Jacobian matrix dynamic stability

Figure 20. The stability diagram. Depending on the trace Tr and determinant A ad be of the Jacobian matrix M, the steady state can be classified intodistinct dynamic regimes. The parabola indicates the line Tr2 4A, corresponding to the occurrence of imaginary eigenvalues. For an interpretation of the different dynamic regimes, see text.

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

A Lyapunov exponent is a generalized measure trf the growth or decay of small perturbations away from a particular dynamical state. For perturbations around a fixed point or steady state, the Lyapunov exponents are identical to the stability eigenvalues of the Jacobian matrix discussed in an earlier section. For a limit cycle, the Lyapunov exponents are called Floquet exponents and are determined by carrying out a stability analysis in which perturbations are applied to the asymptotic, periodic state that characterizes the limit cycle. For chaotic states, at least one of the Lyapunov exponents will mm out to be positive. Algorithms for the calculation of Lyapunov exponents are discussed in a later section in conjunction with the analysis of experimental data. These algorithms can be used for simulations that yield possibly chaotic results as well as for the analysis of experimental data. 

The stability of such a cycle again depends on the eigenvalues of the Jacobian matrix of G. As an example we consider the dynamic system... 

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