In this paper we present a method based on a generalized Hamiltonian formalism to associate to a chaotic system of known dynamics a function of the phase space variables with the characteristics of an energy. Using this formalism we have found energy functions for the Lorenz, Rössler, and Chua families of chaotic oscillators. We have theoretically analyzed the flow of energy in the process of synchronizing two chaotic systems via feedback coupling and used the previously found energy functions for computing the required energy to maintain a synchronized regime between systems of these families. We have calculated the flows of energy at different coupling strengths covering cases of both identical as well as nonidentical synchronization. The energy dissipated by the guided system seems to be sensitive to the transitions in the stability of its equilibrium points induced by the coupling.

The dissociation energy of the Ti(OH,)+ ion-molecule complex was calculated by the multiconfigurational selfconsistent field theory, coupled cluster theory, and two density functional theory based methods, using both all-electron basis sets and effective core potentials. The calculations show that approximate density functional theory gives results in better agreement with experiment than either the multiconfigurational self-consistent field theory or the coupled cluster theory, with both allelectron basis sets and effective core potentials. Nevertheless, the optimized geometries and harmonic vibration frequencies are very similar, irrespective of the level of theory used. The interconfigurational energy ordering of the two valence electronic configurations dn-'s and dn-'s' of the 4~ electronic state of the titanium cation were also calculated and are discussed.Key words: ab initio, dissociation energy, ion-molecule complex, effective core potentials, transition metals.Resume : On a calculC l'knergie de dissociation du complexe ion-molCcule, Ti(OH,)+, a I'aide de la thCorie de champ autocohCrent rnulticonfigurationnel, de la thCorie de I'agrCgat couplt et de deux mCthodes basCes sur la thCorie de la densite fonctionnelle, utilisant des ensembles de base de tous les Clectrons ainsi que tous les potentiels effectifs des noyaux. Les calculs montrent que la theorie approximative de la densit6 fonctionnelle conduit h des rksultats qui sont en meilleur accord avec les donnCes expkrimentales que ceux obtenus a I'aide tant de la thCorie du champ autocohkrent multiconfigurationnel que de la thCorie de l'agrCgat coup16 incorporant les ensembles de base de tous les Clectrons ainsi que tous les potentiels effectifs des noyaux. NCanmoins, les gComCtries optimisCes et les frCquences de vibration harmonique sont trks semblables, quel que soit le niveau de la thCorie utilist. On a aussi calculC l'ordre des Cnergies interconfigurationnelles des deux configurations Clectroniques de valence dn-'s et dn-,s' de 176tat Clectronique 4~ des cations du titane et on en discute.

Feedback coupling through an interaction term proportional to the difference in the value of some behavioral characteristics of two systems is a very common structural setting that leads to synchronization of the behavior of both systems. The degree of synchronization attained depends on the strength of the interaction term and on the mutual interdependency of the structures of both systems. In this paper, we show that two chaotic systems linked through a feedback coupling interaction term of gain parameter k reach a synchronized regime characterized by a vector of variable errors which tends towards zero with parameter k while the interaction term tends towards a finite nonzero permanent regime. This means that maintaining a certain degree of synchronization has a cost. In the limit, complete synchronization occurs at a finite limit cost. We show that feedback coupling in itself brings about conditions permitting that systems with a degree of structural parameter flexibility evolve close towards each other structures in order to facilitate the maintenance of the synchronized regime. In this paper, we deduce parameter adaptive laws for any family of homochaotic systems provided they are previously forced to work, via feedback coupling, within an appropriate degree of synchronization. The laws are global in the space of parameters and lead eventually to identical synchronization at no interaction cost. We illustrate this point with homochaotic systems from the Lorenz, Rössler and Chua families.

We have deduced an energy function for a Hindmarsh-Rose model neuron and we have used it to evaluate the energy consumption of the neuron during its signaling activity. We investigate the balance of energy in the synchronization of two bidirectional linearly coupled neurons at different values of the coupling strength. We show that when two neurons are coupled there is a specific cost associated to the cooperative behavior. We find that the energy consumption of the neurons is incoherent until very near the threshold of identical synchronization, which suggests that cooperative behaviors without complete synchrony could be energetically more advantageous than those with complete synchrony.

Ab initio electronic structure calculations demonstrate that a singlet planar cyclic isomer of P202 with D2h symmetry is more stable than all the previously characterized P202 isomers. A proper description of the electronic structure wavefunction is found to be essential to predict correctly the multiplicity of the ground state.