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.