Short Introduction to Bifurcations · Short Introduction to Bifurcations Yoichiro Mori March 16, 2019 ... Even when xis very large, predation by birds will never increase above A

Short Introduction to Bifurcations

Yoichiro Mori

March 16, 2019

1 Saddle-Node Bifurcation

Suppose we have a differential equation with a certain parameter r:

dx

dt= f(x, r), x ∈ RN , r ∈ R. (1)

We say that we have a bifurcation at r = r∗ if the behavior of the dynamicalsystem changes qualitatively at r = r∗. Rather than discuss generalities, itis easier to take a look at examples. Consider the dynamical system:

dx

dt= r − x2 = f(x). (2)

When given a differential equation, the first thing to do is to look for steadystates. The steady states x = x∗ satisfy:

r − x2∗ = 0 (3)

So,x∗ = ±

√r, r ≥ 0. (4)

Note here that r ≥ 0 so that there is a solution. Now, let us examine thestability of each of the fixed points. We must compute the derivative of thef .

df

dx

∣∣∣∣x=x∗

= −2x∗. (5)

1

x

y

r

x*

Figure 1: Left figure: The intersection of the parabolas y = r − x2 (bluecurves) with the x axis (red line) are the fixed points of the differentialequations (2). For r > 0 there are two fixed points whereas for r < 0there are no fixed points. Right figure: bifurcation diagram, which plots theposition of the fixed points as a function of r.

We thus see the following (see Figure 1):

r < 0 : no fixed points,

r = 0 : x∗ = 0, f ′(0) = 0,

r > 0 : x∗ = −√r, f ′(−

√r) = 2

√r > 0, unstable fixed point,

x∗ =√r, f ′(

√r) = −2

√r < 0, stable fixed point.

(6)

One way to visualize the bifurcation is to draw the bifurcation diagram(Figure 1, right). Here, we plot the fixed points as a function of r. At r = 0,two fixed points emerge at the locations x∗ = ±

√r.

At r = 0, we have a bifurcation. The behavior of the system changesqualitatively at r = 0. For r < 0, we have no fixed point. At r = 0, wehave one fixed point, at which f ′(x∗) = 0. For r > 0, we have two fixedpoints, one of which is stable, and the other of which is unstable. This iscalled a saddle-node bifurcation or a fold bifurcation. The name saddle-nodecomes from the fact that, at the bifurcation point, a saddle and node cometogether and annihilate. In one-dimensional systems, there are no saddles,so with the above example, it may not be clear why it should be called asaddle-node bifurcation. The analogue of the above in higher dimension, infact, always involves a saddle and a node. We shall see examples of thislater.

There are different kinds of bifurcations, some of which we will introducein these notes. One important observation is that, at the parametric value

MATH 4428 2 Yoichiro Mori

r = r∗ at which the bifurcation takes place, at the fixed point, we have:

f ′(x∗) = 0. (7)

This is a general feature of bifurcations. We shall see later how this conditiongeneralizes in higher dimensional systems.

Let us look at another example. Consider:

dx

dt= r − x− e−x = f(x). (8)

We want to find the fixed points of the above. Let the fixed points be x∗.We must solve the equation:

r − x∗ − e−x∗ = 0. (9)

Solving this equation is equivalent to finding the intersection of the graphs:

y = r − x and y = e−x. (10)

As it turns out, when r < 1, the two graphs do not have any common point,and the line y = r − x lies below y = e−x. When r = 1, y = r − x touchesy = e−x at x∗ = 0. For r > 0, the two graphs have two intersection points,x+ and x− where x+ > 0 and x− < 0. Let us examine the stability of thesefixed points. The derivative of f is given by:

f ′(x∗) = −1 + e−x. (11)

Therefore, we have:

r < 1 : no fixed points,

r = 1 : x∗ = 0, f ′(0) = 0,

r > 1 : x∗ = x−, f′(x−) = −1 + exp(−x−) > 0, unstable fixed point,

x∗ = x+, f′(x+) = −1 + exp(−x+) < 0, stable fixed point.

(12)

To determine the signs of the derivative at x±, we used the fact that x+ > 0and x− < 0. We see here that at r = 1, we have a saddle-node bifurcation.

Let us look at an example that is a little more interesting. We consider thefollowing model of population growth of a spruce budworm (adopted fromStrogatz, Nonlinear Dynamics and Chaos). These worms live in spruce


forests in eastern Canada, and occasional outbreaks of this worm decimatesthe spruce forests.

Let x be the population of the spruce budworm. We consider the model:

dx

dt= rx

(1− x

K

)︸︷︷︸

population growth

− Ax2

B + x2︸︷︷︸predation by birds

, r,K,A,B > 0. (13)

There are two terms to the equation. The first term represents the logisticgrowth model that we have already seen. The parameter r is the growthrate and K is the carrying capacity. The second term is new, and modelspredation by birds. Note first that predation increases with x: the moreinsects there are, the greater the number of insects eaten by the birds. Theother important feature is that, as x increases, predation by birds hits aceiling value of A. Even when x is very large, predation by birds will neverincrease above A.

Let us examine the fixed points of this equation. Let x = x∗ be a fixedpoint. We must solve the equation:

rx∗

(1− x∗

K

)=

Ax2∗B + x2∗

. (14)

This means that, we have:

x∗ = 0 or r(

1− x∗K

)=

Ax∗B + x2∗

. (15)

We must therefore examine the latter equation. This amounts to finding theintersection points of the graphs:

y = r(

1− x

K

)and y =

Ax

B + x2. (16)

We consider the specific situation when

A

2√B< r <

3√

3A

8√B. (17)

Then, we will have the following picture as K is increased (See Figure 2).For values of small K, the two graphs have only one intersection point, x−.When K reaches K = K∗, the graphs are tangent to each other at a pointx̂. When K is increased further, two intersection points, x0 and x+ emerge.


x

y

Figure 2: The intersection of the two graphs, Ax/(B + x2) (red curve) andr(1−x/K) (blue lines) for various values of K. As K is increases (x-interceptof the blue lines) the number of intersection points goes from a, increases to3 and decreases back to 1.

At K = K∗, the graphs are tangent again, where the points x− and x0coalesce to form the tangent point x̃. Above K = K∗, we again only haveone intersection point x = x+.

We can now see how the system behaves for different values of K. The sta-bility of each of the fixed points can be determined by taking the derivative,but in this case, it is probably easier to just consider the sign of the righthand side of (13). We have:

0 < K < K∗ : x∗ = 0 unstable, x− stable,

K = K∗ : x∗ = 0, x̂ unstable, x− stable,

K∗ < K < K∗ : x∗ = 0, x0 unstable, x−, x+ stable,

K = K∗ : x∗ = 0, x̃ unstable, x+ stable,

K∗ < K : x∗ = 0 unstable, x+ stable.

(18)

In this case, we have a saddle-node bifurcation, both at K = K∗ and K =K∗. We note that condition (17) ensures that we will indeed have twosaddle-node bifurcations.

Let us consider the biological implications of this analysis. The parameterK is the carrying capacity of the budworm living in the spruce forest. If


the forest is young, K is low, and K becomes larger as the forest matures.When K is small, the there is only one stable steady state, x−. As the forestmatures, the value of K crosses K∗. Here, x− disappears in a saddle-nodebifurcation, and the only stable steady state is x∗ = x+. As K crosses K∗,therefore, we see a sudden increase in the budworm population. This modelthus explains the budworm outbreak.

2 Transcritical/Pitchfork Bifurcations

There are other kinds of bifurcations that can happen in one-dimensionalsystems. Let us look at the following system:

dx

dt= rx− x2 = f(x) (19)

Let us look at the steady states:

rx∗ − x2∗ = 0, x∗ = 0, r. (20)

We see that, at r = 0, the two steady states collide and merge into one (SeeFigure 4). When r 6= 0, however, there are two steady states. In order toexamine the stability of the fixed points, we can look at the derivative off(x) at x∗ or we can just consider the sign of f above and below x∗. Thederivative is given by:

f ′(x∗) = r − 2x, f ′(0) = r, f ′(r) = −r. (21)

We thus see that:

r < 0 : 0 stable, r unstable,

r = 0 : 0 unstable,

r > 0 : 0 unstable, r stable.

(22)

So, as r passes through r = 0, the two steady states cross each other andexchange stability. This is called a transcritical bifurcation.

Finally, let us consider the pitchfork bifurcation. Consider the following:

dx

dt= rx− x3 = f(x) (23)


x

y

r

x*

Figure 3: Left figure: The intersection of the parabolas y = rx − x2 (bluecurves) with the x axis (red line) are the fixed points of the differentialequations (19). Right figure: bifurcation diagram. Blue line indicates thestable steady states whereas the red line indicates the unstable steady states.

Let us consider the steady states:

rx∗ − x3∗ = 0, x∗ = 0,±√r. (24)

Thus, when r > 0, we have three steady states, whereas, when r ≤ 0 thereis only one steady state. Looking at the sign of f ′(x∗) or looking at the signof f , we can see that the stability of the steady states is given by:

r < 0 : 0 stable,

r = 0 : 0 stable,

r > 0 : 0 unstable, ±√r stable.

(25)

We thus see the single stable steady state for r < 0 splits into three steadystates, two stable and one unstable, when r > 0. The above is called asupercritical pitchfork bifurcation to be distinguished from the subcriticalpitchfork bifurcation, which is what happens for the following equation:

dx

dt= rx+ x3 = f(x). (26)


x

y

r

x*

Figure 4: Left figure: The intersection of the parabolas y = rx − x3 (bluecurves) with the x axis (red line) are the fixed points of the differentialequations (23). Right figure: bifurcation diagram. Blue line indicates thestable steady states whereas the red line indicates the unstable steady states.

For this, we have:

r < 0 : 0 stable, ±√−r unstable.

r = 0 : 0 unstable,

r > 0 : 0 unstable.

(27)

The difference is that two unstable steady states emerge instead of the stablesteady states that appear in the supercritical case (see (25)).

Of the three kinds of bifurcations we have seen so far, the saddle-node,transcritical and pitchfork bifurcations, the saddle-node is by far the mostcommonly seen. The pitchfork bifurcation is usually seen in systems thathave a reflection symmetry. Indeed, in the pitchfork bifurcation, we sawabove, the two new steady states that emerge came out symmetrically. Themetal plate buckling problem discussed briefly in class is one such problem(a metal plate under compression is equally likely to buckle left or right).


-2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2r=-1

-2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2r=1

Figure 5: When r < 0 (left figure) we have no steady states. When r > 0,we have two steady states, one of which is a saddle, and another a sink. Atr = 0, we have a saddle node bifurcation.

3 Limit Cycles and the Hopf Bifurcation

For dynamical systems of dimension greater than or equal to 2, we can alsohave saddle-node, transcritical or pitchfork bifurcations. Let us take a lookat one example.

dx

dt= 4r − (x+ y)2 − (x− y) = f(x, y),

dy

dt= 4r − (x+ y)2 + (x− y) = g(x, y).

(28)

An easy computation shows that the above system has 2 fixed points forr > 0:

x+ = (√r,√r), x− = (−

√r,−√r). (29)

When r = 0, the two fixed points merge into one, and for r < 0, there areno fixed points (see Figure 5).

In order to determine the stability of the fixed points, we must examine theeigenvalues of the Jacobian matrix. We have:

J =

(fx fygx gy

)=

(−2(x+ y)− 1 −2(x+ y) + 1−2(x+ y) + 1 −2(x+ y)− 1

). (30)

At x+, the Jacobian matrix is:

J =

(−4√r − 1 −4

√r + 1

−4√r + 1 −4

√r − 1

). (31)


The eigenvalues λ are given by:

λ = −8√r,−2. (32)

Since both eigenvalues are negative, this is a stable fixed point. At x−, wehave:

J =

(4√r − 1 4

√r + 1

4√r + 1 4

√r − 1

). (33)

The eigenvalues λ are given by:

λ = 8√r,−2. (34)

We thus have a saddle. As r approaches 0, the two fixed points come closerand collide, and then annihilate. This is an example of a saddle-node bifur-cation in 2D.

In 2D and higher dimension, it is possible to have a completely different kindof bifurcation, the Hopf bifurcation. This stems in part from the followingfact. In 1D, the only thing that can happen with a solution to a differentialequation as t → ∞ is that it goes to a steady state or goes to infinity. In2D and higher, there is another important possibility. It can approach aperiodic orbit. Consider the following system of differential equations:

dx

dt= µx+ y − x(x2 + y2) = f(x, y),

dy

dt= −x+ µy − y(x2 + y2) = g(x, y).

(35)

As can be seen in Figure 6, when µ > 0, this system happens to have aperiodic solution.

As usual, let us examine the steady states of this system and determine theirstability. We first look for the steady states (x∗, y∗). We have:

f(x∗, y∗) = µx∗ + y∗ − x∗(x2∗ + y2∗) = 0,

g(x∗, y∗) = −x∗ + µy∗ − y∗(x2∗ + y2∗) = 0.(36)

To solve this equation, multiply the first equation by x∗ and multiply thesecond by y∗ and add:

(µ− (x2∗ + y2∗))(x2∗ + y2∗) = 0. (37)

This implies:x2∗ + y2∗ = 0, µ. (38)


-2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2=-1

-2 -1 0 1 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2=1

Figure 6: When µ < 0 (left figure) we have a stable spiral at the origin.When µ > 0, we have an unstable spiral at the origin and a stable periodicorbit. At µ = 0, we have a supercritical Hopf bifurcation.

Suppose x2∗ + y2∗ = 0. In this case, we must have (x∗, y∗) = (0, 0). On theother hand, if x2∗ + y2∗ = µ, we may substitute this into (36) to obtain:

y∗ = 0, −x∗ = 0. (39)

Thus, all in all, the only steady state of this system is:

(x∗, y∗) = (0, 0). (40)

Let us now examine the stability of this steady state. The Jacobian matrixis given by:

J =

(fx fygx gy

)=

(µ− 3x2 − y2 1− 2xy−1− 2xy µ− 3y2 − x2

). (41)

Thus, the Jacobian matrix at (x∗, y∗) = (0, 0) is given by:

J =

(µ 1−1 µ

)(42)

The eigenvalues of this matrix can be found by computing the characteristicequation:

det

(λ− µ −1

1 λ− µ

)= (λ− µ)2 + 1 = 0. (43)

Solving this equation, we obtain:

λ = µ± i. (44)


Stability is determined by the sign of the real part of the eigenvalues. Thus,if µ > 0, the steady state is unstable whereas if µ < 0 we have stability ofthe steady state. In fact, as we see from Figure 6, for µ > 0, we have astable periodic orbit:

µ < 0 : stable steady state (0, 0),

µ > 0 : unstable steady state (0, 0), stable periodic orbit at radius√µ.

(45)

This is an example of a Hopf bifurcation.

In the saddle node, transcritical and pitchfork bifurcations, one of the eigen-values of the Jacobian matrix at the steady state will become 0 as the pa-rameter crosses the bifurcation value. In the case of a Hopf bifurcation, acomplex conjugate pair of eigenvalues crosses the imaginary axis. This is oneof the fundamental differences between the bifurcations we have discussedpreviously and the Hopf bifurcation.

Thus, if you are tracking a steady state, and a complex conjugate eigenvaluepair of the Jacobian matrix crosses the imagrinary axis, that would be acandidate point at which a Hopf bifurcation is likely to present.

Much like the pitchfork bifurcation, the Hopf bifurcation comes in two va-rieties, the supercritical and subcritical Hopf bifurcations. What we sawabove is the supercritical Hopf bifurcation. The following system exhibits asubcritical Hopf bifurcation:

dx

dt= µx+ y + x(x2 + y2) = f(x, y),

dy

dt= −x+ µy + y(x2 + y2) = g(x, y).

(46)

For the above system, we have:

µ < 0 : stable steady state (0, 0), unstable periodic orbit at radius√µ.

µ > 0 : unstable steady state (0, 0),

(47)

This should be compared with (45).

It is beyond the scope of this class to determine whether a given Hopf bi-furcation is supercritical or subcritical.


Figure 7: Solution to the Lorenz system approaches a butterfly shaped at-tractor.

In dimension 1, solutions to differential equations will go to steady states att → ∞ or go to infinity. For dimension 2, in addition to these possibilities,it can go to a periodic solution. Such behavior is certainly possible in higherdimensional systems. Are there other kinds of behaviors that can be seenin higher dimension? The answer to this question is yes, and a solution toa differential equation can approach what is called a strange attractor orchaotic attractor. A famous example of this is the Lorenz system:

dx

dt= σ(y − x),

dy

dt= ρxx− y − xz,

dz

dt= −βz + x+ y.

(48)

When σ = 10, β = 8/3 and ρ = 28, the above system has a strange attractorwhose overall shape can be seen in Figure 7. The butterfly shaped attractornever closes on itself, and is not a periodic orbit.


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Short Introduction to Bifurcations · Short Introduction to Bifurcations Yoichiro Mori March 16, 2019 ... Even when xis very large, predation by birds will never increase above A