Causal path different

While the power line is an acausal concept, i.e., it does not require any organization of the model equations, the causal path needs a causality assignment in the bond graph representation. Its definition is firsf recalled. Then the length and the order of a causal path are introduced, and finally, botii different and disjoint causal paths are defined. The latter concepfs, as for tiie power line, will be used in fhe invertibilify criteria. The concept of different causal paths will also be used to characterize the structure at infinity of a model from its bond graph representation. 

Definition 6.7 (Causal path order) In a causal (or bicausal) bond graph representation, the order, denoted a>jt(i>i -> vj) (or the generalized length), of a causal path k between a variable Vj and another variable vj is defined as the difference between the number of energy storage elements in integral causality and the number of those in derivative causality along this causal path [2,15],... 

Definition 6.8 (Different causal path) In a bond graph representation in preferential integral causality, two causal paths are said to be different if they have no energy storage element in integral causality in common [40,41],... 

First, a series of criteria concerns the invertibility checking of a model. An approach based on different I/O causal paths (see Definition 6.8) and the system matrix determinant has been proposed in [40]. Here the approach based on disjoint I/O causal paths (see Definition 6.9) is presented [15, 24, 25]. It uses two structural criteria which, if not verified, enable the inversion process to be stopped early in the procedure. A third criterion is formulated at a behavioral level. This level is called behavioral in the sense that it requires analytical developments based on the constitutive and conservation laws in the bond graph representation. 

Initially, flows corresponding to inertances and displacements associated with compliances are used to establish the dynamic equations and to find the zero-order causal paths of the system. Two different methods are used to solve the ZCPs [1, 2]. With the first one, Lagrange multipliers are introduced by means of new flows and efforts as break variables of causal paths, adding constraint equations. With the second one, break variables are used directly to open the ZCPs. 

Class 5.1 ZCPs The causal path is established between storage ports with integral and differential causalities. The associated topological loops are flat loops. In MBG systems, the two extreme ports of the path can correspond to different directions. An example of an MGB model with class 5.1 ZCPs is shown in Fig. 9.28. 

Understanding the electronic movement in physical atomic as being driven by the conneeted and correlated functions especially by the (temporally) causal Green-fimction/quantum propagators Describing the physical atom as a semiclassical description of quantum motion, i.e., merely quantum than classical yet with certain orders of Planck constant contributions in electronic orbits in atom Learning the difference between the second and the fourth order of path integral expansion of the quantum amplitude of electronic orbits as quantifies in the associated partition functions ... 

Synthesis involves a complex set of processes and complex systems often show chaos, in particular in autocatalysis. Chaotic processes diverge two systems that follow the same process but have an unnoticed difference in start concentrations follow completely different paths. Chaos is not recognized if the system is locked into what is called a stable attractor or a limit cycle. Then it seems to be simply causal and is usually easy to model. When the system is in a strange attractor it seems to run amok. Chaos in synthesis can be recognized by several indicators ... 

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