Locally topologically equivalent system

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

Definition 8.2. A system X [continuous or discrete) is said to have a modulus if in some subspace B of the space of dynamical systems, where X G B, a continuous, locally non-constant functional h is defined such that if X and X are topologically equivalent, then h[X) = h[X). 

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