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The chaotic behavior of the modified Rayleigh-Duffing oscillator with φ 6 potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damping effect.

This paper addresses the issues on the dynamics of nonlinear damping gyros subjected to a quintic nonlinear parametric excitation. The fixed points and their stability are analyzed for the autonomous gyros equation. The number of fixed points of the system varies from one to six. The approximate equation of gyros is considered by expanding the nonlinear restoring force and parametric excitation for the study of the dynamics of gyros. Amplitude and frequency of possible resonances are found by using the multiple scales method. Also obtained are the principal parametric resonance and orders 4 and 6 subharmonic resonances. The stability conditions for each of these resonances are also obtained. Chaotic oscillations, multistability, hysteresis, and coexisting attractors are found using the bifurcation diagrams, the Lyapunov exponents, the phase portraits, the Poincaré section and the time histories. The effects of the damping parameter, the angular spin velocity and the parametric nonlinear excitation are analyzed. Results obtained by using the approximate gyros equation are compared to the dynamics obtained with the exact equation of gyros. The analytical investigations are complemented by numerical simulations.

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In this work, the linear and nonlinear dynamics of thermal convection in an incompressible Newtonian (alumina-copper)/water hybrid nanofluid each confined in an infinite rectangular cavity and heated from below via the Cattaneo–Christov heat flux model are studied for volume fractions fixed between 0 and 0.05. Firstly, using the governed flow equations and free boundary conditions, we proceed to the classical theory of stationary and oscillatory convection which help us to find the expressions of critical Rayleigh numbers, Cattaneo numbers and wavenumber of base fluid as a function of mono or hybrid nanofluid thermophysical properties. The results show that when the Cattaneo number increases, the critical Rayleigh number is constant in the case of stationary convection but decreases in the case of oscillatory convection for pure base fluid and hybrid nanofluids. It is noted that beyond the threshold Cattaneo number, the critical Rayleigh number of stationary convection is greater than those of oscillatory convection and the critical wavenumber shifts discontinuously from stationary to oscillatory convection. Secondly, the use of the truncated Galerkin approximation made it possible to find a five low-dimensional system in order to study numerically the transition from natural convection to chaotic behavior of nanofluid. We noticed that the addition of hybrid nanoparticle in the heat fluid transfer increases the domain of stationary convection by delaying the oscillatory convection with increasing normalized Rayleigh number. Also, the bifurcation diagrams show that the use of hybrid nanoparticles allows further control of the chaos in base fluid by expanding the convective flow. Furthermore, in the presence of thermal relaxation time, the chaos in the hybrid nanofluid lasts longer compared to the ordinary fluid.

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