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In this paper, the Lie symmetry theory of discrete singular systems is studied in phase space. Firstly, the discrete canonical equations and the energy evolution equations of the constrained Hamilton systems are established based on the discrete difference variational principle. Secondly, the Lie point transformation of discrete group is applied to the difference equations and constraint restriction, and the Lie symmetry determination equations of the discrete constrained Hamilton systems are obtained; Meanwhile, the Lie symmetries of singular systems lead to the discrete Noehter type conserved quantities when the structure condition equations (discrete Noether identity) are established. Finally,an example is given to illustrate the application, the results show that the conservative constrained Hamilton systems also have the discrete energy conservation.
In the article the combined algorithm for finding conservation laws and implectic operators has been proposed. Using the Novikov-Bogoyavlensky method the finite dimensional reductions have been found. The structure of invariant submanifolds has been examined. Having analyzed phase portraits of Hamiltonian systems, partial periodical solutions have been found.
Interval arithmetic techniques such as VALENCIA-IVP allow calculating guaranteed enclosures of all reachable states of continuous-time dynamical systems with bounded uncertainties of both initial conditions and system parameters. Considering the fact that, in naive implementations of interval algorithms, overestimation might lead to unnecessarily conservative results, suitable consistency tests are essential to obtain the tightest possible enclosures. In this contribution, a general framework for the use of constraints based on physically motivated conservation properties is presented. The use of these constraints in verified simulations of dynamical systems provides a computationally efficient procedure which restricts the state enclosures to regions that are physically eaningful. A branch and prune algorithm is modified to a consistency test, which is based on these constraints. Two application scenarios are studied in detail. First, the total energy is employed as a conservation property for the analysis of mechanical systems. It is shown that conservation properties, such as the energy, are applicable to any Hamiltonian system. The second scenario is based on constraints that are derived from decoupling properties, which are considered for a high-dimensional compartment model of granulopoiesis in human blood cell dynamics.
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