In the present work a transition from the spin-$0$ Duffin-Kemmer-Petiau equation to the Dirac equation is described. This transformation occurs when a crossed field changes into a certain longitudinal field. An experimental setup to carry out the transition is proposed.
We study relation between the Duffin-Kemmer-Petiau algebras and some representations of Tzou algebras. Working in the setting of relativistic wave equations we reduce, via a similarity transformation, five and ten dimensional Duffin-Kemmer-Petiau alg
ebras to three and seven dimensional Tzou algebras, respectively.
We carry out a constructive review of non-standard solutions of relativistic wave equations. Such solutions are obtained via splitting of relativistic wave equations written in spinor form. All these solutions are also solutions of the Dirac equation
and are non-standard because they involve higher-order spinors. The main finding is that non-standard solutions describe decaying states.
We study the 7x7 Hagen-Hurley equations describing spin 1 particles. We split these equations, in the interacting case, into two Dirac equations with non-standard solutions. It is argued that these solutions describe decay of a virtual W boson in beta decay.
We study fermion-boson transitions. Our approach is based on the $3times 3$ subequations of Dirac and Duffin-Kemmer-Petiau equations, which link these equations. We demonstrate that free Dirac equation can be invertibly converted to spin-$0$ Duffin-K
emmer-Petiau equation in presence of a neutrino field. We also show that in special external fields, upon assuming again existence of a neutrino (Weyl) spinor, the Dirac equation can be transformed reversibly to spin-$0$ Duffin-Kemmer-Petiau equation. We argue that such boson-fermions transitions are consistent with the main channel of pion decay.
We study several formulations of zero-mass relativistic equations, stressing similarities between different frameworks. It is shown that all these massless wave equations have fermionic as well as bosonic solutions.
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the table i
s approximated in one period by four cubic polynomials. Results obtained for this model are used to elucidate dynamics of the standard model of bouncing ball with sinusoidal motion of the limiter.
Dynamics near the grazing manifold and basins of attraction for a motion of a material point in a gravitational field, colliding with a moving motion-limiting stop, are investigated. The Poincare map, describing evolution from an impact to the next i
mpact, is derived. Periodic points are found and their stability is determined. The grazing manifold is computed and dynamics is approximated in its vicinity. It is shown that on the grazing manifold there are trapping as well as forbidden regions. Finally, basins of attraction are studied.
We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are dete
rmined within the Krylov-Bogoliubov-Mitropolsky approach - we compute the amplitude profiles, which from mathematical point of view are algebraic curves. In the present paper we investigate metamorphoses of amplitude profiles induced by changes of control parameters near singular points of these curves. It follows that dynamics changes qualitatively in the neighbourhood of a singular point.
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter
is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiters motion making analysis of chattering possible.
