dynamical systems chaos

8/3/2019 Dynamical Systems Chaos

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ure 32.

x

Figure 32: The trajectory is homoclinic to the equilibrium x. Thismeans that x is both its -limit set and its -limit set. Note that isnot a closed curve, but a closed curve minus one point.

Example: Heteroclinic orbits . A trajectory which has an -limit set A and an -limit set B is said to be aheteroclinic orbit . In Figure 33 a heteroclinic orbit between two equilibria x and y is shown.

y

x

Figure 33: The trajectory is heteroclinic between the equilibria yand x. x is its -limit set and y is its -limit set. is an open linesegment.

4.9.2 Poincar e-Bendixson Theory in R 2

An elegant result in 2 D tells us that if a limit set doesnt contain any equilibria, then it must be a periodicorbit. This means that behaviour in R 2 is much simpler than general behaviour in higher dimensions.

[NOTE: The full statement of the Poincar e-Bendixson Theorem also deals with the possibilities wherelimit sets do contain equilibria, and deals with 2D spaces other than R 2 . In fact all possible limit sets in2D are described, but we wont go into so much detail here.]

Theorem (Poincar e-Bendixson Theorem) . Consider a di ff erentiable vector eld x = f ( x) on R 2 . Sup-pose that the forward trajectory of a point x0 , i.e. + ( x0) enters a bounded region E

R 2 , and never leavesit. Suppose in addition that the region E contains no equilibria. Then ( x0), the -limit set of x0 , is aperiodic orbit.

This theorem tells us that any bounded limit set in R 2 which doesnt contain an equilibrium is a periodicorbit. So if we know that a trajectory enters and never leaves an area which has no equilibria, then it must tend to a periodic orbit. This allows us to nd periodic orbits for dynamical systems on R 2 as follows:

1. Find a trapping region a region trajectories enter, but can never leave2. If this region doesnt contain any equilibria (or the equilibria it contains dont attract all trajectories),

then it must contain a periodic orbit.

Our work on the geometry of orbits tells us how to nd a trapping region if trajectories cross some closedsurface in a particular direction, then this surface is the boundary of a trapping region.

Outline of the proof: The proof of the Poincar e-Bendixson Theorem is not easy, but has some elegantgeometrical ideas, so we will sketch the stages. First we need a couple of denitions

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Denition: A point which is not a xed point of a ow is called a regular point.

Denition: A nite closed line segment l, is called a transversal for a vector eld if there are no equilibriaon l, and if the vector eld is not tangent to l at any point.

The following ideas add up to a proof of the Poincar e-Bendixson theorem. Some of them are quite intu-itively obvious, while others take more work. Remember that we are only talking about R 2 , and not higherdimensional spaces.

1. Every regular point of a vector eld is an interior point of some transversal

2. If a trajectory intersects a transversal at a number of points then these points move monotonicallyalong the transversal.

3. If a trajectory and its -limit set () have a point in common, then is either an equilibrium ora periodic orbit.

4. If () contains no equilibria, then () is a periodic orbit.

The fourth point is of course the Poincar e-Bendixson theorem. We will also show that:

A periodic orbit intersects a transversal l at no more than one point (follows directly from point 2above)

Any limit set can intersect a transversal l in at most one point (follows from the nal point above).

Sketch proof:

1. Point 1 is intuitively obvious and you can believe it. If a point is not an equilibrium we can simplytake as our transversal any straight line segment perpendicular to the vector eld at that point. Aslong as the segment is short enough, then it will be a transversal.

2. The most fundamental idea is the second point above: If a trajectory intersects a local transversalseveral times, then successive crossing points move monotonically along the transversal. This isconnected very much with the geometry of R 2 , (and is hence the reason the proof wont work inhigher dimensions, or on more complicated spaces.) The idea is easiest to see via the pictures inFigure 34. In each case the unshaded region is a positively invariant region, so any further crossingswith l must lie in the unshaded region. A complete proof of this part would require use of the Jordan

curve theorem .

3. The third point if the limit set and trajectory of any point x0 share a point in common, then the limitset is a closed curve follows directly from the second point. Let x be a shared point, both on + ( x0 )and in ( x0 ). Clearly the point x must be in its own limit set, i.e. x

( x). Choose a transversalthrough x. By the denition of a limit point, there must be a sequence of intersections betweenthe trajectory of x and this transversal, which converge to x. But suppose the rst intersection isdiff erent from x. Then the second intersection cannot get closer to x. And so forth. This implies thatthe trajectory can never come back close to x. So the only possibility is that the rst intersectionmust be equal to x, and the trajectory is a closed curve.

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ll

Figure 34: When a trajectory ( x) intersects a transversal l more than once it does so monotonically. Ineach of the above pictures the trajectory leaves the shaded region and can never return (because it cannotcross l going the wrong way, and it cannot cross itself).

This proof also shows us that closed curves (periodic orbits) can intersect any transversal at no more

than one point. Put simply, what we have just shown is that if a trajectory shares any point with itsown limit set, then it must intersect any transversal at most once, and so it must be a closed curve.(Assuming of course it is not an equilibrium.)

4. We have almost nished sketching the proof of the Poincar e-Bendixson theorem. It remains to showthat the only possible limit sets containing no equilibria are periodic orbits. Take any point x0 whosetrajectory enters and never leaves a bounded region E

R 2 . Take a point x in ( x0). There are now

three possibilities:

If x is also in the trajectory of x0 , then from point 3, the trajectory is a closed curve and we arenished.

Otherwise consider the trajectory through x. Note that this whole trajectory is in the limit setof x0 . Take any limit point y of this trajectory and construct a local transversal l through y. If this limit point y is in the trajectory through x, then again from point 2 the trajectory is periodicand we are done.

Otherwise y is a limit point of the trajectory through x but is not on the trajectory through x.This means that the trajectory of x must move monotonically along the transversal l, intersect-ing it at points p1 , p2 , etc. These points must accumulate monotonically on y. But each of these p i is also in the limit set of x0 (because the whole trajectory of x is in the limit set of x0 ), so the trajectory of x0 must be able to pass arbitrarily close to p i and then p i+ 1 and then p iagain. but this implies that the intersections of + ( x0 ) do not move monotonically along l acontradiction. So the only possibility is that y

+ ( x) and hence that ( y) is a closed curve.

In case youve got lost, we can rephrase the above as The trajectory of x0 must either itself be aclosed curve, or must tend towards a closed curve. This is because if the trajectory tends towardsan object which is not a closed curve then it must intersect some transversal in a non-monotonicsequence, which is impossible.

Note that we have also just shown that the only kind of limit set in R 2 which contains no equilibriais a closed curve.

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Example: Using the Poincar e-Bendixson theorem . Show that the vector eld

x = x y x( x2 + y2 )

y = x + y y( x2 + y2)

has a closed orbit for > 0.

In polar coordinates x = r cos , y = r sin this transforms to

r = r ( r 2)

= 1

Exercise: Carry out the coordinate transformation.

The only equilibrium of the system is at r = 0. Let > 0. Then on the circle r = / 2 we haver = 3 / 8 > 0. On the circle r = 2 , r = 6 < 0. These two circles thus bound a positivelyinvariant region containing no equilibria. Thus this region must contain a periodic orbit. In fact any twocircles r = r 1 < and r = r 2 > bound an invariant region. Of course in this case it is also simple tond the periodic orbit directly from the di ff erential equations in polar form.

4.10 Return maps (Poincar e maps)

These are several ways in which it is possible to generate maps discrete-time dynamical systems from di ff erential equations. The simplest way is to note that the time- t maps generated by a ow form adiscrete-time dynamical system. But there are other particularly useful ways of generating maps in specialcircumstances.

4.10.1 Using Poincar e maps to study periodic orbits of autonomous systems

One particularly important idea is that of a Poincar e map or a return map. The basic idea is as follows:Suppose you have a dynamical system which generates ow t , and have some codimension 1 surface in the phase space (i.e. a surface with one less dimension than the phase space itself), and suppose all ow

lines intersect

transversely. Suppose we also know for some reason that ow lines which intersect someregion of will denitely return to intersect again. Then we might be able to construct a map whichtakes the points where ow lines rst intersect to where they next intersect . There are certain situationswhere this idea makes a lot of sense, because we can guarantee that all ow lines in some neighbourhoodwill intersect again for example in the neighbourhood of special solutions such as periodic orbits andhomoclinic orbits.

The situation in the neighbourhood of a point on a periodic orbit is shown in Figure 35. Note that the ideaof the surface is much like the idea of a transversal, except that it makes sense in any dimension.

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x

Figure 35: It is always possible to construct a section transverse to a periodic orbit . If we choose smallenough then it is also transverse to the ow near . More-over, by continuity, all ow lines intersecting sufficientlynear to must intersect again (even if the periodic orbitis unstable).

If the section is small enough, then the periodic orbit intersects it in a unique point x. If T is the periodof the periodic orbit, then T ( x) = x. Now consider a small neighbourhood U

containing x, where U is chosen small enough so that the trajectory of any point y

U will eventually intersect again. Let T y

be the time at which the trajectory of y intersects again. T y is close to T by continuity. It makes sense todene a map P from U to , where P ( y) = T y( y) the point of rst return to .

The reason this idea is important because it provides the simplest way of studying stability and bifurcationsof periodic orbits. We will not prove this result but it is not hard to believe:

Theorem: A periodic orbit is orbitally stable / unstable i ff the xed point of the associated Poincar e map isstable / unstable.

Example: Consider the dynamical system in polar co-ordinates

r = r (1

r )

= 1

The ray dened by = 0, r > 0 forms a suitable Poincar e section. Lets call this section R. We can conrmthis by checking that the dot product of the vector eld with a vector normal to R (0, 1) for example isnever zero:

0, 1 r (1 r )1

= 1

The ODE system can be solved explicitly because it is already separated. Carrying out this procedure

gives:

r (t ) =r 0et

1 r 0 + r 0et (t ) = 0 + t mod 2

From the second equation it is obvious that any initial condition on the ray = 0 (except the origin) returnsto this ray at t = 2. So it makes sense to study the map P : R R where

P (r ) =re 2

1 r + re 2=

re 2

1 + r (e2 1)

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It is easy to conrm that the map r n+ 1 = P (r n) has an equilibrium at r = 1. Moreover we can calculate that

P (r ) =e2

1 + r (e2 1) re 2(e2 1)

(1 + r (e2 1)) 2=

e2

(1 + r (e2 1)) 2and hence |P (1)| < 1 so the equilibrium is stable.Of course in this example because of the circular symmetry of the system, a Poincar e map was not nec-essary to understand the dynamics of this map. However in more complicated situations for examplewhere numerical computation is the only way of calculating solutions Poincar e maps can be very usefulways of studying periodic orbits.

4.10.2 Using Poincar e maps to study time-periodic ODEs

Another situation in which Poincar e maps can arise is from time-periodic ODEs. These are systems of theform

x = f ( x, t )

where f ( x, t + T ) = f ( x, t ) for all x i.e. f is periodic in time with period T . Such systems arise in manyareas of physics and biology, such as in the study of forced oscillators. For example, the unforced systemmight be a biological system, and the periodic forcing might arise from periodic external processes theday-night cycle or a seasonal cycle.

Note that these are non-autonomous systems, and so the construction of a Poincar e map is somewhatdiff erent from what we have seen above. We have not said much about nonautonomous systems so rstwe make some general comments.

First of all, note that for nonautonomous systems we cannot dene the ow in the same way as for au-tonomous systems. For example, given a point x, it would not make sense to talk about t ( x), withoutspecifying explicitly at what time we take x as an initial condition. It would make sense to talk about t ,0( x) meaning the solution which passes through x at time t = 0. We can of course iterate an initialcondition x forward in time in discrete time steps. But this does not in general give us a discrete timedynamical system, because at time t = t 1 we have to apply a di ff erent map to the space from the one weapply at some other time t 2 . Suppose y = t ,0( x) the point where x gets to after time t , then t ,0 ( y) is notthe next iterate of x under the time t map. The next iterate of x is in fact t ,t 0 ( y), and because the system isnonautonomous, t ,t 0 is in general di ff erent from t ,0 .

A second point to note is that if (t ) is a solution to a nonautonomous ODE x = f ( x, t ), then (t + s) is a

solution to the ODE x = f ( x, t + s) for any s. This follows because by denitionddt (t + s) means

dd t (t ) t = t + sand this in turn is equal to f ( x, t + s). One way of summarising this idea is: If we time-shift an ODE

we get time-shifted solutions. For autonomous ODEs of course the time-shifted ODE is identical to theoriginal ODE, and so any time-shift of a solution gives us another solution.

For periodic ODEs the above arguments have the following specic consequence: The time T map of aperiodic ODE with period T gives us a discrete time dynamical system.

Suppose that (t ) is a solution to the periodic ODE x = f ( x, t ). Then

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If (t ) is a solution, then so is (t + kT ) for any integer k . This follows because from above (t + kT )solves the ODE x = f ( x, t + kT ), but f ( x, t + kT ) = f ( x, t ) by denition.

If we start iterating a point forward at time t or time t + T or time t + kT for any k , it makes nodiff erence we get to the same place after a given time. This is pretty obvious given a bit of thought.Note that if (t ) is the solution that takes initial condition x at time t = 0 and (t ) (t kT ) is thesolution which takes initial condition x at time kT . So if we take initial condition x at time t = 0and map forward for time s we get to (s). And if we take initial condition x at time t = T and mapforward for time s we get to (T + s

T ) = (s) the same place.

We can also construct this map in a more geometrical way (which also explains why it is called a Poincar emap). We start by a useful little trick which involves converting the nonautonomous system x = f ( x, t )into the autonomous system

x = f ( x, )

= 1

Note that is dened modulo T by denition, or geometrically lives on a circle. So the dynamical systemis an autonomous system on R n S 1 . So we can take the plane = 0 (or indeed = 0 for any other 0)as a Poincare section for this dynamical system. We can guarantee that the ow lines always cross thissection transversely because of the second equation. The Poincar e map we generate is still just the time- T map for the original system. A schematic picture is shown in Figure 36.

t

x

x

1

2

t=0

t=TFigure 36: A schematic picture of the two sec-tions at time t = 0 and t = T in the extendedphase space. All ow lines always cross these

sections transversally since t 0. Any initialcondition specied at time t = 0 is mapped bythe ow to a unique point at time t = T .

All of this is easiest to understand via an example.

Example: Consider the periodically forced nonautonomous 1D system

x = (a + b cos t ) x

Assume that > 0. Note that this system is linear and it is periodic with period 2 / . Moreover it isseparable, and we can nd the solution for any initial condition at time t = 0 by separating the variablesand integrating:

x(t ) x(0) d x x = t

0(a + b cos s)d s

This gives us:

x(t ) = x(0) exp( at +b

sin( t ))

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Note that this system has an equilibrium at x = 0. Since the system was periodic with period 2 / we canconstruct the time-2 / map of the system (since > 0, this a map in forward time). This gives us

x(2/ ) = x(0) exp(2 a / )

(since sin(2 ) = 0). Writing this as a discrete dynamical system gives us

xn+ 1 = kxn

where k = exp(2 a / ). We see that if a < 0, then exp(2 a / ) < 1, and from our study of discretetime systems, all solutions converge to the zero solution xn = 0. On the other hand if a > 0, thenexp(2 a / ) > 1, then all solutions tend to innity except the zero solution. If a = 0, then exp(2 a / ) = 1,and we get the identity map.

Thus in this example, even though the system is forced we can see that the equilibrium at 0 is stable whena < 0, and unstable when a > 0, and stability doesnt depend on b or on the size and freqency of theperiodic component of the forcing.

In this example we used explicit computation of the solutions to construct the Poincar e map. Of coursein general this is impossible. But even in situations where we can only solve a periodically forced ODEnumerically the Poincar e map is very useful. We simply follow the evolution of initial conditions intime-steps of size T and this gives us a complete picture of the dynamics of the system.

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5 An introduction to bifurcation theory

We now study the beginnings of a very large and elegant body of theory with a huge number of applicationsand real world consequences.

Broadly speaking bifurcation theory is the theory of when we get qualitative changes in families of mapsor ows. Imagine applying increasing force to a metal rod, forcing it to bend. In general a small increasein force (the parameter) causes a small increase in the bending (the dynamic variable). But there mightcome a point where a small increase in the force causes the rod to snap a dramatic event. Such eventsare the physical manifestations of bifurcations .

Although the idea is intuitively obvious there are a lot of di ff erent forms that bifurcations can take. Tobegin to see this we rst need to put the argument into mathematical form. When we study bifurcationswe are interested in parametrised families of discrete or continuous time dynamical systems of the form

xn+ 1 = f ( xn , ) or x = f ( x, )

is a parameter, and for the bifurcations we will be interested in we can assume that it is real. The questionwe are asking is whether as we change , there are values of where something dramatic happens.

Bifurcations fall into two categories:

Local bifurcations are changes which occur in some small neighbourhood in phase space. For example anobject might change stability or even disappear. But outside of a neighbourhood of the object there maybe no major changes.

Global bifurcations involve qualitative changes in the orbit structure of the dynamical system which arenot restricted to some small region in phase space. Although they are fascinating, they are harder to study.It is possible for limit sets which are not restricted to small regions of phase space (for example chaotic

sets) to be born in such bifurcations.

5.1 The Saddle-Node (SN) bifurcation

We start with a very simple continuous time example. Consider the 1D ODE:

x = x2

For < 0 there are no equilibria, whereas for > 0 there are two one at x+ = and the other at x = . At = 0 there is a single nonhyperbolic equilibrium. Clearly at = 0 an important eventoccurs. This event is in fact a saddle-node bifurcation as we shall see below. It is the only bifurcationwhich we shall study in some detail.

A very useful way of visualising the behaviour of the above ODE is to plot the set dened by x2 = 0 in x space. At any xed value of this plot tells us about the equilibria at that value of . See Figure 37.Looking at Figure 37 we see that the equation x2 = 0 denes a single curve in x space, and the point(0, 0) is a special point on this curve.

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x

Figure 37: The bifurcation diagram for theODE x = x2 . Arrows indicate the direc-tion of the ow. At = 0 a pair of equilibria areborn. It is this event we want to study.

Now well look at the situation more abstractly. Note that our discussion applies equally to continuous ordiscrete time systems. To nd the equilibria of a ow or a discrete time system, we always have to solvean algebraic equation of the form

g( x, ) = 0

where x

R n and

R and g : R n R R n . In the continuous time case g is just the right hand side of the ODE system. In the discrete time case ( xn

+

1=

f ( xn , )), g( x, )=

f ( x, ) x. So how we construct gmight be di ff erent in discrete and continuous time, but we end up with an equation of the same form.What does a function of the form g( x, ) = 0 dene? Another way of stating that g : R n R R n is that itis a set of n equations for n + 1 unknowns. In general what this denes is a set of one dimensional curvesin R n R . If we x = 0 we get a cross section R n 0 . The 1 D curves will in general intersect thiscross section in a set of points. These points will be the solutions of g( x, ) = 0 for = 0 . This picture isshown in Figure 38.

f(x, )=0

f(x, )=0

x

1 2

Figure 38: The equation f ( x, ) = 0 denes a set of curves in x space. If we choose constant sections = 1 and = 2 we can nd out how many solutions the equation f ( x, ) = 0 has at these values of .In the picture shown the equation has ve solutions at = 1 and four solutions at = 2 . At = 2 wecan see that a bifurcation is taking place on one of the solutions branches.

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5.1.1 Characterisation of the SN bifurcation

How can we characterise what happens near the bifurcation point geometrically? Imagine a section = 0moving slowly along. Suppose we pass through a value of where a pair of solutions are born, then thiswould imply a saddle-node bifurcation has taken place. Thus saddle-node bifurcations are points where acurve of solutions takes a U-turn in the -direction.

Saddle-node bifurcations are fundamentally the same in any dimension, so well make the argument pre-

cise for a 1D system. So we start with an equation:

g( x, ) = 0

where now x

R . We will say that there is a saddle-node bifurcation, at x0 , 0 if locally the curve denedby g( x, ) = 0 looks like the curve in Figure 39. Our task is to write down conditions on the functiong( x, ) near ( x0 , 0), which will ensure that it looks like this.

g(x, )=0

x

x0

0

Figure 39: Local behaviour of the curve g( x, ) near( x0 , 0).

The rst thing we note is that the slope of the curve becomes is innite, or rotating the picture and thinkingof as a function of x, the slope of the graph becomes zero. We will shortly justify this idea of thinkingof as a function of x. But for the time being assume that we can think of as a function of x. Then the

slope of the curve = ( x) is obtained by di ff erentiating g( x, ( x)) = 0 with respect to x along the solutioncurve. This gives:

g x + g d d x

= 0

[We have used the notation g x =g x and g =

g to simplify things since we will often be needing these

partial derivatives. Since we will later need higher order partial derivatives, note that g xx =2 g x2 and

g x = 2 g

x . The generalisation of this notation to third and higher order derivatives is obvious.]

The previous equation solves to give:

d d x

= g xg

This slope is zero if g x = 0, as long as g 0. The rst condition, g x = 0, is our main bifurcation condition.It tells us that the point x0 is nonhyperbolic. The second condition, g 0, is called a genericity condition.If g = 0 then we cannot guarantee that

d d x = 0 in fact the question may not make any sense because

there may be no curve of solutions at all locally as we shall see below. There is one further thing that wewant to ensure, and this is that we dont have the situation shown in Figure 40 where the curve has a pointof inection, but not a genuine turning point.

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g(x, )=0

x

x 0

0

Figure 40: Possible degenerate behaviour of thecurve g( x, ) near ( x0 , 0).

From basic calculus we know that this degenerate behaviour occurs if d2

d x2 = 0. Just as we could calculatethe slope of the curve ( x) in terms of the partial derivatives of g we can do the same for the secondderivative. We di ff erentiate g( x, ( x)) = 0 twice to get:

g xx + g xd d x

+ g d2 d x2

= 0

But at bifurcation d d x = 0, so this reduces to:

g xx + g d2 d x2 = 0

solving for d2

d x2 gives:

d2 d x2

= g xxg

Since we have already insisted that g 0, we see thatd2 d x2 = 0 iff g xx = 0. Thus our second genericity

condition for a saddle-node bifurcation to occur is that g xx 0. This condition ensures that we wont getthe situation in Figure 40.

Using only geometrical ideas, we have enumerated four conditions which together mean that the equationg( x, ) = 0 has a saddle-node bifurcation at ( x0 , 0). These are:

1. g( x0 , 0 ) = 0 (there is an equilibrium at ( x0 , 0))

2. g x( x0 , 0) = 0 (the condition for nonhyperbolicity)

3. g ( x0 , 0) 0 (a genericity condition)

4. g xx( x0 , 0) 0 (a second genericity condition)

Note that the two genericity conditions are so called because they are in general true we have to beunlucky to violate either of them in a typical family. We will return to this idea a little later. First we ll ahole in the argument we presented above.

5.1.2 Completing the argument: the Implicit Function Theorem

There is only one part of the above argument which is incomplete. We have not proved that there must bea curve of solutions = ( x) through the bifurcation point in the rst place. To prove this we need the

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Implicit Function Theorem (IFT). Large parts of bifurcation theory rest on repeated and creative use of the IFT. Stated for this context we have:

IFT in 2D

Suppose g : R 2 R 1 is a C 1 -function (i.e. both partial derivatives exist and are continuous.) Supposefurther that1. g( x0 , y0) = 02. g

y( x

0, y

0) 0

Then there exist open intervals I about x0 and J about y0 and a C 1 function p : I J satisfying:1. p( x0 ) = y02. g( x, p( x)) = 0 for all x

I .

We apply the IFT to tighten up our argument as follows. Above we had a function with an equilibriumat ( x0 , 0 ), i.e. g( x0 , 0) = 0. Our rst genericity condition stated that g 0. By the IFT this ensuresthat in a small enough neighbourhood of ( x0 , 0 ) there is a function ( x) satisfying g( x, ( x)) = 0. Thiswas the only part missing from our argument above we simply assumed that in the neighbourhood of

the bifurcation point we could nd a function ( x) such that g( x, ( x)) = 0. As long as we insist on thecondition g 0, the IFT says that this function exists.

Example: Applying the theory to the ODE system

x = x2Let g( x, ) = x2 . We suspect from our arguments above that there is a SN bifurcation at (0 , 0). All wehave to do to conrm it is to check all the conditions. First the main conditions.

1. g(0, 0) = 0 (there is a solution at (0 , 0)).2. g x(0, 0) = 2 x|(0 ,0) = 0

Now we have to check the genericity conditions

1. g (0, 0) = 1 0

2. g xx(0, 0) = 2 0

Since all the bifurcation conditions and genericity conditions are fullled, a SN bifurcation occurs at (0 , 0).

Example: SN bifurcation in a map . Consider the 1D map

xn+ 1 = + x x2

Let g( x, ) = + x x2 x = x2 . All our arguments in the continuous time case now apply since wehave obtained the same function g, and we hence know that the map xn+ 1 = + x x2 has a SN bifurcationat (0 , 0).

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It is not generally the case that the study of a bifurcation for maps and ows is so similar, but in the caseof the saddle-node bifurcation this is true.

A nongeneric example : Consider the di ff erential equation in 1D:

x = g( x, ) x( x)We can check that there is a nonhyperbolic equilibrium at (0 , 0), because g(0, 0) = 0 and g x(0, 0) =0. However g (0, 0) = 0, violating the rst genericity condition. Thus although there appears to be abifurcation at (0 , 0) it is not a saddle-node bifurcation. In fact it is a non-generic bifurcation called atranscritical bifurcation in which two solution branches meet and then diverge again. This is simple tosee from the bifurcation diagram in Figure 41.

g(x, )=0

0

x 0

x

Figure 41: The local picture near a transcritical bifur-cation. Two lines of solutions cross each other at thebifurcation point i.e. two equilibria merge and thenseparate again. Note that the transcritical bifurcation

is a nongeneric bifurcation.

General implicit function theorem

The complete implicit function theorem is awkward to state, but the idea is very much the same as in the1D case. The basic question is as follows: We have a map f : R n R m where m < n and we are interstedin nding solutions to f ( x, y) = 0 where x

R nm and y

R m. When can we solve for variables y in termsof variables x?

Remember that an equation of the form f : R n R m = 0 can be thought of as a set of m equations for nunknowns. So it is plausible that we might be able to use these equations to solve for m of the unknownsin terms of the rest. Is this always possible?

The answer is no, but the IFT gives us conditions when it is possible. Suppose that we know a solution,i.e. values x0 and y0 such that f ( x0 , y0) = 0. We are looking for a function p( x) such that f ( x, p( x)) = 0for values of x near x0 ? In 2D as we have seen, we can do this provided the slope of the curve dened by f ( x, y) = 0 at the point ( x0 , y0) is not innite.

The general IFT tells us that if D y f is non-singular at x = x0 , then we can solve for y in terms of x near

x = x0 . D y f means the Jacobian of partial derivatives of f with respect to the variables y. If y

Rthen itsimply means f y . Since y is an m-dimensional vector, and f consists of m equations, the Jacobian D y f is

a square matrix. To say that it is non-singular means that det( D y f ) 0.

Example: Consider the equations

F 1( x, y, z, u, v, w) = 0F 2( x, y, z, u, v, w) = 0F 3( x, y, z, u, v, w) = 0

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If x, y, z, u, v, w

R , then this describes a map from R 6 to R 3 . The IFT tells us that we can solve for u, vand w in terms of x, y and z as long as

F 1u

F 1v

F 1w

F 2u

F 2v

F 2w

F 3u

F 3v

F 3w

0

The above Jacobian is often abbreviated to

(F 1 , F 2 , F 3)(u, v, w)

The next example shows that though the IFT is not easy to state or prove, it is actually quite easy to apply.

Example : Consider the above example with F 1 = 3 x+ u(1 + 2 y) + zv, F 2 = xy + (3 + u)vw, F 3 = w( z+ 1).This has a solution at ( x, y, z, u, v, w) = (0, 0, 0, 0, 0, 0). The question is whether for small values of x, y and z we can solve for u, v and w in terms of x, y and z. To nd out the answer we evaluate the Jacobian

(F 1 , F 2 , F 3)(u, v, w) (0 ,0,0,0,0,0)

=1 + 2 y z 0

v 3 + u 10 0 1 + z (0 ,0,0,0,0,0)

=1 0 00 3 10 0 1

This matrix is non-singular (i.e. its determinant is non-zero), so the answer is yes, we can solve for u, vand w in terms of x, y and z near the origin.

5.1.3 The SN bifurcation in higher dimensions

We will not fully answer the question of how to characterise an SN bifurcation in a higher dimensionalsystem, because this would require more theory than we have developed. But the basic geometrical argu-ment we presented above is the same in any dimension. If, as is varied a solution curve takes a U-turn,

then at that point we have a SN bifurcation. Suppose g( x, ) = 0 where now x

Rn

. The basic bifurcationconditions for a saddle-node bifurcation to occur in a dynamical system of any dimension are:

1. There must be a solution at ( x0 , 0), i.e. g( x0 , 0) = 0

2. The Jacobian at ( x0 , 0) must have a zero eigenvalue i.e. det( Dg( x0 , 0)) = 0

The genericity conditions have the same geometrical meanings as in the 1D case but to state them oneneeds some extra ideas. Essentially, all interesting behaviour near the bifurcation point occurs on a 1Dcurve called the centre manifold, and on this curve the description of what happens reduces to the

discussion in the 1D case. But calculating the centre-manifold can be tedious, so checking the genericityconditions in examples is not always simple. Often we have to be content to hope that they are notviolated.

5.2 When do equilibria of maps and ows undergo local bifurcations?

We now leave the SN bifurcation, and ask the much more general question about how we know when abifurcation is taking place. We have foreshadowed the answer to this question at several points in these

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notes. It is very easy to identify in general when a xed point of a map or ow is undergoing a localbifurcation. If the xed point is hyperbolic then by the Hartman-Grobman theorem, behaviour near thexed point is quite simple. In fact it can be proved that if a we have a xed point which is hyperbolic, thena small perturbation to the vector eld or the map wont change the essential orbit structure.

Since small perturbations to maps or ows with hyperbolic xed points produce no dramatic changes, itis not surprising that bifurcations of equilibria sudden qualitative changes near the xed point occurwhen we have a nonhyperbolic xed point.

The basic geometrical picture for a ow and a map are shown in Figure 42. We see that a xed point of a ow undergoes a bifurcation when the real parts of some eigenvalues of the Jacobian pass through theimaginary axis i.e. the line Re() = 0, and that a xed point of a map undergoes a bifurcation when thereal parts of some eigenvalues of the Jacobian pass through the unit circle ||= 1.

)

) = 0

)

Im(

Re(

Re(

unstablestable

0

flows

Re( ) < 0 Re( ) > 0

)

)

Im(

Re(

unstable

0

stable

maps

| | < 1

| | > 1

| | = 1

Figure 42: Bifurcations in ows and maps occur when eigenvalues of the Jacobian evaluated at the xedpoint pass through the imaginary axis (for ows) and the circle || = 1 (for maps). Eigenvalues are eitherreal or come in complex conjugate pairs. So for ows there are two simple scenarios a single realeigenvalue passes through 0, or a pair of complex conjugate eigenvalues with non-zero real parts passthough the imaginary axis. For maps there are three scenarios a single real eigenvalue passes through 1,a single real eigenvalue passes through 1 or a pair of complex conjugate eigenvalues with nonzero realparts pass through the unit circle.

From Figure 42 we can work out that there are two simple scenarios for ows, and three scenarios formaps. We list the bifurcations associated with these scenarios for reference:

1. For ows:

a single real eigenvalue passes through 0 saddle-node bifurcation a pair of complex conjugate eigenvalues with non-zero real parts pass though the imaginary

axis Hopf bifurcation

2. For maps:

a single real eigenvalue passes through 1 saddle-node bifurcation

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a single real eigenvalue passes through 1 period doubling (ip) bifurcation a pair of complex conjugate eigenvalues with non-zero real parts pass though the unit circle

Hopf bifurcation

Note that for both maps and for ows, a Hopf bifurcation can only occur in a system with dimensiongreater than 1 since the Jacobian must have at least two eigenvalues.

Since the conditions for a xed point to be nonhyperbolic are quite di ff erent for maps and for ows itmight seem strange that we could study the saddle-node bifurcation in maps and ows at the same time.The reason we could do this is the following observation: Consider a discrete time dynamical system

xn+ 1 = f ( xn)

Fixed points are solutions of g( x) f ( x) x = 0. Saddle-node bifurcations occur when det( Dg) = 0,i.e. det( D f I ) = 0 where I is the identity matrix. Now it is not hard to convince yourself that D f I has an eigenvalue of 0 if and only if D f has an eigenvalue of 1. Thus the local study of the saddle-nodebifurcation in maps and ows is identical.

Before briey discussing bifurcations other than the SN bifurcation, we return to an idea that keeps arisingin this section.

5.2.1 Genericity

When we examined the SN bifurcation in some detail, we saw that to ensure that this bifurcation reallytakes place it was not enough simply to get a 0 eigenvalue at the xed point. To avoid pathologicalsituations, we needed to ensure that some genericity conditions held. The form of these conditions wassomething 0, and from this form we see immediately that we have to be unlucky to violate one of

these conditions. Unlucky is of course a loose word. What we mean is, roughly speaking, if you pick aone-parameter family of ows / maps out of all possible families of ows / maps, and if there is a bifurcationin this family (i.e. some eigenvalues lie on the imaginary axis or the unit circle respectively), then thereis probability 1 that the bifurcation will be of the kinds described above. To make these ideas precise youneed to construct a space of one-parameter families and put some structure on it.

When genericity conditions are violated:

A bifurcation might not really occur (e.g. when there is a point of inection in the curve of equilibriain x

space)

There are other more exotic local bifurcations of equilibria which can occur. These do occur inapplications, but almost always because there is some special structure in the system being studied e.g. some symmetry properties. We wont discuss these in detail, but the transcritical bifurcationwe encountered in an example earlier was a nongeneric bifurcation.

[NOTE: This is not a complete discussion of the notion of genericity, which is a powerful and importantidea in dynamical systems!]

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x x

Figure 43: The two possible scenarios in the Hopf bifurcation. The rst is a supercritical Hopf bifurcation.At bifurcation the stable equilibrium becomes unstable, and a stable periodic orbit is born. The secondis a subcritical Hopf bifurcation. An unstable equilibrium becomes stable, and an unstable periodic orbitis born. In both cases the newly born periodic orbit grows in size as we move away from the bifurcationpoint.

Assuming that this equation has an equilibrium at x = x0 , y = y0 , the basic condition for a Hopf bifurcationto occur at = 0 is that the Jacobian

f x f yg x

g y

has a pair of purely imaginary eigenvalues when evaluated at x = x0 , y = y0 , = 0 . This condition is easyto check. The full analysis of the system is quite complicated, and as before leads to genericity conditions,which will ensure that the bifurcation really does take place. The rst condition is simple enough, but thesecond is quite hard to state and check! They are

1. f x + g y 0 and

2.1

16 ( f xxx + g xxy + f xyy + g yyy) +1

16 ( f xy( f xx + f yy) g xy(g xx + g yy) f xxg xx + f yyg yy) 0where is the size of the imaginary part of the eigenvalues evaluated at bifurcation point.

Note that although the second genericity condition looks very complicated, it is still almost always goingto be true we would have to be unlucky to violate it. In 2D applications genericity is straightforward buttedious to check.

In higher dimensional ODE systems a Hopf bifurcation generally occurs when the Jacobian evaluated at acritical point has a pair of purely imaginary eigenvalues. To check genericity conditions one would haveto calculate an invariant 2D subspace the centre manifold and nd a form for the system restricted tothis subspace. Since this is in practice very complicated one often simply checks for the main bifurcationcondition (a pair of purely imaginary eigenvalues) and then states that provided the genericity conditionsarent violated we have a Hopf bifurcation.

5.3.3 The Hopf bifurcation for maps

The Hopf bifurcation for xed points of maps has some similarity to the Hopf bifurcation for ows. Itoccurs when the Jacobian at the xed point has a pair of eigenvalues on the unit circle, but not at 1 or 1.

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There are a variety of complicated possibilities that can arise, when the eigenvalues are roots of unity, butthe general situation is that shortly after bifurcation the map has an invariant circle near to the xed point.This circle may be attracting or repelling depending of whether we have the supercritical or subcriticalcase.

Note that for a map, an invariant circle is not the same as a periodic orbit for a ow it is not a single orbit.Dynamics on the circle could be periodic or could be aperiodic (e.g. a point could have a dense orbit onthe circle). In general as the bifurcation parameter moves away from bifurcation there will be parametervalues at which there is periodic behaviour interspersed with others at which there is aperiodic behaviour.

If the map in question is really a Poincar e map constructed near a periodic orbit of a ow, then what woulda Hopf bifurcation mean? The invariant circle would actually be a torus. The dynamics on this torus couldbe either periodic or quasiperiodic.

5.4 Global bifurcations

We have nished our classication of generic local bifurcations of xed points of ows and maps. Of

course there are many other kinds of local bifurcations non-generic bifurcations, and bifurcations whichoccur in 2-parameter families for example but we will not study these. Instead we nish this course byglancing briey at global bifurcations this is a taster, not a rigorous discussion.

5.4.1 Homoclinic bifurcation in a ow

Homoclinic orbits were dened in Section 4.9.1. They are important in many global bifurcations, and wewill look at a continuous and a discrete example. The rst example is of a homoclinic bifurcation in aplanar ODE system. This bifurcation is illustrated in Figure 44. As some parameter passes through 0, a

homoclinic orbit is created and destroyed immediately. On one side of = 0 this gives rise to a periodicorbit. If approaches 0 from the side where there is a periodic orbit, the period of the periodic orbitincreases towards innity.

= 0 > 0< 0Figure 44: Periodic orbits can be destroyed in a rather exotic way - they come closer and closer to anequilibrium of saddle type. Their period tends towards innity. At bifurcation there is a homoclinic orbitand no periodic orbit.

One way of looking at this homoclinic bifurcation is as another way that periodic orbits can be born ordie in families of ows, apart from in Hopf bifurcations. This homoclinic bifurcation can be studiedusing Poincare maps dened on appropriate sections.

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5.4.2 Homoclinic bifurcation in a map

Our second example of a homoclinic bifurcation is a wonderful situation which can occur in 2D maps, andwhich is still the subject of a lot of research. First we need to mention that a homoclinic orbit of a xedpoint of a map is dened identically to a that of a xed point of a ow given a xed point p, if thereis some point x which has p as its -limit set and as its -limit set, then x is a homoclinic point, and theorbit of x is homoclinic to p. Note that a homoclinic orbit in a map (like any orbit in a map) is a discretesequence of points, and not a line as for a ow.

Suppose an invertible 2D map has a xed point of saddle-type i.e. the Jacobian has two real eigenvalues,one inside the unit circle, and one outside. The local situation is shown in Figure 45. The lines shownleading towards the xed point are the stable and unstable manifolds of the xed point sets of pointswhich tend towards the xed point in forward and backward time respectively. (Of course, most pointdont go towards a saddle in either forward or backward time, but instead tend away from it along theunstable and stable manifolds respectively). We havent proved that these manifolds must exist, but this isa basic result in dynamical systems.

Figure 45: The situation near a saddle point is shown. Sinceorbits of a map consist of discrete points, the stable and unstablemanifolds consist of many orbits, not a single orbit.

When iterated, a point on the stable manifold jumps to another point on the stable manifold and if wekeep iterating for long enough it gets closer and closer to the xed point. Points on the unstable manifolddo the same in negative time.

Now consider the following situation imagine the stable manifold and the unstable manifold crossingin a point (see Figure 46 a). This situation is quite possible since orbits consist of discrete points. Thecrossing point is now part of a homoclinic orbit because its iterates must tend towards the xed pointboth in forward and in backward time. But this fact forces the stable and unstable manifolds to intersectagain, and again, and again... The geometry of this situation is shown in Figure 46 b). Apart from thepoints on the original homoclinic orbit new homoclinic orbits must exists because the stable and unstablemanifolds both get squeezed in one direction and expanded in the other if we continue them backwardsand forwards in time respectively. In fact it isnt very di fficult to show that this geometrical crossing of thestable and unstable manifold must imply chaos i.e. the existence of closed bounded sets which containcountably many periodic orbits and uncountably many aperiodic orbits. In fact there are innitely many

such sets.

To see where a global bifurcation comes in, consider varying a parameter so that the situations are asshown in Figure 47. As passes through 0 we create a homoclinic tangle.

The homoclinic tangency is a very complicated bifurcation indeed. Before it occurs the dynamics of themap are quite simple (in the region shown at least) and then suddenly very complicated dynamics arecreated. And yet this bifurcation is not rare it occurs in some very simple 2D systems.

Well end these course-notes with these exotic pictures. They show that even in simple looking dynamical

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homoclinic points

homoclinic pointssecondaryhomoclinic point

b)a)

Figure 46: A saddle with a single homoclinic point has wild implications. It means that there must becountlessly many homoclinic orbits. It is not hard to show that this situation implies chaos. The situationshown in b) is often referred to as a homoclinic tangle .

= 0 > 0< 0

Figure 47: For < 0 there are no homoclinic points. At = 0 there is a homoclinic tangency andhomoclinic points are created. For > 0 we have a homoclinic tangle. Thus, as passes through 0,immensely complicated dynamics are suddenly created.

systems, we can get complicated and strange behaviour, and moreover this behaviour can be born suddenly.Before the eld arose, when such behaviour was observed in real-world situations, it was often dismissedas noise. The pictures also illustrate how we can use purely geometric arguments to give us a huge amountof information about a dynamical system. Although the arguments look loose all of them can be madeprecise and exact.

What weve studied in this course are only some of the many beautiful phenomena which occur in dy-namical systems. If you want to explore further there are a number of excellent books ranging from theelementary to the advanced. You might want to look at some of these they are arranged in approximateorder of di fficulty.

An Introduction to Dynamical Systems by D. K. Arrowsmith and C. M. PlaceAn Introduction to Chaotic Dynamical Systems by Robert DevaneyStability, Instability and Chaos by Paul GlendinningIntroduction to Applied Nonlinear Dynamical Systems and Chaos by Stephen Wiggins

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Indexasymptotic stability, 72asymptotically stable, 18

basin of attraction, 8bifurcation, 39, 45, 89

global, 100homoclinic, 100, 101Hopf

ows, 96, 98maps, 97, 99

period doubling, 40, 97, 98pitchfork, 41saddle-node, 39, 89, 91, 95, 96transcritical, 94

bijection, 14

Cantor set, 2931Cantors intersection theorem, 23, 24, 55chain rule, 19chaos, 3335, 54, 101, 102cobweb diagram, see graphical analysisconjugacy, see topological conjugacycontinuous, 14coordinate transformations, 64

dense, 25diff eomorphism, 15doubling map, 26

energy functions, 77eventually periodic point, 10existence (of solutions to an ODE), 50

xed point, 10ip bifurcation, see period doubling bifurcationow (generated by an ODE), 52

gradient systems, 79graphical analysis, 17

Hamiltonian systems, 78heteroclinic orbit, 81homeomorphism, 15homoclinic orbit, 80, 100, 101homoclinic tangency, 102homoclinic tangle, 102homogeneous ODE, 58hyperbolicity

bifurcations, 96

equilibria of ows, 70xed points of 1D maps, 19, 20xed points of general maps, 45

IFT, see implicit function theoremimplicit function theorem, 92inhomogeneous ODE, 58intermediate value theorem, 15invariant set, 8, 29, 54, 75invertible, 14itinerary, 23, 24, 34IVT, see intermediate value theorem

Liapunov function, 73, 75Liapunov stability, 72, 73limit set, 8, 28, 54, 80, 81linearly stable, 18linearly unstable, 18logistic family, 36, 37logistic map, see logistic family

mean value theorem, 16MVT, see mean value theorem

ODE, see ordinary di ff erential equationsorbital stability, 73ordinary di ff erential equations, 47

linear autonomous ODEs, 59, 62, 66, 68linear homogeneous ODEs, 58linear inhomogeneous ODEs, 59time periodic, 86

period 3 point, 20periodic orbit

ows, 53, 73, 8185, 98101maps, 23, 24, 37

periodic point of a map, 10, 18Poincar e map, 8487Poincar e-Bendixson theorem, 8184

quadratic family, see logistic family

recoordinatisation, see coordinate transformationsreturn map, see Poincar e map

Sarkovskiis theorem, 22scalar function preserving systems, 77semi-conjugacy, 32sensitive dependence on initial conditions, 33

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sequence space, 28shift map, 28solution

ODEchaotic, 54constant, 53periodic, 53, 81, 84quasiperiodic, 53, 100

stabilityows, 61, 62, 66, 701D ows, 57global stability, 72, 73

maps1D maps, 18higher dimensional maps, 43

state space, 6superstable, 37symbolic dynamics, 28

Taylor expansions, 59topological conjugacy, 31topological transitivity, 33

uniqueness (of solutions to an ODE), 50

vector eld, 48

dynamical systems chaos

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