1 Chapter 4B

4.7 Topological Conjugacy and Equivalence

This section is concerned with classification of dynamical systems. First we need some notions from analysis and topology.

A map is

 surjective or onto if for all there is at least one such that ,  injective or one-to-one if implies ,  bijective if both surjective and injective.

Surjective or Onto: sketch a surjective map from to 

Injective or One-to-one: sketch an injective map from to 

Bijective: sketch a bijective map from to 

A bijective map establishes a one-to-one correspondence between elements and has an inverse.

A map is a homeomorphism if it is continuous, bijective and has a continuous inverse.

A map is a diffeomorphism if it is , bijective and has a inverse.

Example. defined by . sketch

Inverse . sketch.

is a homeomorphism but not a diffeomorphism. Note that is not at . ■

The flows and are topologically conjugate or conjugate if there is a homeomorphism such that

. [1]

We can also write or , a form which recalls similarity transformations from linear algebra. A commutative diagram for [1] is

2 Chapter 4B

Both pathways around the diagram give the same result.

Example. Show that the flow of is topologically conjugate to that of .

We need to find a homeomorphism between the two flows such that [1] holds.

Find and .

Then [1] requires .

For find that works. This does not give a homeomorphism if extended to .

For find that works.

Thus from the previous example is the required homeomorphism. ■

Topological conjugacy requires that orbits be put into one-to-one correspondence. For the previous 1D example we have the picture.

 -, arrows, show correspondence between the three orbits of this with -, arrows

In 2D we have sketch a figure like 4.12.

Topological conjugacy requires more than that the trajectories of one map map continuously into those of the other. It also requires that the temporal parameterization be the same.

Two flows and are topologically equivalent if there exists a homeomorphism that maps the orbits of onto the orbits of and preserves the direction of time. That is, look for an increasing map such that

. [2]

Example. On the flow of is topologically equivalent to that of .

We need to find a homeomorphism between the two flows and a map such that [2] holds.

Find .

To find the second flow, solve the initial value problem

.

3 Chapter 4B

Separate variables and use the initial condition to find .

In the notation used for flows, .

Then [2] becomes .

sketch -, -,   -, -, 

The flows on are qualitatively similar. Try . Then [2] becomes

, solve for to get

For fixed, is an increasing function of . ■

Theorem 4.10 (One Dimensional Equivalence). Two flows and in are topologically equivalent iff their equilibria, ordered on the line, can be put into one-to-one correspondence and have the same topological type (sink, source or semistable).

Do not give proof.

Example. On the flow of is not topologically equivalent to that of . Sketch the flows

Two flows and are diffeomorphic when there is a diffeomorphism such that [1] is satisfied.

Example. Show that the flow of

,

Is diffeomorphic to that of

, .

First find the flows. Denote , etc. Obtain . Then the equation for becomes:

,

Multiply by an integrating factor and integrate to obtain

.

4 Chapter 4B

In the notation of flows we have

and .

We now seek to satisfy [1]. The first component may be written

.

Note that is a solution. The second component of [1] may be written

.

A short calculation shows that is a solution. The required diffeomorphism is

.

To sketch the two flows, note that the unstable eigenspace is the image of and the stable eigenspace is the image of .

axes, , and , axes,  ■

Consider diffeomorphic flows with a pair of corresponding equilibria: and with and , with and . Take a time derivative of the diffeomorphism to obtain

,

.

Differentiate again with respect to

.

Evaluate at and use to obtain

. 5 Chapter 4B

Let and . These two matrices appear in the linearizations of the flows about the equilibrium points:

and .

Let . Then the matrices of the linearized systems are related by

or .

The matrices are related by a similarity transformation. They have the same eigenvalues, and these have the same algebraic and geometric multiplicities.

Theorem 4.11 (Linear Conjugacy). The flow and of the linear systems and are diffeomorphic iff the matrix is similar to the matrix .

Proof. Note and .

Assume is similar to , i.e., there is a nonsingular matrix such that . Then is a diffeomorphism and

, which implies and are diffeomorphic.

Assume and are diffeomorphic. Then there is such that and

for all and . Differentiate to obtain

.

Evaluate this at and set to find . Differentiate with respect to and set to finally obtain . Thus and are similar. □

Example. The matrices and are not similar.

The matrix is similar to no other matrix than itself. To see this, suppose that is any invertible matrix. Then since commutes with . It follows that the flows and of and cannot be diffeomorphic.

More generally, recall the classification of linear systems. Let and , and .

 -, -, the parabola  6 Chapter 4B

Similarity transformations preserve both the algebraic and geometric multiplicities of eigenvalues. Therefore the flows of linear systems on the parabola that have only a single eigenspace cannot be diffeomorphic to flows that have two eigenspaces. In particular, the only eigenspace of is . ■

The flows associated with the matrices and in this example are topologically conjugate.

Theorem 4.12. Suppose and are two real, hyperbolic matrices and and the corresponding flows. Then and are topologically conjugate iff the dimensions of the stable and unstable spaces of are equal to the corresponding dimensions for .

Proof. Don’t give.

Show topological equivalence classes of linear systems from Hale and Kocak

4.8 Hartman-Grobman Theorem

Recall linearization of an ODE about an equilibrium point. Let and suppose . Then

, where and we have used . Let . The linearization of the system at is

.

Theorem 4.13 (Hartman-Grobman) Let be a hyperbolic equilibrium point of the vector field

with flow . Then there is a neighborhood of such that is topologically conjugate to its linearization on .

Example. Consider the system

, [1]

.

Study this system as the parameter varies. The system linearized about the origin is 7 Chapter 4B

, where . [2]

The eigenvalues of are . The linearized system is a stable focus for , an unstable focus for , and a center for . The Hartman-Grobman theorem guarantees that the flow of the nonlinear system is topologically conjugate to that of the linearized system in some neighborhood of the origin so long as the equilibrium point is hyperbolic, that is

At , there is a bifurcation, a qualitative change in the character of the solution. The Hartman-Grobman theorem does not apply because the equilibrium point is not hyperbolic. At the linearized system [2] is a center. What is the nature of the flow for the system [1] at We can study this by transforming to polar coordinates

,

.

Differentiating, one finds

,

.

From the second equation obtain . At the system is described by

and .

Study the dynamics of by plotting as a function of

0…r-, 0… -, arrows

The system is an unstable nonhyperbolic focus.

sketch and compare with the flow of the center.

At the flow in any open neighborhood of the origin of the nonlinear system [1] is qualitatively different from that of the linearized system [2]. ■

Discussion of Proof. Meiss’ discussion of the proof of the Hartman-Grobman theorem is interesting and the main point is easy to understand. Write in the form

8 Chapter 4B

where represents the nonlinear terms, so that . Let be the flow of the nonlinear ODE and the flow for the linear part. We wish to find a homeomorphism satisfying

or

. [3]

Suppose is a homeomorphism that satisfies this equation for one value of time, say . Then

. [4]

Now let

. [5]

Then it follows from the group property of the flow and [4] that

Thus also satisfies [4].

In the actual proof, is the flow of the system , with in some neighborhood of the origin and outside some larger neighborhood . Then, under the hypotheses of the theorem, it is possible to show that the solution to [4] is unique. It follows that . Therefore, by [5]

, satisfying [3] and is the homeomorphism that we originally sought!

The time 1 homeomorphism can be found iteratively, by starting with the , and then successively solving

, as described by Miess. In general, convergence of the iteration scheme requires outside of a neighborhood of the origin. 9 Chapter 4B

Remark. If in the above theorem we have , then is diffeomorphic to its linearization in some neighborhood of a hyperbolic equilibrium point. See Perko, p. 127.

4.9 Omega-Limit Sets

Consider a complete flow , point with forward and backward orbits and

.

is a limit point of if there is a sequence of times such that as . The -limit set of is

.

limits sets are important in applications because they characterize the steady state behavior of solutions of differential equations.

is a limit point of if there is a sequence of times such that as . The -limit set of is

.

Example. An asymptotically stable equilibrium point is an -limit set of all points in some neighborhood of . sketch point, nearby trajectories

A source point is an -limit set of all points in some neighborhood of . sketch point, nearby trajectories. ■

A limit cycle is a periodic orbit that is the - or - limit set of some point .

Example. The planar system ( and are real).

,

.

Polar coordinates sketch axes, 

,

.

Take time derivatives 10 Chapter 4B

where

.

sketch -, -, 

sketch -, -, circle, flow arrows, nearby trajectory

is the -limit set of every point in ■

[Revert to ]

Lemma 4.14 (Closure) , where is the closure of the forward orbit of Hence, is closed.

Proof. If , then for all , since this includes all limit points. Therefore is in the intersection of all of these sets.

Conversely, let . If then it is in the closure of but not in the orbit itself, so is a limit point of the forward orbit. Thus . Otherwise for some

and for some . It follows that infinitely often, so .

Finally recall that the intersection of a closed family of sets is closed. □

Examples.

 An asymptotically stable equilibrium point is a closed set.  A limit cycle is a closed set.

 for some . Then , a closed set! ■

Lemma 4.15 (Invariance) -limit sets are invariant.

Proof. If , then there is a sequence such that . Continuity and the group property then imply that for any fixed :

.

Therefore □

Remark. By the same reasoning, -limit sets are invariant. 11 Chapter 4B

A set is connected if it cannot be partitioned into two nonempty subsets such that each subset has no points in common with the closure of the other.

Example. Let . This set is connected. and partition , but has points in common with the closure of . Let . This set is not connected. and partition . The closure of has no points in common with and conversely. ■

Lemma 4.16 (Compact and Connected). If the forward orbit of is contained in a compact set, then is nonempty, compact and connected. Furthermore, as .

Proof. Let be compact and . Then is an infinite sequence contained in , which must have a convergent subsequence. Let this be .

Then . Thus , and is nonempty.

Note (from lemma 4.14) that is closed and contained in a compact set and is therefore itself compact.

Suppose is not connected. Then it must consist of two disjoint closed components and . Since these are closed, they must be some finite distance apart. Since both , there must be arbitrarily large times such that and other arbitrarily large times such that . Since these distances are continuous functions of time, there must be an infinite sequence of times such that The sequence

is contained in and must have a subsequence converging to some such that . By definition, . But and . This is a contradiction, so must be connected.

Finally, suppose . Then for some there must be a sequence of times

such that . This must have a subsequence converging to some

, which would be a limit point of not in , a contradiction. Thus . □

Example. Consider the mechanical system

, .

There are 3 equilibrium points and . The potential energy is

. The associated Hamiltonian is

12 Chapter 4B

. The energy is constant along trajectories. We can sketch the phase portrait using .

 -, -, orbits corresponding to 

 -, -, , and energy levels 

compare with figure generated by XPP on handout; note regions where 

Modify the system

Then

.

compare with figure generated by XPP on handout

Consider the trajectory inside the right lobe of the heteroclinic figure eight. Then and

, except where it intersects . For the -limit set is the entire right lobe of the figure eight. The analysis of trajectories inside the left lobe is similar.

For a trajectory that start outside the figure eight, and . The -limit set is the entire figure eight.

However, if the initial condition is on the figure eight, then . ■

A point is nonwandering if for every neighborhood of and every time there is a

such that .

Examples

 Every equilibrium point is nonwandering.  Every periodic orbit is nonwandering.  Every point on the heteroclinic figure eight of the previous example is nonwandering. ■

13 Chapter 4B

A set S is minimal if it is closed, nonempty, invariant and does not contain any such set as a proper subset.

Theorem 4.17. Suppose is compact; then is minimal iff for each we have .

Proof. Assume that for each we have , but is not minimal. Then there is a closed set that is invariant. This implies that if then , which is a contradiction.

Conversely, assume is minimal but that there is for which Since is compact, so is and lemma 4.15 implies that is invariant, so has an invariant subset, which is a contradiction. □

4.10 Attractors and Basins

Recall the distance between a point and a set: .

Define the closed ball of radius around a compact set :

.

is a neighborhood of a compact set if contains an open set containing .

A compact invariant set is stable if such that

.

A compact invariant set is asymptotically stable if it is stable and, in addition, there is a neighborhood such that for each as .

Example. The limit cycle defined in polar coordinates by and . [1]

sketch the limit cycle and the nearby flow

is compact and invariant. For , the ball is a tube of radius enclosing .

Add to sketch.

is asymptotically stable. ■

A set is a trapping region of the flow if is compact and for .

Example. is trapping region if the vector field of points inwards everywhere on the boundary of . sketch ■ 14 Chapter 4B

A collection of closed sets is nested if whenever .

Lemma 4a. If is a nested collection of nonempty compact sets then

is compact and nonempty.

Proof. is closed since it is an intersection of closed sets. is bounded since it is contained in the compact set . To see that it is nonempty choose a sequence of points

. Since this sequence is contained in it must have a convergent subsequence . The limit is contained in every set and therefore it is in the intersection. Therefore is nonempty. □

Lemma 4b If is a trapping region of the flow , then for .

Proof. The statement is clearly true for . Let . Then . Operate on both sides by and use the group property to obtain the result. □

Lemma 4c If is a trapping region then is a nested collection of compact sets.

Proof. Since is compact it is clear* that is compact for . To show the collection is nested consider times and . Since is a trapping region and we have

. Operate on both sides of this relation with and use the group property of flows to get . □

*[Certainly is bounded. To see that is closed consider a Cauchy sequence in . It must converge to some limit . Since is continuous on a compact set, it is uniformly continuous. By the uniform continuity of the corresponding sequence in will also be Cauchy. It then converges to .]

A set is an attracting set if there is a trapping region containing such that

. [2]

By lemma 4a and lemma 4c, is compact and nonempty.

Lemma 4.18’ An attracting set is asymptotically stable. If a compact set is asymptotically stable it contains an attracting set. 15 Chapter 4B

Proof. First suppose is an attracting set; then by definition every trajectory in any associated trapping region stays in and approaches -- so is asymptotically stable.

Now assume that is compact and asymptotically stable. Construct a trapping region. Since

is asymptotically stable there is a ball of initial conditions from which all trajectories stay in some larger ball and eventually approach . There exists a least time for each such that for all . Let

.

is a trapping region. □

Remark. The argument that is a trapping region does not seem complete. How do we know that

for ? But surely the statement of the lemma is true. Can anyone complete the argument?

The basin of attraction or stable set of an invariant set is the set of all points for which as . If is an attracting set with trapping region then

A set is an attractor if it is an attracting set and there is some point such that .

Example. The limit cycle from [1]. For , the toroidal is a trapping region. Let

. is compact and asymptotically stable. Consistent with theorem 4.18, , which is compact and asymptotically stable, contains the attracting set . . The basin of attraction of is . Finally, is an attractor as it is the -limit set of every point in . ■

Example. The attracting figure eight structure from figure 4.18. Show transparency or handout Let be the set of points that constitute the figure eight. The Hamiltonian,

, associated with this structure has values and

Let and . Then is a trapping region. The vector fields points inward everywhere on , except on the set

. If , the dynamics immediately take off of to where the vector fields points inward. Note that and satisfy [2]. is an attracting set. is also an attractor since it is the -limit set of any point outside of the figure eight. ■

16 Chapter 4B

4.11 Stability of Periodic Orbits

Consider and . Let be a -periodic orbit: . Let , where is a deviation away from that orbit. Expand about the periodic orbit: and give

.

Linearize:

, [1] where is -periodic: . Solutions are the subject of Floquet theory.

The fundamental matrix solution of [1] satisfies

, [2]

Any solution of [1] may be expressed in the form

. [3]

The monodromy matrix is .

Geometric Interpretation. is the deviation from a periodic orbit. is the change in that deviation after one period. If is an eigenvector of then , where is an eigenvalue. If then the deviation from the periodic orbit grows. If then the deviation decays. The eigenvalues of are the Floquet multipliers.

Theorem 4.19. The monodromy matrix for the linearization of about a periodic orbit has at least one unit eigenvalue.

Proof. . Differentiate with respect to time:

. [4]

Thus is a solution of [1]. Use [3] to represent in terms of the fundamental matrix:

.

is -periodic: we have . Thus is an eigenvector of with unit eigenvalue. □ 17 Chapter 4B

Geometric Interpretation: points in the direction of a phase shift. periodic orbit , based at a point on that orbit. Another point phase shifted slightly in front The phase shift remains constant after one orbit.

A periodic orbit is linearly stable if all of its Floquet multipliers have magnitude at most one

.

A periodic orbit is asymptotically linearly stable if all of its Floquet multipliers, apart from the trivial unit multiplier, have magnitude less than one

.

Example (Strogatz, 1994). Find the fundamental matrix and discuss the stability of the periodic orbit of and for all values of .

The dynamics take place on the cylinder . sketch. Let and The Jacobian matrix of [1] is

.

The periodic orbit is . It’s time derivative is a solution of the linearized system [1]. We can find another solution by finding the difference between a solution with initial conditions and . Let

,

and . These are two solutions of the linearized system with the property

. Then the Fundamental Matrix Solution of [2] is given by

and the monodromy matrix is .

Alternatively, we may take advantage of the fact that, for this simple example, is time independent and calculate directly

.

The system is asymptotically linearly stable for , stable for and unstable for . ■ 18 Chapter 4B

Example. Find the fundamental matrix and discuss the stability of the periodic orbit (in polar coordinates) and .

Our first goal is to calculate the Jacobian matrix for this system. Since it is given in polar coordinates, we rewrite it in Cartesian coordinates using and . Then

In this way we find

,

.

Calculate the partial derivatives and evaluate them at the periodic orbit to find

[5]

One can now explicitly verify that is a solution of [1] (see[4]). Geometrically, points in the direction of the tangent vector

sketch the orbit, a point on the orbit, and the tangent vector .

Find a second solution of [1] that is orthogonal to . Set and let be the deviation from the orbit in the radial direction. Then

.

Substitute and and drop the higher order term to obtain , whose solution is . Now find the corresponding and deviations:

, 19 Chapter 4B and similarly,

.

A vector that expresses deviations from the periodic orbit in the radial direction is

.

add to sketch One can now explicitly verify that this is a second solution of [1].

Put and notice that and . We can express the solution of [2] as

.

The monodromy matrix is

.

The periodic orbit is linearly asymptotically stable. ■

Periodic orbits in 2D have two Floquet multipliers and one of these is given by theorem 4.19.

Abel’s theorem (Chapter 2) gives , which can be used to determine the stability of an orbit in 2D without solving [2].

Example. From [5] we have and it follows ■

4.12 Poincaré Maps

Consider a periodic orbit in and a section intersecting and such that the vector field is perpendicular to at the point of intersection and transverse to at every point.

sketch , 

Choose sufficiently close to . add  Follow the flow to the first return to . Let be the time required to return to from the same side. The Poincaré map is defined by

. 20 Chapter 4B

If for all eventually crosses and then returns to at a later time, then is a global section.

Example. Consider the skew-product system

,

,

where and . This has global section . ■

If and are two global sections, the corresponding Poincaré maps are topologically conjugate.

sketch , trajectory from to 

Let be the time required for a trajectory with initial point to arrive at .

Consider the homeomorphism defined by . Then the topological conjugacy corresponds to the diagram

Thus or . In principle, knowledge of one Poincaré map allows us to reconstruct the other. and contain the same information about the flow.

A locally defined Poincaré map always exists in a neighborhood of a periodic orbit .

sketch 

Let . The section is perpendicular to . By continuity of , there is some neighborhood for which is transverse to . Let be the period of the orbit, then . Let near . Then by continuity of as a function of , for some time close to .

21 Chapter 4B

Example. Find the Poincaré map for the system on :

.

Solve to find , then . Let to write . The Poincare map is graphed below for (blue line). Also shown is the line (red).

Successive intersections of a trajectory with the Poincaré map can be constructed by cobwebbing. Start at some initial value on the Poincaré section. Move vertically to the map to get the new intersection of the trajectory with the section at . Then move horizontally to the line , giving . Repeat to generate new values in the sequence of intersections between the trajectory and the section.

Places where the graph of the Poincaré map intersects the line with a slope less than 1 correspond to stable fixed points, as shown above. Intersections with slope greater than 1 correspond to unstable fixed points. ■

Example. Find the Poincaré map for the system in polar coordinates

, where . Separate variables, integrate and exponentiate both sides to get

,

where is an undetermined constant. Let at and solve for to obtain

.

The period is . Let and to obtain

.

22 Chapter 4B

A Poincaré map is given below for (blue). The cobweb construction can again be used to generate successive iterates.

The fixed point at corresponds to an unstable equilibrium point at , and the fixed point at corresponds to a stable periodic orbit. ■

Stability calculated using the Poincaré map is the same as stability calculated using Floquet multipliers.

Theorem 4.20 Let be a periodic orbit of a flow , be a local section through a point

and be the Poincaré map. If the monodromy matrix of is , then

, where is the spectrum of matrix

Proof. For any in a neighborhood of , let be the time of first return to and let

. Then . Differentiate :

.

The last term is an outer product. Note that . Let the period of be .

Without loss of generality, let be the origin. Then , ,

) and , giving

. [1]

Recall that is a vector tangent to and is also an eigenvector of with unit eigenvalue.

sketch ,  23 Chapter 4B

Let be an orthonormal basis for and let be the unit vector in the direction of . Using these coordinates, write in components as

,

and thus must have the structure

.

Then write [1] as

.

Then and the result follows. □

Theorem 4.21 If is a periodic orbit of a flow that is linearly asymptotically stable (the spectrum of the Poincaré map is inside the unit circle), then it is asymptotically stable.

Do not give proof.