Dynamical systems Jacobian matrix

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. 

To highlight the relationship of the matrices A and to the quantities discussed in Section VILA (Dynamics of Metabolic Systems) and Section VII.B (Metabolic Control Analysis), we briefly outline an alternative approach to the parameterization of the Jacobian matrix. Note the correspondence between the saturation parameter and the scaled elasticity ... 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

C.l) has the form (B.9) where F and G are independent oft and (B.12) holds in D, then w is a monotone dynamical system with respect to If in addition, the Jacobian matrix of f is irreducible at every point ofD, then w is strongly monotone with respect to... 

In this experiment, a constant inflow of one species at a time is added to the system at steady state near a supercritical Hopf bifurcation. This inflow additional to the inflow terms in eq. (11.2) should not be large enough to shift the system from one dynamics regime to another, for example, from a stationary state to an oscillatory state. The response of the concentrations of as many species as possible should be determined after the addition of each species and compared to the steady-state concentrations of the unperturbed system. These measurements allow the construction of an experimental shift matrix, which is directly related to the Jacobian matrix. If we approximate the... 

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... 

Torque-based impedance controller, x is the robot actual pose in the task space computed from the actual joint configuration q with the forward kinematics (FK) block J is the robot Jacobian is the desired pose in the task space x is the equilibrium pose of the environment is the net stiffness of the sensor and of the envirotunent f j and are the external enviroiunent forces expressed in the task space and in the joint space, respectively fj is the desired force vector is the desired torque vector computed from the force equilibrium r is the torque input vector of the inner torque control loop and is the commanded motor torque vector. The command force f is defined as f = Z(x - x), where Z is the impedance matrix. When the environmental forces are available (dotted lines), the measurements are used to decouple the dynamic of the system. 

---