The Grobman-Hartman theorem

The behaviour of trajectories of the system (5.2) in the neighbourhood of the stationary point x, y is generally determined by the properties of the 

The Grobman-Hartman theorem provides a criterion permitting to establish the equivalence (nonequivalence) of two non-linear autonomous systems when conditions (l)-(3) are satisfied. In this way, a counterpart to the relation of equivalence of potential functions of elementary catastrophe theory (see Chapter 2) is obtained in the area of autonomous systems. 

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

When conditions (1)—(3) are not fulfilled, the Grobman-Hartman theorem is not valid. As will be shown later, then we have to deal with the sensitive state of a dynamical system (this corresponds to a degenerate critical point in elementary catastrophe theory). A generalization of the Grobman-Hartman theorem, the centre manifold theorem which may be regarded as a counterpart to the splitting lemma of elementary catastrophe theory, has been found to be very convenient in that case. 

When condition (3) is not fulfilled, we deal in the case of system (5.2) with a structurally unstable centre. In real physical systems this kind of 

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When we deal with a dynamical system in which a sensitive state occurs (assumptions of the Grobman-Hartman theorem are not fulfilled), it may turn out that the sensitive state is associated only with a part of state variables. The variables related to the sensitive state may then be separated and the catastrophes occurring in a system dependent on a smaller number of state variables examined. 

A counterpart to the splitting lemma of elementary catastrophe theory is the centre manifold theorem (also called a neutral manifold), generalizing the Grobman-Hartman theorem to the case of occurrence of sensitive states (5.28). The centre manifold theorem allows us to establish an equivalence (nonequivalence) of two autonomous systems. In this sense it is also a generalization to the case of autonomous systems of the equivalence relationship introduced in Chapter 2 for potential functions. 

The centre manifold method permits us to split the system (5.29) into the subsystem having a sensitive state and the subsystem fulfilling assumptions of the Grobman-Hartman theorem. Note that in the linearized system (5.30) the separation took place automatically as a result of linearity. The phase portrait of the linearized system is of the form shown in Fig. 73. 

Recall that the condition for an occurrence of the catastrophe of a change in the phase portrait nearby a stationary point derives from the Grobman-Hartman theorem if, on changing a control parameter, the Re (A) value does not pass through zero, then the qualitative alteration of the phase portrait in the vicinity of a stationary point cannot occur. 

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. 

This is referred to as the Hartman-Grobman theorem (for more discussion, see [177], where this result is referred to as the Linearization Theorem a proof may be found in [362]). 

That is, the solution started near to z stays near z for all time. The Hartman-Grobman theorem clearly implies that the stability of a given hyperbolic equilibrium point z of a nonlinear system can be inferred from the stability of the origin for the linearization of the system around z. ... 

In the case of Hamiltonian systems, the eigenvalues are, very often, on the imaginary axis, and the Hartman-Grobman theorem is not of much use. However, there is another result which is relevant in this case. Observe that an equilibrium point z = (q, p ) of a Hamiltonian system in kinetic plus potential form... 

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