Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExact nonlinear fourth-order equation for two coupled nonlinear oscillators: metamorphoses of resonance curves
180
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 determined 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.
rate research
Read More
We study dynamics of two coupled periodically driven oscillators. Important example of such a system is a dynamic vibration absorber which consists of a small mass attached to the primary vibrating system of a large mass. Periodic solutions of the approximate effective equation are determined within the Krylov-Bogoliubov-Mitropolsky approach to get the amplitude profiles $AOmega) $. Dependence of the amplitude $A$ of nonlinear resonances on the frequency $ Omega $ is much more complicated than in the case of one Duffing oscillator and hence new nonlinear phenomena are possible. In the present paper we study metamorphoses of the function $A(Omega) $ induced by changes of the control parameters.
We study dynamics of two coupled periodically driven oscillators. An important example of such a system is a dynamic vibration absorber which consists of a small mass attached to the primary vibrating system of a large mass. Periodic solutions of the approximate effective equation (derived in our earlier papers) are determined within the Krylov-Bogoliubov-Mitropolsky approach to compute the amplitude profiles $A(Omega)$. In the present paper we investigate metamorphoses of the function $A(Omega)$ induced by changes of the control parameters in the case of 1:3 resonances.
Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.
We show that the nonlinear stage of modulational instability induced by parametric driving in the {em defocusing} nonlinear Schrodinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis.
Using the generalized perturbation reduction method the Hirota equation is transformed to the coupled nonlinear Schrodinger equations for auxiliary functions. A solution in the form of a two-component vector nonlinear pulse is obtained. The components of the pulse oscillate with the sum and difference of the frequencies and the wave numbers. Explicit analytical expressions for the shape and parameters of the two-component nonlinear pulse are presented.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
