arXiv subjects
Publications and source records attributed to Andrzej Okniński.
The dynamics of the driven van der Pol oscillator are investigated. We study birth and metamorphoses of $1:1$ and $1:3$ resonances within the formalism of differential properties of amplitude-frequency response implicit functions.
The dynamics of nonlinear oscillators are investigated. We study the formation of $1:2$ resonance in nonlinear periodically forced oscillators due to period doubling of the primary $1:1$ resonance, or born independently. We compute the amplitude-frequency implicit function, the steady-state asymptotic solution, for the effective equation approximating coupled oscillators. Working in the framework of differential properties of implicit functions, we demonstrate that birth of $1:2$ resonances corresponds to singular isolated points of the implicit functions. We provide numerical examples illustrating our theoretical findings.
We investigate the $1: 2$ resonance in the periodically forced asymmetric Duffing oscillator due to the period-doubling of the primary $1: 1$ resonance or forming independently, coexisting with the primary resonance. We compute the steady-state asymptotic solution - the amplitude-frequency implicit function. Working in the differential properties of implicit functions framework, we describe complicated metamorphoses of the $1:2$ resonance and its interaction with the primary resonance.
In this work, we investigate the period doubling phenomenon in the periodically forced asymmetric Duffing oscillator. We use the known steady-state asymptotic solution -- the amplitude-frequency implicit function -- and known criterion for the existence of period doubling. Working in the framework of differential properties of implicit functions we derive analytical formulas for the birth of period-doubled solutions.
We proceed with our study of the Hagen-Hurley equations describing spin one bosons. In this work, we describe a general decay of the Hagen-Hurley boson. It is important that the transformation conserves spin $s=1$ of the decaying boson and provides information about the kinematics of the decay. We explain the high instability of the Hagen-Hurley particle and identify it as the $W$ boson.
We study the jump phenomenon present in the forced asymmetric Duffing oscillator using the known steady-state asymptotic solution. The major result is the computation of the jump manifold, which encodes global information about all possible jumps.
Dynamics of nonlinear coupled driven oscillators is investigated. Recently, we have demonstrated that the amplitude profiles -- dependence of the amplitude $A$ on frequency $Ω$ of the driving force, computed by asymptotic methods in implicit form as $F\left( A,Ω\right) =0$, permit prediction of metamorphoses of dynamics which occur at singular points of the implicit curve $F\left( A,Ω\right) =0$. In the present study we strive at a global view of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point.
We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( Ω,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(Ω,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $Ω_{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.
We study decays of a spin $1$ boson within the formalism of Hagen-Hurley equations. Such particle can decay into two spin $\frac{1}{2}$ particles, a Weyl neutrino and a massive fermion, whose spins can couple to $S=0$ or $S=1$. Since spin $0$ and spin $1$ bosons can be described by the Dirac equation within the same representation of $γ^{μ}$ matrices mixing of $S=0$ and $S=1$ states is possible. We argue that the Hagen-Hurley equations describe $W$ boson with spin $S\in 0\oplus 1$ space and analyse mixed beta decays as well as top quark decays from this perspective.
Dynamics of the Duffing--Van der Pol driven oscillator is investigated. Periodic steady-state solutions of the corresponding equation are computed within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude $A$ on forcing frequency $Ω$ as an implicit function, $F\left( A,Ω\right) =0$, referred to as resonance curve or amplitude profile. In singular points of the amplitude curve the conditions $\frac{\partial F}{\partial A}=0$, $\frac{\partial F}{\partial Ω}=0$ are fulfilled, i.e. in such points neither of the functions $A=f\left( Ω\right) $, $Ω=g\left( A\right) $, continuous with continuous first derivative, exists. Near such points metamorphoses of the dynamics can occur. In the present work the bifurcation set, i.e. the set in the parameter space, such that every point in this set corresponds to a singular point of the amplitude profile, is computed. Several examples of singular points and the corresponding metamorphoses of dynamics are presented.
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 Poincaré map, describing evolution from an impact to the next impact, is described. Displacement of the table is 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.