Eigenvalue analysis dynamic stability

Although the linear complex eigenvalue analysis method is useful in establishing the local stability boundaries of the equilibrium point in the system s parameter space, it does not reveal any information regarding dynamic behavior of the system. Further investigations such as numerical simulations or nonlinear analysis methods may be utilized to study the amplitude and frequency of the resulting vibrations under the mode coupling instability mechanism. 

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. 

For the low-temperature steady state y in Figure 4 (A-2) a similar analysis shows that this steady state is stable as well. However, for the intermediate steady-state temperature yi and Sy > 0 the heat generation is larger than the heat removal and therefore the system will heat up and move away from y2. On the other hand, if 5y < 0 then the heat removal exceeds the heat generation and thus the system will cool down away from 2/2 -We conclude that yi is an unstable steady state. For 2/2, computing the eigenvalues of the linearized dynamic model is not necessary since any violation of a necessary condition for stability is sufficient for instability. 

Linear stability analysis is carried out on dynamical equations linearized about the steady-state solution (Mj,Vj). The steady-state solution is stable if the eigenvalues of the system of linearized equations are negative, unstable if those are positive, and indeterminate otherwise. 

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 predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

C.4. 53. Below we present (following [185]) a list of asymptotic normal forms which describe the trajectory behavior of a triply-degenerate equilibrium state near a stability boundary in systems with discrete symmetry. We say there is a triple instability when a dynamical system has an equilibrium state such that the associated linearized problem has a triplet of zero eigenvalues. In such a case, the analysis is reduced to a three-dimensional system on the center manifold. Assuming that (x, y, z) are the coordinates in the three-dimensional center manifold and a bifurcating equilibrium state resides at the origin, we suppose also that our system is equivariant with respect to the transformation (x,y,z) <- (-X, -y, z). 

The linearized equations of motion of the dynamical systems are presented in Sect. 3.1. An introduction to the modeling of dynamical systems that include frictional constraints is given in Sect. 3.2. In Sect. 3.3, a classification of the linearized equations of motion is given and the consequences of the nonconservative forces such as friction is discussed. The eigenvalue stability analysis method is reviewed briefly in Sects. 3.4 and 3.5 for the general case and the imdamped case, respectively. 

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