orbits of homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in into classes of flows sharing the same dynamics, again from the topological viewpoint. Topological equivalence We say that two flows and are topologically equivalent, if there is a homeomorphism , mapping orbits of to orbits of homeomorphically, and preserving orientation of the orbits. In other words, letting denote an orbit, one has for each . In addition, one must line up the flow of time: for each , there exists a such that, if , and if is such that , then . Overall, topological equivalence is a weaker equivalence criterion than topological conjugacy, as it does not require that the time term is mapped along with the orbits and their orientation. An example of a topologically equivalent but not topologically conjugate system would be the non-hyperbolic class of two dimensional systems of differential equations that have closed orbits. While the orbits can be transformed to each other to overlap in the spatial sense, the periods of such systems cannot be analogously matched, thus failing to satisfy the topological conjugacy criterion while satisfying the topological equivalence criterion. Smooth and orbital equivalence More equivalence criteria can be studied if the flows, and , arise from differential equations. Two dynamical systems defined by the differential equations, and , are said to be smoothly equivalent if there is a diffeomorphism, , such that In that case, the dynamical systems can be transformed into each other by the coordinate transformation, . Two dynamical systems on the same state space, defined by and , are said to be orbitally equivalent if there is a positive function, , such that . Orbitally equivalent system differ only in the time parametrization. Systems that are smoothly equivalent or orbitally equivalent are also topologically equivalent. However, the reverse is not true. For example, consider linear systems in two dimensions of the form . If the matrix, , has two positive real eigenvalues, the system has an unstable node; if the matrix has two complex eigenvalues with positive real part, the system has an unstable focus (or spiral). Nodes and foci are topologically equivalent but not orbitally equivalent or smoothly equivalent, because their eigenvalues are different (notice that the Jacobians of two locally smoothly equivalent systems must be similar, so their eigenvalues, as well as algebraic and geometric multiplicities, must be