain nullclines.
  • Find their intersections to get the coordinates of potential equilibrium points.
  • Analyze these points with the discriminant \(D\) to understand their nature and stability.
Equilibrium points are critical for predicting long-term behavior and are central to constructing the system's phase portrait. They explain why, for example, predators and prey populations stabilize in ecology or why an economy might return to a steady state after a disruption.

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Most popular questions from this chapter

(Irrational flow yields dense orbits) Consider the flow on the torus given by \(\dot{\theta}_{1}=\omega_{1}, \dot{\theta}_{2}=\omega_{2}\), where \(\omega_{1} / \omega_{2}\) is irrational. Show each trajectory is dense; i.e., given any point \(p\) on the torus, any initial condition \(q\), and any \(\varepsilon>0\), there is some \(t<\infty\) such that the trajectory starting at \(q\) passes within a distance \(\varepsilon\) of \(p\).

(Plotting Lissajous figures) Using a computer, plot the curve whose parametric equations are \(x(t)=\sin t, y(t)=\sin \omega t\), for the following rational and irrational values of the parameter \(\omega\) : (a) \(\omega=3\) (b) \(\omega=\\}\) (c) \(\omega=\frac{5}{3}\) (d) \(\omega=\sqrt{2}\) (e) \(\omega=\pi\) (f) \(\omega=\frac{1}{2}(1+\sqrt{5})\). The resulting curves are called Lissajous figures. In the old days they were displayed on oscilloscopes by using two ac signals of different frequencies as inputs.

(Logistic equation with periodically varying carrying capacity) Consider the logistic equation \(\dot{N}=r N(1-N / K(t))\), where the carrying capacity is positive, smooth, and \(T\)-periodic in \(t .\) a) Using a Poincar茅 map argument like that in the text, show that the system has at least one stable limit cycle of period \(T\), contained in the strip \(K_{\min } \leq N \leq K_{\max ^{\prime}}\) b) Is the cycle necessarily unique?

(Brusselator) The Brusselator is a simple model of a hypothetical chemical oscillator, named after the home of the scientists who proposed it. (This is a common joke played by the chemical oscillator community; there is also the "Oregonator," "Palo Altonator," etc.) In dimensionless form, its kinetics are $$ \begin{aligned} &\dot{x}=1-(b+1) x+a x^{2} y \\ &\dot{y}=b x-a x^{2} y \end{aligned} $$ where \(a, b>0\) are parameters and \(x, y \geq 0\) are dimensionless concentrations. a) Find all the fixed points, and use the Jacobian to classify them. b) Sketch the nullclines, and thereby construct a trapping region for the flow. c) Show that a Hopf bifurcation occurs at some parameter value \(b=b_{c}\), where \(b_{e}\) is to be determined. d) Does the limit cycle cxist for \(b>b_{c}\) or \(b

(Predator-prey model) Odell (1980) considered the system $$ \dot{x}=x[x(1-x)-y], \quad \hat{y}=y(x-a) $$ where \(x \geq 0\) is the dimensionless population of the prey, \(y \geq 0\) is the dimension- less population of the predator, and \(a \geq 0\) is a control parameter. a) Sketch the nullelines in the first quadrant \(x, y \geq 0\). b) Show that the fixed points are \((0,0),(1,0)\), and \(\left(a, a-a^{2}\right)\), and classify them. c) Sketch the phase portrait for \(a>1\), and show that the predators go extinct. d) Show that a Hopf bifurcation occurs at \(a_{e}=\pm .\) Is it subcritical or supercritical? e) Estimate the frequency of limit cycle oscillations for \(a\) near the bifurcation. f) Sketch all the topologically different phase portraits for \(0
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