Fractional Damping Term Research Articles

Chaotic vibrations of the duffing system with fractional damping

We examined the Duffing system with a fractional damping term. Calculating the basins of attraction, we demonstrate a broad spectrum of non-linear behaviour connected with sensitivity to the initial conditions and chaos. To quantify dynamical response of the system, we propose the statistical 0-1 test as well as the maximal Lyapunov exponent; the application of the latter encounter a few difficulties because of the memory effect due to the fractional derivative. The results are confirmed by bifurcation diagrams, phase portraits, and Poincaré sections.

NEW DECAY RATES FOR A PROBLEM OF PLATE DYNAMICS WITH FRACTIONAL DAMPING

In this paper, we obtain decay rates for the total energy associated to the linear plate equation with effects of rotational inertia and a fractional damping term depending on a number θ ∈ [0, 1]. We observe that the dissipative structure of the equation with θ = 0 is of the regularity-loss type. This decay structure still remains true in the plate equation with a power of fractional damping θ > 0, but it becomes more weak when θ increase. This means that we can have an optimal decay estimate of solutions under an additional regularity assumption on the initial data. Our results generalize previous results by Luz and Charão and some of recent results due to Sugitani and Kawashima. We use a special method in the Fourier space which we developed in a previous work for the wave equation. So, our approach shows to be very effective to study decay properties for several problems in Rn.

Vibration of generalized double well oscillators

AbstractWe have applied the Melnikov criterion to examine a global homoclinic bifurcation and a transition to chaos in the case of the double well dynamical system with a nonlinear fractional damping term and external excitation. The usual double well Duffing potential having one negative square term and one positive quartic term has been generalized to a double well potential with a negative square term and a positive one with an arbitrary real exponent q > 2. We have also used a fractional damping term with an arbitrary power p applied to velocity which enables one to cover a wide range of realistic damping factors: from dry friction p → 0 to turbulent resistance phenomena p = 2. Using perturbation methods we have found a critical forcing amplitude μc above which the system may behave chaotically. Our results show that the vibrating system is less stable in transition to chaos for smaller p satisfying an exponential scaling low. The critical amplitude μc is an exponential function of p. The analytical results have been illustrated by numerical simulations using standard nonlinear tools such as Poincare maps and the maximal Lyapunov exponent. As usual for chosen system parameters we have identified a chaotic motion above the critical Melnikov amplitude μc.