Hopf Limit Cycle Research Articles

Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion

This paper focuses on the impact of cross-diffusion for Turing-Hopf instability in a morphochemical model for electrodeposition (DIB) and completes the analysis on the role of cross-diffusion on pattern formation in electrodeposition we recently carried out in [Lacitignola et al. 2018]. We derive and discuss conditions for the destabilization a´la Turing of the Hopf limit cycle in the neighborhood of the codimension-two Turing-Hopf bifurcation point, resulting in oscillatory Turing patterns. Numerical validation of the above theoretical findings is provided both in the case of Weak and Strong Turing-Hopf instability. Experimental evidence of this kind of phenomenology is then discussed also in relation to the charging and discharging process of a battery.

Limit cycles and bifurcations in a nonlinear MEMS resonator with a 1:3 internal resonance

This work investigates the behavior of an AlGaAs/GaAs piezoelectric nonlinear MEMS oscillator exhibiting a 1:3 internal resonance. The device is explored in an open-loop configuration, i.e., as a driven resonator, where depending on the drive conditions we observe energy transfer between the first and third modes, and the emergence of supercritical Hopf limit cycles. We examine the dependence of these bifurcations on the oscillator's frequency and amplitude and reproduce the observed behavior using a system of nonlinearly coupled equations which show interesting scaling behavior.

A normal form method for the determination of oscillations characteristics near the primary Hopf bifurcation in bandpass optoelectronic oscillators: Theory and experiment.

We propose a framework for the analysis of the integro-differential delay Ikeda equations ruling the dynamics of bandpass optoelectronic oscillators (OEOs). Our framework is based on the normal form reduction of OEOs and helps in the determination of the amplitude and the frequency of the primary Hopf limit-cycles as a function of the time delay and other parameters. The study is carried for both the negative and the positive slopes of the sinusoidal transfer function, and our analytical results are confirmed by the numerical and experimental data.

Anti-control of Hopf bifurcation in the Shimizu–Morioka system using an explicit criterion

We consider anti-control of Hopf bifurcation for the Shimizu–Morioka system by using an explicit criterion. We first provide the two conditions for the existence of Hopf bifurcation, that is, eigenvalue assignment and transversality conditions, which could be formulated through the coefficients of characteristic equation, and the obtained conditions do not need to calculate the eigenvalue and eigenvalue’s derivatives. The center manifold theory and normal form reduction are utilized to derive the nonlinear gains for controlling the stability of the created limit circle. In addition, we further improve the computing formulas of amplitude and frequency of Hopf limit cycle. Numerical analysis also verifies the effectiveness of the proposed results.

Hopf and torus bifurcations, torus destruction and chaos in population biology

One of the simplest population biological models displaying a Hopf bifurcation is the Rosenzweig–MacArthur model with Holling type II response function as essential ingredient. In seasonally forced versions the fixed point on one side of the Hopf bifurcation becomes a limit cycle and the Hopf limit cycle on the other hand becomes a torus, hence the Hopf bifurcation becomes a torus bifurcation, and via torus destruction by further increasing relevant parameters can follow deterministic chaos. We investigate this route to chaos also in view of stochastic versions, since in real world systems only such stochastic processes would be observed.However, the Holling type II response function is not directly related to a transition from one to another population class which would allow a stochastic version straight away. Instead, a time scale separation argument leads from a more complex model to the simple 2 dimensional Rosenzweig–MacArthur model, via additional classes of food handling and predators searching for prey. This extended model allows a stochastic generalization with the stochastic version of a Hopf bifurcation, and ultimately also with additional seasonality allowing a torus bifurcation under stochasticity.Our study shows that the torus destruction into chaos with positive Lyapunov exponents can occur in parameter regions where also the time scale separation and hence stochastic versions of the model are possible. The chaotic motion is observed inside Arnol’d tongues of rational ratio of the forcing frequency and the eigenfrequency of the unforced Hopf limit cycle.Such torus bifurcations and torus destruction into chaos are also observed in other population biological systems, and were for example found in extended multi-strain epidemiological models on dengue fever. To understand such dynamical scenarios better also under noise the present low dimensional system can serve as a good study case.

Anti-controlling Hopf bifurcation in a type of centrifugal governor system

Anti-controlling Hopf bifurcation is considered as one way to design Hopf limit circle into a dynamical system, and the oscillatory behavior of Hopf limit circle can be beneficial in many practical applications such as mixing, low-energy navigation control and fault diagnosis. In this paper, the feedback control problem of designing Hopf bifurcation in a centrifugal governor system is addressed. A feedback control method is proposed to achieve three aspects of controlling problem including existence, stability, and adjusting amplitude and frequency of the limit cycle to be designed. An explicit criterion of Hopf bifurcation including eigenvalue assignment and transversality conditions, without using eigenvalue computation, is utilized to derive the linear gains responsible for control of the bifurcation existence. The center manifold theory and normal form reduction is utilized to derive the nonlinear gains responsible for control of the stability of the created limit circle. The expressions of the approximate amplitude and frequency of the limit cycle are developed to derive the nonlinear gains responsible for controls of amplitude and frequency of the limit cycle. Numerical simulations for a centrifugal governor system show that Hopf limit cycle with desired properties can be created at any a pre-specified parameter point.

A predator-prey system involving five limit cycles

In this paper we consider the multiparameter system introduced in \cite{sch}, which corresponds to an extension of the classic minimal Daphnia-algae model. It is shown that there is a neighborhood in the parameter space where the system in the realistic quadrant has a unique equilibrium point which is a repelling weak focus of order four enclosed by a global attractor hyperbolic limit cycle. For a small enough change of the parameters in this neighborhood, bifurcation occurs from the weak focus four infinitesimal Hopf limit cycles (alternating the type of stability) such that the last bifurcated limit cycle is an attractor. Moreover, for certain values of parameters, we concluded that this applied model has five concentric limit cycles, three of them being stable hyperbolic limit cycles. This gives a positive answer to a question raised in \cite{col, Llo}.

Theoretical and experimental study of slow-scale Hopf limit-cycles in laser-based wideband optoelectronic oscillators

Optoelectronic oscillators are the subject of extensive research because of the wide variety of associated applications, which include chaos cryptography, ultrastable microwave generation, and neuromorphic computing. The wideband optoelectronic oscillator presents a particular feature allowing for two dynamical time scales to be superimposed, namely, a slow one and a fast one. In this paper, we fully characterize the onset of the slow-scale oscillation in the wideband optoelectronic oscillator. We investigate the dynamics associated to the first Hopf bifurcation and calculate analytically both the amplitude and period of the induced limit-cycle. In particular, we show how the dynamics of the zero-delay case can be used to provide insight into the infinite-dimensional dynamics of the delayed system. Our theoretical results are in very good agreement with the experimental measurements.

Hopf bifurcation and analysis of equilibrium for a third-order differential equation in a model of competition

In this paper, a mathematical model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor is investigated. The model is in the form of a system of nonlinear differential equations. By using qualitative methods, the conditions for the existence and local stability of the equilibria are obtained. The existence and stability of periodic solutions of the Hopf type are studied. Numerical simulations about the Hopf bifurcation value and Hopf limit cycle are also given.

Application of normal forms to the Lorenz model in the subcritical region

It is shown that, within a certain parameter interval below the Hopf bifurcation, the Lorenz equations can be decoucpled in the normal form representation. We normalize only up to monomials of finite order and combine the unnormalized part of the flow in a remainder. The formalism avoids small denominators and is well defined if the inverse of a finite polynomial map exists. Numerical results are presented for the basin of attraction in some neighborhood of the stable fixed points and for the Hopf limit cycle.