arXiv subjects
Publications and source records attributed to Jan Kyzioł.
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.
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 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.
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.