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A model which allows a double impacting regime for a particle undergoing simple harmonic motion is considered in some detail. The behavior of the particle in the weak spring limit is considered. Symmetries of the motion are found and the extent of the resonant dynamical behavior is considered. Control equations are developed and strategies are described for both the preservation and the annihilation of experimental and analytical resonant periodic orbits.

We analyze an important class of engineering systems characterized by the discontinuous motion of a spring-mass constrained by the motion of a feedback-assisted actuator. We show that the combined effects of mechanical restitution coefficient and displacement feedback can be exactly represented by a single equivalent dissipation coefficient. We also show that the topological properties of the surfaces of section of orbits generated by impact oscillators which possess differing proportions of restitution and feedback levels, but whose equivalent dissipation coefficients are equal, are equivalent and universally scalable. The scaling law allows us to interchange the effects of restitution and feedback coefficients and so, effectively, eliminate one of these parameters from the equations of motion. Thus, the topological properties of dissipative feedback-assisted systems can be seen as scaled versions of either purely dissipative, or purely feedback-assisted, oscillators.

This paper deals with master–slave synchronization for Lur'e systems subject to a more general sector condition by using time delay feedback control. A new Lyapunov–Krasovskii functional and a new Lur'e–Postnikov Lyapunov functional are proposed to obtain some new delay-dependent synchronization criteria, which are formulated in the form of linear matrix inequalities (LMIs). These criteria cover some existing results as their special cases. An example shows that the result derived in this paper significantly improves some existing ones.

The work investigates the influence of spike-timing dependent plasticity (STDP) mechanisms on the dynamics of two synaptically coupled neurons driven by additive external noise. In this setting, the noise signal models synaptic inputs that the pair receives from other neurons in a larger network. We show that in the absence of STDP feedbacks the pair of neurons exhibit oscillations and intermittent synchronization. When the synapse connecting the neurons is supplied with a phase selective feedback mechanism simulating STDP, induced dynamics of spikes in the coupled system resembles a phase locked mode with time lags between spikes oscillating about a specific value. This value, as we show by extensive numerical simulations, can be set arbitrary within a broad interval by tuning parameters of the STDP feedback.

Gene expression is inherently noisy, implying that the number of mRNAs or proteins is not invariant rather than follows a distribution. This distribution can not only provide the exact information on the dynamics of gene expression but also describe cell-to-cell variability in a genetically identical cell population. Here, we systematically investigate a two-state model of gene expression, a model paradigm used to study expression dynamics, focusing on the effect of feedback on the type of mRNA or protein distribution. If there is no feedback, then the distribution may be bimodal, power-law tailed, or Poisson-like, depending on gene switching rates. However, we find that feedback can tune or change the type of the distribution in each case and tends to unimodalize the distribution as its strength increases. Specifically, positive feedback can change not only a power-law tailed distribution into a bimodal or Poisson-like distribution but also a bimodal distribution into a Poisson-like distribution (implying that stochastic bifurcation can take place). In addition, it can make a Poisson-like distribution become more peaked but does not change the type of this distribution. In contrast to positive feedback, negative feedback has less influence on the shape of the distributions except for the bimodal case. In all cases, the noise-feedback curve used extensively in previous studies cannot well reflect the feedback-induced changes in the shape of distributions. Feedback-induced variations in distribution would be important for cell survival in fluctuating environments.

This paper presents examples of hysteresis from a broad range of scientific disciplines and demonstrates a variety of forms including clockwise, counterclockwise, butterfly, pinched and kiss-and-go, respectively. These examples include mechanical systems made up of springs and dampers which have been the main components of muscle models for nearly one hundred years. For the first time, as far as the authors are aware, hysteresis is demonstrated in single fibre muscle when subjected to both lengthening and shortening periodic contractions. The hysteresis observed in the experiments is of two forms. Without any relaxation at the end of lengthening or shortening, the hysteresis loop is a convex clockwise loop, whereas a concave clockwise hysteresis loop (labeled as kiss-and-go) is formed when the muscle is relaxed at the end of lengthening and shortening. This paper also presents a mathematical model which reproduces the hysteresis curves in the same form as the experimental data.