Non-autonomous system

By autonomous we mean that F, Fit). This is not really a restriction since any non-autonomous system can always be transformed into an autonomous one by the addition of extra variables. Similarly, an N -order differential system can always be transformed to a first order one by introducing additional variables. 

Nonlinearity In addition, it is well known that the process kinetics shows a highly nonlinear behavior. This a serious drawback in instrumentation and automatic control because, in contrast to linear systems where the observability can be established independently of the process inputs, the nonlinear systems must accomplish with the detectability condition depending on the available on-line measurements, including process inputs in the case of non autonomous systems [23]. 

In view of the above discussion on the transformation properties, it needs to be noted that Eq. (8.30) and thus the resulting equations of motion contain time-dependent matrices. The mathematical theory for such non-autonomous systems has primarily been developed with regard to a periodic dependence on time. The most prominent approaches to solve these systems are the methods of Floquet [75] and of Hill [95]. For further details, see Prothmann [146], Gasch and Knothe [78], or Meirovitch [125]. To pave the way for an eventually autonomous system with a less expensive theoretical framework, not to exceed the scope of the study at hand, we will abstain from a time-dependent orientation of the clamping of the beam. 

Let us prove that the same holds true for the trajectories of the non-autonomous system (10.5.10). First, let us consider more carefully the rescaling of time given by (10.5.13). The meaning of this formula is that the old time t, which parametrizes the trajectories of (10.5.12), is a function of R (p) and of the new time r, which parametrizes the trajectories of (10.5.14), and it is defined by... 

So, we have that the non-autonomous system (10.5.20) is well-defined at R = 0 where it assumes the form... 

By the results of Secs. 5.2 and 5.3, to prove the existence and uniqueness of the stable separatrix of the saddle equilibrium state of the non-autonomous system it is sufficient to check that in a small neighborhood of the equilibrium, for all positive times, the non-linearities remain small along with all derivatives. Thus, we must check that the functions i,2(-R> i R ))/R 3. (x 1) the right-hand side of (10.5.20) are small along with all derivatives, provided that for some small <5... 

This map coincides with the shift map over time f = 4 of some non-autonomous system of the form... 

The closed invariant curve Wq for the Poincare map on the cross-section is the loci of intersection of an invariant two-dimensional torus W with the cross-section. The torus is smooth if the invariant curve is smooth, and it is non-smooth otherwise. If the original non-autonomous system does not have a global cross-section, then other configurations of W are also possible, as... 

As a first step towards the analytical study of the dynamics of the mixed PTV-SHV regime in the vicinity of the SHV base flow, we consider a non-autonomous perturbation of system (4.4.3) as follows ... 

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. 

In the analysis of autonomous non-gradient systems the methods of gradient system theory have proved useful. The notions such as a stationary (critical) point, degenerate stationary point, structural stability (instability), morsification, phase portrait can be directly transferred to autonomous systems. A qualitative description of dynamical autonomous systems is constructed analogously with the description of gradient systems. 

Inspection of the autonomous system (6.145) will being with finding the stationary points, z1 = 0,...,z4 = 0. The system (6.145) has the following stationary points having the non-negative coordinates zx, z3 ... 

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

The surface energy depends not only on the composition of the surface layer, but also on the compositions of the bulk phases. Bulk phases can be declared autonomous, while surface phases are non-autonomous. This distinction is the origin for dynamic surface tension, e.g. for liquid two component systems, as intensively studied in the classical monograph by Defay et al. (1966) and demonstrated in Fig. 2C.1. 

In addition to showing that X changed with respect to H/P, the results indicated also that it depended explictly on time, so that the differential equations describing the predator-prey system are probably non-autonomous. 

All operations, from initial startup to final shutdown, must be as autonomous as possible. Operator functions could be limited to pressing a button to start the nuclear systan and perftnming those duties necessary to operate tire power conversion or other non-nuclear systems. For the most... 

Autonomous control describes processes of decentralized decision making in heterarchical structures. It presumes interacting elements in non-deterministic systems, which possess the capability and possibility to render decisions independently. The objective of autonomous control is the achievement of increased robustness and positive emergence of the total system due to distributed and flexible coping with dynamics and complexity... 

To the theory of periodic oscillations of quasi-linear non-autonomous periodic systems with periodic lags. - In Proceedings of V Intemat. Conf. on Nonlinear Oscillations, vol. 1, 617-622, Inst. Matematiki Akad. Nauk Ukrain. SSR, Kiev, 1970. 

Non-autonomous components essential for rehable operation of the apparatus and protection systems defined above. 

A non autonomous differential equation x = f t,x) can be written in autonomous form where the right hand side of the differential equation is not explicitly depending on time, by augmenting the system by the trivial equation t = 1 ... 

We have described a method for obtaining a high-level safety monitor specification, taking into account the specific features of autonomous systems. We base it on hazard analysis, which is non-formal. Thanks to formal methods, we ensure that the derivation from formal safety invariants to safety rules is correct, provided the modeling of safety invariants is valid. Safety invariants are modeled separately in order to maintain model validability and to ensure scalability. 

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