Abstract:
The effect of the shape of six different periodic forces and second periodic forces on the onset of horseshoe chaos are studied both analytically and numerically in a Duffing oscillator. The external periodic forces considered are sine wave, square wave, symmetric saw-tooth wave, asymmetric saw-tooth wave, rectified sine wave, and modulus of sine wave. An analytical threshold condition for the onset of horseshoe chaos is obtained in the Duffing oscillator driven by various periodic forces using the Melnikov method. Melnikov threshold curve is drawn in a parameter space. For all the forces except modulus of sine wave, the onset of cross-well asymptotic chaos is observed just above the Melnikov threshold curve for onset of horseshoe chaos. For the modulus of sine wave long time transient motion followed by a periodic attractor is realized. The possibility of controlling of horseshoe and asymptotic chaos in the Duffing oscillator by an addition of second periodic force is then analyzed. Parametric regimes where suppression of horseshoe chaos occurs are predicted. Analytical prediction is demonstrated through direct numerical simulations. Starting from asymptotic chaos we show the recovery of periodic motion for a range of values of amplitude and phase of the second periodic force. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.

Abstract:
We show that the three-dimensional primitive equations admit a strong time-periodic solution of period $T>0$, provided the forcing term $f\in L^2(0,\mathcal T; L^2(\Omega))$ is a time-periodic function of the same period. No restriction on the magnitude of $f$ is assumed. As a corollary, if, in particular, $f$ is time-independent, the corresponding solution is steady-state.

Abstract:
We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision (Brownian motion with adhesion). This is another form of a stochastic oscillator, different from oscillator usually studied that is subject to a random force or having random frequency or random damping. We calculated first two moments for different form of a random force, and studied different resonance phenomena (stochastic resonance, vibration resonance and “erratic” behavior) interposed between order and chaos.

Abstract:
We study the dynamics of a mechanical oscillator with linear and cubic forces -the Duffing oscillator- subject to a feedback mechanism that allows the system to sustain autonomous periodic motion with well-defined amplitude and frequency. First, we characterize the autonomous motion for both hardening and softening nonlinearities. Then, we analyze the oscillator's synchronizability by an external periodic force. We find a regime where, unexpectedly, the frequency range where synchronized motion is possible becomes wider as the amplitude of oscillations grows. This effect of nonlinearities may find application in technological uses of mechanical Duffing oscillators -for instance, in the design of time-keeping devices at the microscale- which we briefly review.

Abstract:
The effects of disorder in external forces on the dynamical behavior of coupled nonlinear oscillator networks are studied. When driven synchronously, i.e., all driving forces have the same phase, the networks display chaotic dynamics. We show that random phases in the driving forces result in regular, periodic network behavior. Intermediate phase disorder can produce network synchrony. Specifically, there is an optimal amount of phase disorder, which can induce the highest level of synchrony. These results demonstrate that the spatiotemporal structure of external influences can control chaos and lead to synchronization in nonlinear systems.

Abstract:
In this paper, we present a spatial version of phytoplankton-zooplankton model that includes some important factors such as external periodic forces, noise, and diffusion processes. The spatially extended phytoplankton-zooplankton system is from the original study by Scheffer [M Scheffer, Fish and nutrients interplay determines algal biomass: a minimal model, Oikos \textbf{62} (1991) 271-282]. Our results show that the spatially extended system exhibit a resonant patterns and frequency-locking phenomena. The system also shows that the noise and the external periodic forces play a constructive role in the Scheffer's model: first, the noise can enhance the oscillation of phytoplankton species' density and format a large clusters in the space when the noise intensity is within certain interval. Second, the external periodic forces can induce 4:1 and 1:1 frequency-locking and spatially homogeneous oscillation phenomena to appear. Finally, the resonant patterns are observed in the system when the spatial noises and external periodic forces are both turned on. Moreover, we found that the 4:1 frequency-locking transform into 1:1 frequency-locking when the noise intensity increased. In addition to elucidating our results outside the domain of Turing instability, we provide further analysis of Turing linear stability with the help of the numerical calculation by using the Maple software. Significantly, oscillations are enhanced in the system when the noise term presents. These results indicate that the oceanic plankton bloom may partly due to interplay between the stochastic factors and external forces instead of deterministic factors. These results also may help us to understand the effects arising from undeniable subject to random fluctuations in oceanic plankton bloom.

Abstract:
本文研究了周期调制噪声驱动的具有质量涨落的欠阻尼谐振子的随机共振，其中的振子质量的涨落为对称双态噪声,而内噪声为高斯噪声.通过Shapiro-Loginov公式和Laplace变换,本文得到了系统稳态响应的一阶矩的解析表达式,接着利用Routh-Hurwitz判据推导了系统响应的一阶矩的稳定性条件,进而通过数值仿真研究了系统响应的一阶矩与系统各参变量间的依赖关系. 仿真结果表明,稳态响应振幅与周期输入信号频率、涨落噪声参数及系统固有参数均呈非单调变化关系, 即模型出现真实共振、广义随机共振和参数诱导共振等丰富的随机共振现象. 进一步地,本文的研究还表明质量涨落噪声和周期信号调制噪声的相互协作将导致系统的一些新的共振效应出现, 比如关于系统稳态响应振幅与驱动频率的双峰共振及关于某些噪声参数的单谷共振行为. In this paper, stochastic resonance of an underdamped harmonic oscillator with random mass and driven by periodic modulated noise is investigated. The fluctuation of oscillator mass is modeled by a dichotomous noise while the internal noise is assumed to be Gaussian. Using the Shapiro-Loginov formula and the Laplace transform technique, exact expressions of first moment of the steady-state response and output of the system are presented. Then some simulations are implemented to study the dependence of long-time behavior of the first moment on variety of the system parameters. It is shown that the output amplitude non-monotonically depends on the signal frequency, the noise parameters and the system parameters, which indicates the occurrences of bona fide stochastic resonance, generalized stochastic resonance and parameter-induced stochastic resonance. Furthermore, based on the exact expressions it is demonstrated that interplay of the mass fluctuation and the periodic modulated noise can generate some novel cooperation effects, such as double-peak resonance as well as one-valley resonance

Abstract:
We study the long time behaviour of a nonlinear oscillator subject to a random multiplicative noise with a spectral density (or power-spectrum) that decays as a power law at high frequencies. When the dissipation is negligible, physical observables, such as the amplitude, the velocity and the energy of the oscillator grow as power-laws with time. We calculate the associated scaling exponents and we show that their values depend on the asymptotic behaviour of the external potential and on the high frequencies of the noise. Our results are generalized to include dissipative effects and additive noise.

Abstract:
We generalize the previously considered cases of a harmonic oscillator subject to a random force (Brownian motion), or having random frequency, or random damping. We consider here a random mass which corresponds to an oscillator where the particles of the surrounding medium adhere to the oscillator for some (random) time after collision, thereby changing the oscillator mass. Such a model is appropriate to chemical and biological solutions as well as to some nano-technological devices. The first moment and stability conditions for white and dichotomous noise are analyzed.

Abstract:
We consider a damped impact oscillator subject to the action of a biharmonic force. The conditions for the existence and stability of almost periodic resonance solutions are investigated.