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Chapter 3: Problem 12

With tongue in cheek, we pointed out that the pitchfork bifurcation could be called a "trifurcation," since three branches of fixed points appear for \(r>0 .\) Can you construct an example of a "quadfurcation," in which \(\dot{x}=f(x, r)\) has no fixed points for \(r<0\) and four branches of fixed points for \(r>0\) ? Extend your results to the case of an arbitrary number of branches, if possible.

Short Answer

Expert verified
The function \(\dot{x} = f(x, r) = x(x^2 - a - r)(x^2 - b - r)\), with \(a\) and \(b\) being positive constants and \(a < b\), represents a "quadfurcation" because it has no fixed points for \(r < 0\) and four branches of fixed points for \(r > 0\). To generalize this to an arbitrary number of branches, \(n\), we can use the following function form: \(\dot{x} = f(x, r) = x(x^2 - a_1 - r)(x^2 - a_2 - r)...(x^2 - a_n - r)\), where \(a_1, a_2, ..., a_n\) are positive constants satisfying \(a_1 < a_2 < ... < a_n\). This generalized function has no fixed points for \(r < 0\) and \(n + 1\) branches of fixed points when \(r > 0\).

Step by step solution

01

Define a function for the quadfurcation

Let us consider a function of the following form: \[\dot{x} = f(x, r) = x(x^2 - a - r)(x^2 - b - r)\] where \(a\) and \(b\) are positive constants such that \(a < b\). This function has the following properties: if \(r < 0\), the function has no fixed points, i.e., there are no values for \(x\) where the function equals zero. However, if \(r > 0\), there are four branches of fixed points depending on the values of \(a\), \(b\), and \(r\) used.
02

Show that the function has no fixed points for \(r < 0\)

Let's show that the function has no fixed points when \(r < 0\): When \(r < 0\), we can see that \(x^2 - a - r > x^2 - a\) and \(x^2 - b - r > x^2 - b\). Since \(a\) and \(b\) are positive, it means that \(x^2 - a - r > 0\) and \(x^2 - b - r > 0\) for all \(x\). Now, let's consider our function: \[\dot{x} = x(x^2 - a - r)(x^2 - b - r)\] Since all three factors (i.e., \(x\), \(x^2 - a - r\) and \(x^2 - b - r\)) are positive or negative, their product will never be zero. Therefore, there are no fixed points for \(r < 0\).
03

Show that the function has four branches of fixed points for \(r > 0\)

Now let's show that there are four branches of fixed points when \(r > 0\). When \(r > 0\), there will be a fixed point at \(x = 0\), and three other fixed points when \(x^2 = a + r\) and when \(x^2 = b + r\). Hence, there are four branches of fixed points as \(r\) increases.
04

Generalize for an arbitrary number of branches (Optional)

If we want to create a function that generates an arbitrary number of branches (\(n\)) when \(r > 0\), we can consider the following form: \[\dot{x} = f(x, r) = x(x^2 - a_1 - r)(x^2 - a_2 - r)...(x^2 - a_n - r)\] where \(a_1, a_2, ..., a_n\) are positive constants such that \(a_1 < a_2 < ... < a_n\). This generalized function will share the same properties discussed earlier. It will have no fixed points for \(r < 0\) and will have \(n + 1\) branches of fixed points when \(r > 0\), since there is a fixed point at \(x=0\) and each term of the form \(x^2 - a_i - r\) will contribute to a new set of fixed points.

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Key Concepts

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

Fixed Points
Imagine plotting the position of a pendulum at each moment and analyzing its behavior over time. In the context of differential equations, particularly those that govern dynamic systems, we come upon a concept known as fixed points. Fixed points are values at which the system doesn't change; it's essentially 'fixed' in place. These points are critical to understanding system behavior because they often represent the system鈥檚 equilibrium states.

Fixed points are calculated by setting the time derivative of the state variable to zero, which yields the points in the state space where the system has no tendency to move. So, in the provided problem, fixed points are the values of x for which \(\f(x, r)\) is equal to zero, indicating that the system is at rest, neither growing nor shrinking. The solution clearly defines how to determine these points for a given parameter 'r', dividing the scenarios where 'r' is negative or positive.
Quadfurcation
In the mysterious realm of nonlinear dynamics, quadfurcation appears as a marvel where a system splits into four distinct pathways, similar to a road that divides into four separate paths. This term takes inspiration from 'bifurcation', where a system transitions from one state to another, leading to multiple possible outcomes. In our exercise, quadfurcation is thought of in a playful manner, where instead of the more common 'trifurcation' in a pitchfork bifurcation, four separate branches of fixed points emerge when control parameter r crosses from negative to positive.

The solution provided offers a beautifully crafted mathematical equation that assures us of no fixed points when r < 0, and the grand emergence of four fixed point branches once r turns positive. The excitement lies in the realization that through quadfurcation, we can witness the birth of multiple equilibria that didn't exist before, revealing the enchanting complexity of nonlinear systems.
Nonlinear Dynamics
Nonlinear dynamics is a domain that unfurls the complexities of systems that do not follow a straight line 鈥� quite literally. These systems, governed by nonlinear equations, can exhibit behavior that's vastly different and more intricate than their linear counterparts. This field bestows upon us phenomena like chaos, complex patterns, and unpredictable yet deterministic behavior that can mesmerize and baffle even the most astute minds.

In our exercise, the nonlinear equation that describes the quadfurcation is a quintessential example of how nonlinear behavior can result in multiple outcomes, depending on the system's parameters. Beyond just discovering fixed points and noting their existence, nonlinear dynamics delves deeper into their stability and how tiny perturbations can dramatically alter the system's trajectory 鈥� a thrilling dance between chaos and order, sensitively dependent on initial conditions and system parameters.

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Most popular questions from this chapter

In pioneering work in epidemiology, Kermack and McKendrick ( 1927 ) proposed the following simple model for the evolution of an epidemic. Suppose that the population can be divided into three classes: \(x(t)=\) number of healthy people; \(y(t)=\) number of sick people; \(z(t)=\) number of dead people. Assume that the total population remains constant in size, except for deaths due to the epidemic. (That is, the epidemic evolves so rapidly that we can ignore the slower changes in the populations due to births, emigration, or deaths by other causes.) Then the model is $$ \begin{aligned} &\dot{x}=-k x y \\ &\dot{y}=k x y-\ell y \\ &\dot{z}=\ell y \end{aligned} $$ where \(k\) and \(\ell\) are positive constants. The equations are based on two assumptions: (i) Healthy people get sick at a rate proportional to the product of \(x\) and \(y\). This would be true if healthy and sick people encounter each other at a rate proportional to their numbers, and if there were a constant probability that each such encounter would lead to transmission of the disease. (ii) Sick people die at a constant rate \(\ell\). The goal of this exercise is to reduce the model, which is a third-order system, to a first-order system that can analyzed by our methods. (In Chapter 6 we will see a simpler analysis.) a) Show that \(x+y+z=N\), where \(N\) is constant. b) Use the \(\dot{x}\) and \(\dot{z}\) equation to show that \(x(t)=x_{0} \exp (-k z(t) / \ell)\), where \(x_{0}=x(0)\) c) Show that \(z\) satisfies the first-order equation \(\dot{z}=\ell\left[N-z- x_{0} \exp (-k z / \ell)\right]\). d) Show that this equation can be nondimensionalized to $$ \frac{d u}{d \tau}=a-b u-e^{-4} $$ by an appropriate rescaling. e) Show that \(a \geq 1\) and \(b>0\). f) Determine the number of fixed points \(u^{*}\) and classify their stability. g) Show that the maximum of \(\dot{u}(t)\) occurs at the same time as the maximum of both \(\tilde{z}(t)\) and \(y(t)\), (This time is called the peak of the epidemic, denoted \(t_{\text {peak }}\). At this time, there are more sick people and a higher daily death rate than at any other time.) h) Show that if \(b<1\), then \(\dot{u}(t)\) is increasing at \(t=0\) and reaches its maximum at some time \(t_{\text {peak }}>0\). Thus things get worse before they get better. (The term epidemic is reserved for this case.) Show that \(\dot{u}(t)\) eventually decreases to 0 . i) On the other hand, show that \(t_{\text {peak }}=0\) if \(b>1\). (Hence no epidemic occurs if \(b>1 .\) j) The condition \(b=1\) is the threshold condition for an epidemic to occur. Can you give a biological interpretation of this condition? k) Kermack and McKendrick showed that their model gave a good fit to data from the Bombay plague of 1906 . How would you improve the model to make it more appropriate for AIDS? Which assumptions need revising? For an introduction to models of epidemics, see Murray (1989), Chapter 19, or Edelstein-Keshet (1988). Models of AIDS are discussed by Murray (1989) and May and Anderson (1987). An excellent review and commentary on the Kermack- McKendrick papers is given by Anderson (1991).

A magnet can be modeled as an enormous collection of electronic spins. In the simplest model, known as the Ising model, the spins can point only up or down, and are assigned the values \(S_{i}=\pm 1\), for \(l=1, \ldots, N>>1 .\) For quantum mechanical reasons, the spins like to point in the same direction as their neighbors; on the other hand, the randomizing effects of temperature tend to disrupt any such alignment. An important macroscopic property of the magnet is its average spin or magnetization $$ m=\left|\frac{I}{N} \sum_{i=1}^{N} S_{i}\right| $$ At high temperature the spins point in random directions and so \(m \approx 0 ;\) the material is in the paramagnetic state. As the temperature is lowered, \(m\) remains near zero until a critical temperature \(T_{e}\) is reached. Then a phase transition occurs and the material spontaneously magnetizes. Now \(m>0 ;\) we have a ferromagnet. But the symmetry between up and down spins means that there are two possible ferromagnetic states. This symmetry can be broken by applying an external magnetic field \(h\), which favors either the up or down direction. Then, in an approximation called mean-field theory, the equation governing the equilibrium value of \(m\) is $$ h=T \tanh ^{-1} \mathrm{~m}-\mathrm{Jnm} $$ where \(J\) and \(n\) are constants; \(J>0\) is the ferromagnetic coupling strength and \(n\) is the number of neighbors of each spin (Ma 1985, p. 459 ). a) Analyze the solutions \(m^{*}\) of \(h=T \tanh ^{-1} m-I n m\), using a graphical approach. b) For the special case \(h=0\), find the critical temperature \(T_{c}\) at which a phase transition occurs.

Zebra stripes and butterfly wing patterns are two of the most spectacular examples of biological pattern formation. Explaining the development of these patterns is one of the outstanding problems of biology; see Murray (1989) for an excellent review of our current knowledge. As one ingredient in a model of pattern formation, Lewis et al. (1977) considered a simple example of a biochemical switch, in which a gene \(G\) is activated by a biochemical signal substance \(S\). For example, the gene may normally be inactive but can be "switched on" to produce a pigment or other gene product when the concentration of \(S\) exceeds a certain threshold. Let \(g(t)\) denote the concentration of the gene product, and assume that the concentration \(s_{0}\) of \(S\) is fixed. The model is $$ \dot{g}=k_{1} s_{0}-k_{2} g+\frac{k_{3} g^{2}}{k_{4}{ }^{2}+g^{2}} $$ where the \(k\) 's are positive constants. The production of \(g\) is stimulated by \(s_{0}\) at a rate \(k_{1}\), and by an autocatalytic or positive feedback process (the nonlinear term). There is also a linear degradation of \(g\) at a rate \(k_{2}\). a) Show that the system can be put in the dimensionless form $$ \frac{d x}{d \tau}=s-r x+\frac{x^{2}}{1+x^{2}} $$ where \(r>0\) and \(s \geq 0\) are dimensionless groups. b) Show that if \(s=0\), there are two positive fixed points \(x\) * if \(r

The first-order system \(\dot{u}=a u+b u^{3}-c u^{5}\), where \(b, c>0\), has a subcritical pitchfork bifurcation at \(a=0\). Show that this equation can be rewritten as $$ \frac{d x}{d \tau}=r x+x^{3}-x^{3} $$ where \(x=u / U, \tau=t / T\), and \(U, T\), and \(r\) are to be determined in terms of \(a, b\), and \(c\).

The next exercises are designed to test your ability to distinguish among the various types of bifurcations - it's easy to confuse them! In each case, find the values of \(r\) at which bifurcations occur, and classify those as saddle- node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifureation diagram of fixed points \(x^{*}\) vs. \(r\). $$ \dot{x}=r x+\frac{x^{3}}{1+x^{2}} $$
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