Algebraic approach for the exploration of the onset of chaos in discrete nonlinear dynamical systems

An algebraic approach based on the rank of a sequence is proposed for the exploration of the onset of chaos in discrete nonlinear dynamical systems. The rank of the partial solution is identified and a special technique based on Hankel matrices is used to decompose the solution into algebraic primitives comprising roots of the modified characteristic equation. The distribution of roots describes the dynamical complexity of a solution and is used to explore properties of the nonlinear system and the onset of chaos.     

The explosive divergence in iterative maps of matrices

The effect of explosive divergence in generalized iterative maps of matrices is defined and described using formal algebraic techniques. It is shown that the effect of explosive divergence can be observed in an iterative map of square matrices of order 2 if and only if the matrix of initial conditions is a nilpotent matrix and the Lyapunov exponent of the corresponding scalar iterative map is greater than zero. Computational experiments with the logistic map and the circle map are used to illustrate the effect of explosive divergence occurring in iterative maps of matrices.     

Algebraic operator method for the construction of solitary solutions to nonlinear differential equations

Solutions of the KdV equation are derived by the algebraic operator method based on generalized operators of differentiation. The algebraic operator method based on the generalized operator of differentiation is exploited for the derivation of analytic solutions to the KdV equation. The structure of solitary solutions and explicit conditions of existence of these solutions in the subspace of initial conditions are derived. It is shown that special solitary solutions exist only on a line in the parameter plane of initial and boundary conditions. This new theoretical result may lead to important findings in a variety of practical applications.     

Operator Method of Solving Nonlinear Differential Equations

We discuss the possibility of applying linear structures for solving nonlinear differential equations.     

Total synthesis and biological evaluation of (-)-9-deoxy-englerin A

An effective total synthesis of (-)-9-deoxy-englerin (4), an analogue of the natural guaiane sesquiterpene englerin A (1), has been achieved. The synthesis features a transannular epoxide opening to construct the 5,7-fused ring system followed by transannular ether formation with mercury(II) trifluoroacetate.     

Synthesis of the proposed structure of queenslandon

The proposed structure of the benzolactone queenslandon (6) was synthesized utilizing a triol containing building block prepared from d-ribose. While a ring-closing metathesis approach did not lead to the macrocycle, alkylation of a benzyl(phenyl)selane, elimination to generate the styrene double bond, followed by Mitsunobu macrolactonization proved to be successful. Spectral data suggest that the structure of queenslandon should be revised, probably to the C11 epimer.     

Construction of solutions of a partial differential equation with constant coefficients using the form of formal series. III

Kaunas Technological University, Studentų 50, 3028 Kaunas, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 3, pp. 274–287, July—September, 1994.