Heating and thermal squeezing in parametrically-driven oscillators with added noise

In this paper we report a theoretical model based on Green functions, Floquet theory and averaging techniques up to second order that describes the dynamics of parametrically-driven oscillators with added thermal noise. Quantitative estimates for heating and quadrature thermal noise squeezing near and below the transition line of the first parametric instability zone of the oscillator are given. Furthermore, we give an intuitive explanation as to why heating and thermal squeezing occur. For small amplitudes of the parametric pump the Floquet multipliers are complex conjugate of each other with a constant magnitude. As the pump amplitude is increased past a threshold value in the stable zone near the first parametric instability, the two Floquet multipliers become real and have different magnitudes. This creates two different effective dissipation rates (one smaller and the other larger than the real dissipation rate) along the stable manifolds of the first-return Poincare map. We also show that the statistical average of the input power due to thermal noise is constant and independent of the pump amplitude and frequency. The combination of these effects cause most of heating and thermal squeezing. Very good agreement between analytical and numerical estimates of the thermal fluctuations is achieved.Comment: Submitted to Phys. Rev. E, 29 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1108.484

Mixed-mode oscillations in a multiple time scale phantom bursting system

In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow nullcline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion.Comment: 38 pages, 16 figure

Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems

Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending the Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to `hidden' attractors inside the switching surface.Comment: 12 page

Time-Scale and Noise Optimality in Self-Organized Critical Adaptive Networks

Recent studies have shown that adaptive networks driven by simple local rules can organize into "critical" global steady states, providing another framework for self-organized criticality (SOC). We focus on the important convergence to criticality and show that noise and time-scale optimality are reached at finite values. This is in sharp contrast to the previously believed optimal zero noise and infinite time scale separation case. Furthermore, we discover a noise induced phase transition for the breakdown of SOC. We also investigate each of the three new effects separately by developing models. These models reveal three generically low-dimensional dynamical behaviors: time-scale resonance (TR), a new simplified version of stochastic resonance - which we call steady state stochastic resonance (SSR) - as well as noise-induced phase transitions.Comment: 4 pages, 6 figures; several changes in exposition and focus on applications in revised versio

The phase-space of generalized Gauss-Bonnet dark energy

The generalized Gauss-Bonnet theory, introduced by Lagrangian F(R,G), has been considered as a general modified gravity for explanation of the dark energy. G is the Gauss-Bonnet invariant. For this model, we seek the situations under which the late-time behavior of the theory is the de-Sitter space-time. This is done by studying the two dimensional phase space of this theory, i.e. the R-H plane. By obtaining the conditions under which the de-Sitter space-time is the stable attractor of this theory, several aspects of this problem have been investigated. It has been shown that there exist at least two classes of stable attractors : the singularities of the F(R,G), and the cases in which the model has a critical curve, instead of critical points. This curve is R=12H^2 in R-H plane. Several examples, including their numerical calculations, have been discussed.Comment: 19 pages, 11 figures, typos corrected, a reference adde

Simplification of the tug-of-war model for cellular transport in cells

The transport of organelles and vesicles in living cells can be well described by a kinetic tug-of-war model advanced by M\"uller, Klumpp and Lipowsky. In which, the cargo is attached by two motor species, kinesin and dynein, and the direction of motion is determined by the number of motors which bind to the track. In recent work [Phys. Rev. E 79, 061918 (2009)], this model was studied by mean field theory, and it was found that, usually the tug-of-war model has one, two, or three distinct stable stationary points. However, the results there are mostly obtained by numerical calculations, since it is hard to do detailed theoretical studies to a two-dimensional nonlinear system. In this paper, we will carry out further detailed analysis about this model, and try to find more properties theoretically. Firstly, the tug-of-war model is simplified to a one-dimensional equation. Then we claim that the stationary points of the tug-of-war model correspond to the roots of the simplified equation, and the stable stationary points correspond to the roots with positive derivative. Bifurcation occurs at the corresponding parameters, under which the simplified one-dimensional equation exists root with zero derivative. Using the simplified equation, not only more properties of the tug-of-war model can be obtained analytically, the related numerical calculations will become more accurate and more efficient. This simplification will be helpful to future studies of the tug-of-war model

Anomalous exponents at the onset of an instability

Critical exponents are calculated exactly at the onset of an instability, using asymptotic expansiontechniques. When the unstable mode is subject to multiplicative noise whose spectrum at zero frequency vanishes, we show that the critical behavior can be anomalous, i.e. the mode amplitude X scales with departure from onset \mu as $~ \mu^\beta$ with an exponent $\beta$ different from its deterministic value. This behavior is observed in a direct numerical simulation of the dynamo instability and our results provide a possible explanation to recent experimental observations

Exploring constrained quantum control landscapes

The broad success of optimally controlling quantum systems with external fields has been attributed to the favorable topology of the underlying control landscape, where the landscape is the physical observable as a function of the controls. The control landscape can be shown to contain no suboptimal trapping extrema upon satisfaction of reasonable physical assumptions, but this topological analysis does not hold when significant constraints are placed on the control resources. This work employs simulations to explore the topology and features of the control landscape for pure-state population transfer with a constrained class of control fields. The fields are parameterized in terms of a set of uniformly spaced spectral frequencies, with the associated phases acting as the controls. Optimization results reveal that the minimum number of phase controls necessary to assure a high yield in the target state has a special dependence on the number of accessible energy levels in the quantum system, revealed from an analysis of the first- and second-order variation of the yield with respect to the controls. When an insufficient number of controls and/or a weak control fluence are employed, trapping extrema and saddle points are observed on the landscape. When the control resources are sufficiently flexible, solutions producing the globally maximal yield are found to form connected `level sets' of continuously variable control fields that preserve the yield. These optimal yield level sets are found to shrink to isolated points on the top of the landscape as the control field fluence is decreased, and further reduction of the fluence turns these points into suboptimal trapping extrema on the landscape. Although constrained control fields can come in many forms beyond the cases explored here, the behavior found in this paper is illustrative of the impacts that constraints can introduce.Comment: 10 figure

Delay-induced multistability near a global bifurcation

We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.Comment: Int. J. Bif. Chaos (2007), in prin