Eigenvalue analysis dynamic system

The above equation is called the characteristic equation of the system. It is the system s most important dynamic feature. The values of s that satisfy Eq. (6.67) are called the roots of the characteristic equation (they are also called the eigenvalues of the system). Their values, as we will shortly show, will dictate if the system is fast or slow, stable or unstable, overdamped or underdamped. Dynamic analysis and controller design consists of finding out the values of the roots of the characteristic equation of the system and changing their values to give the desired response. The rest of this book is devoted to looking at roots of characteristic equations. They are an extremely important concept that you should fully understand. ... 

We will show in Sec. 15.3 that the eigenvalues of the 4 matrix are the roots of the characteristic equation of the system. Thus the eigenvalues tell us whether the system is stable or unstable, fast or slow, overdamped or underdamped. They are essential for the analysis of dynamic systems. 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

The theoretical analysis here in the present section clearly indicates that the localized delta function excitation in the physical space is supported by the essential singularity (a —> oo) in the image plane. This is made possible because 4> y, a) does not satisfy the condition required for the satisfaction of Jordan s lemma. As any arbitrary function can be shown as a convolution of delta functions with the function depicting the input to the dynamical system. The present analysis indicates that any arbitrary disturbances can be expressed in terms of a few discrete eigenvalues and the essential singularity. In any flow, in addition to these singularities there can be contributions from continuous spectra and branch points - if these are present. 

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. 

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. 

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. 

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]. 

Figure P14.8 also shows the error norm, ej, versus the number of Ritz vectors (from Problem 14.7). The error is smaller when Ritz vectors are used, because they are derived from the force distribution. Ritz vectors are useful for dynamic analysis of large systems with classical damping, since the vibration properties of the system can be obtained by solving, a smaller eigenvalue problem of order 7, instead of original eigenvalue problem of size N. It must be noted that the resulting frequencies and mode shapes are approximations to the...

Previous work using eigenvalue tracking (ET) as a method of spectral association has been successfully applied for the purposes of dynamic analysis and model reduction. ET uses homo-topy methods that transform a system with known eigenvalue-to-state association into the final system and track the eigenvalue associations as the system is transformed. 

In each case, we first studied the laser driven dynamics of the system in the framework of the Floquet formalism, described in Sect. 6.5 of Chap. 6, which provides a geometrical interpretation of the laser driven dynamics and its dependence on the frequency and amplitude of the laser field, through the analysis of the eigenvalues of the Floquet operator, called quasienergies. Various effective models were used for that purpose. This analysis allowed us to explain the shape of the relevant quasienergy curves as a function of the laser parameters, and to obtain the parameters of the laser field that induce the CDT. We then used the MCTDH method to solve the TDSE for the molecule in interaction with the laser field and compare these results with those obtained from the effective Hamiltonian described in Sect. 8.2.3 above. 

---