МОДЕЛИРОВАНИЕ ЭВОЛЮЦИИ ХИЩНИКА В СООБЩЕСТВЕ ВЗАИМОДЕЙСТВУЮЩИХ ВИДОВ

Anti-control and synchronization of period doubling and chaos is a method for bifurcation control. It can be used to detect the occurrence or periodic behavior of a bifurcation at the specified position to meet the requirements of brushless direct current (BLDCM). Antichaotic control can be implemented through the use of an external periodic term or constant. The study of the parametric singularities of brushless direct current (BLDCM) allows the identification of a complex bifurcation structure, namely the limit point (LP), the Hopf (H) and the Bogdanov-Takens (BT) bifurcations, the period-doubling bifurcation and path to chaos . By adjusting the control parameters of the controller, the period doubling bifurcation can be generated or suppressed at the specified position to realize the anti-control of period doubling and chaos bifurcation. Parametric singularities are analyzed using a variety of anti-control signals, including constant voltage, periodic square, sawtooth wave, triangle, etc. The simulation results show that adding constant or periodic factors improves chaos and anti-control effects.

Chapter 11 - PD Bifurcation and Chaos Behavior in a Predator-Prey Model With Allee Effect and Seasonal Perturbation

A family of local difleomorphisms of ${\bf R}^n$ can undergo a period doubling (flip) bifurcation as an eigenvalue of a fixed point passes through $ - 1$. This bifurcation is either supercritical or subcritical, depending on the sign of a coefficient determined by higher-order terms. If this coefficient is zero, the resulting bifurcation is “degenerate.” The period doubling bifurcation with a single higher-order degeneracy is treated, as well as the more general degenerate period doubling bifurcation where a fixed point has $ - 1$ eigenvalue and any number of higher-order degeneracies. The main procedure is a Lyapunov–Schmidt reduction: period-2 orbits are shown to be in one-to-one correspondence with roots of the reduced “bifurcation function,“ which has ${\bf Z}^2$ symmetry. Illustrative examples of the occurrence of the singly degenerate period doubling in the context of periodically forced planar oscillators are also presented.

In this paper we present an experimental analysis of the effect of RF modulation injection current in DFB and FP lasers nearthe first period doubling bifurcation (PDB). By increasing the amplitude of the modulation frequency at the onset of PDB thefirst results of a virtual Hopf phenomenon like behavior has been observed in both types of lasers. We characterized thisobservation by increasing the amplitude of the current modulation where two peaks appear in the spectrum, approaching oneto the other until fmally merging the onset of period doubling bifurcation as predicted by theory. The same virtual Hopfbehavior was observed when we set the amplitude modulation at a fixed value and we sweep the frequency modulation at theonset of the PDB point.Keywords: Laser Diode, Nonlinear Dynamics, Period Doubling Bifurcations. 1. INTRODUCTION Nonlinear dynamics in diode lasers has been the subject of many theoretical and experimental investigations in the lastdecade as they show a rich variety of irregular behavior in their route to chaos1. A single frequency diode laser is commonlystudied by a rate equation model which describes the time evolution of the photon and carrier densities. In these equations,the inclusion of an additional degree of freedom as modulation of the injection current, external light injection or opticalfeedback is needed for nonlinear dynamics to appear2.To our knowledge, the nonlinear behavior in semiconductor diode lasers has been extensively studied by many authors usingdelayed optical feedback and strong current modulation. In the former, the observed route to chaos was through successivesubharmonic bifurcations or by quasi periodic route3'4. In the strong current modulation case the period sequence wastruncated by period tripling before the onset of chaos5'6.On the other hand, the onset of a period doubling bifurcation point in many physical systems brings to nonlinear phenomenasuch as: 1) small signal amplification, 2) shift and suppression of the bifurcation point and 3) the emergence of a closelyspaced set of peaks in the frequency response7.In semiconductor lasers the parametric amplification was observed by Harth and co-workers8'9 which was analyzed byWiesenfeld few years after'°. In the study developed by Wiesenfeld in ref. 10, semiconductor lasers were proposed as goodcandidate to study the rich variety of nonlinear phenomena near period doubling bifurcations. A recent study has shown thatthe effect of Langevin noise in period doubling semiconductor lasers has a virtual Hopf ur12 previously predicted byWiesenfeld'34 as a virtual Hopf phenomenon in systems undergoing a period doubling bifurcations in the presence ofexternal noise.In the present study we have experimentally investigated the nonlinear dynamics of diode lasers in the period doublingbifurcation point. We study the influence of the amplitude modulation in Distributed Feedback (DFB) lasers near perioddoubling bifurcations. In Fabry-Perot (FP) lasers we performed as well the same study as DFB lasers and the analysis of thefrequency modulation near period doubling bifurcation. So in section II we described the experimental procedure for theonset of the first period doubling bifurcation performed in commercial DFB and FP InGaAsP diode lasers at 1,3 tm emission

SummaryPower converter circuits, such as current‐controlled or voltage‐controlled converters and inverters often have multiple inputs in the controller. The multiple inputs cause high‐frequency and low‐frequency oscillations. In earlier studies, the characteristics of circuits in fast‐scale and slow‐scale dynamics have been investigated. However, in many cases, circuits with multiple inputs have three or more dimensional topology which makes detailed analysis difficult. In this paper, we analyze a simple interrupted electric circuit in order to understand essential characteristics of fast‐scale and slow‐scale dynamics. The advantage of this simple interrupted circuit is that it is possible to derive a 1‐dimensional map, which facilitates rigorous studies. Based on the structure of the return map and the characteristic multiplier, we explain the characteristics of the system. We report the occurrence of pitchfork, period doubling, and border collision bifurcations in slow scale, and period doubling bifurcation in fast scale. We found that local bifurcation, which appears in fast‐scale dynamics, does not significantly affect the global behavior of the system while instabilities in the slow‐scale dynamics strongly affect the system behavior. Copyright © 2016 John Wiley & Sons, Ltd.

The model of two species competing for a resource proposed by R. May and A.P. Shapiro has not yet been fully explored. We study its dynamic modes. The model reveals complex dynamics: multistable in-phase and out-of-phase cycles, and their bifurcations occur. The multistable out-of-phase dynamic modes can bifurcate via the Neimark–Sacker scenario. A value variation of interspecific competition coefficients changes the number of in-phase and out-of-phase modes. We have suggested an approach to identify the bifurcation (period-doubling, pitchfork, or saddle-node bifurcations) due to which in-phase and out-of-phase periodic points appear. With strong interspecific competition, the population’s survival depends on its growth rate. However, with a specific initial condition, a species with a lower birth rate can displace its competitor with a higher one. With weak interspecific competition and sufficiently high population growth rates, the species coexist. At the same time, the observed dynamic mode or the oscillation phase can change due to altering of the initial condition values. The influence of external factors can be considered as an initial condition modification, leading to dynamics shift due to the coexistence of several stable attractors.

This work demonstrates, for the first time, frequency-to-pulse density modulation (PDM) functionality in micromechanical resoswitches based on the period doubling bifurcation mechanism. Unlike the previously demonstrated resoswitch-based squegging clock generator of [1] that operates at the squegging oscillation state (low-frequency ringing state), of which the squegging frequency is controlled by the contact structural or material stiffness, this work demonstrates much nonlinear rich dynamical switching behaviors controlled by the period-doubling (PD) bifurcation cascade along with the driving frequency at the tapping region, featuring various M/N impact/motion (M impacts among N motions) combinations. Implemented on a CMOS-MEMS folded-beam comb-driven resoswitch, the period-doubling bifurcation derived M/N impact/motion is directly transferred to a series of pulse trains with a density determined by the M/N ratio. This work not only verifies the nonlinear period-doubled operation in micromechanical resoswitches but also opens a new possibility of designing resoswitch-based communication receivers with more sophisticated modulation schemes based on the frequency controlled PDM in addition to FSK/OOK [2].

Purpose is to study the mechanisms leading to genetic divergence (stable genetic differences between two adjacent populations). We considered the following classical model situation. Populations are panmictic with Mendelian rules of inheritance. The action of natural selection (differences in fitness) on each of population is the same and is determined by the genotypes of only one diallel locus. We assume that adjacent generations do not overlap and genetic transformations can be described by a discrete time model. This model describes the change in the concentration of one of the alleles in each population and the ratio (weight) of first population to the total size. Methods. We used the analogue of saddle charts to construct parametric portraits showing the domains of qualitatively different dynamic modes. The study is supplemented with phase portraits, basins of attraction and bifurcation diagrams. Results. We found that the model dynamic regimes qualitatively coincide with the regimes of a similar model with continuous time, but only for a weak migration. With a strong coupling, fluctuations of the phase variables are possible. We showed that the genetic divergence is possible only with reduced fitness of heterozygotes and is the result of a series of bifurcations: pitchfork bifurcation, period doubling, or saddle-node bifurcation. After these qualitative changes, the dynamics become bi- or quadstable. In the first case, the solutions corresponding to the genetic divergence are unstable and are just a part of the transient process to monomorphic state. In the second case, the divergence is stable and appears as 2-cycle for a strong migration coupling. Conclusion. In neighboring populations, movement towards an asymptotic genetic structure (monomorphism, polymorphism or divergence) can be strictly monotonous or in the form of damped unstable or undamped stable fluctuations with a period of 2 for biologically significant parameters. For insignificant parameters, we found a complex dynamics (chaos) that consist of divergent fluctuations around fixed points and quasi-random transitions between them.

In this chapter,We investigate a non-dimensionalized Nicholson and Bailey discrete-time Host-Parasitoid model. For various parameter ranges, phase portraits are made to show the system's complex dynamics. With regard to intrinsic growth rate r and searching efficiency a, we perform the bifurcation analysis. We see a wide variety of complex dynamics, including chaos and periodic windows. Period- doubling bifurcations are used to build a path to chaotic dynamics. Conditions of occurrence of the period-doubling, Neimark-Sacker and saddle-node bifurcations are analyzed for b \(\neq\) a where a, b are searching efficiency. At this non-dimensionalize system, we investigate stable and unstable manifolds for various equilibrium points as well as the presence of various attractors. The host population behaves according to the Ricker model's dynamics in the absence of the parasitoid. We were able to better comprehend the dynamical behaviour of host-parasitoid interactions with intraspecific knowledge from the current work, which can be employed to enhance the traditional biological management of parasitoids.

In this paper closed-form conditions for predicting the boundary of period-doubling (PD) bifurcation or saddle-node (SN) bifurcation in a class of PWM piecewise linear systems are obtained from a time-domain asymptotic approach. Examples of switched system considered in this study are switching dc-dc power electronics converters, temperature control systems and hydraulic valve control systems among others. These conditions are obtained from the steady-state discrete-time model using an asymptotic approach without resorting to frequency-domain Fourier analysis and without using the monodromy or the Jacobian matrix of the discrete-time model as it was recently reported in the existing literature on this topic. The availability of such design-oriented boundary expressions allows to understand the effect of the different parameters of the system upon its stability and its dynamical behavior.

Mode interactions in iterated maps with a Z2 symmetry involving a period doubling bifurcation and a symmetry breaking bifurcation of fixed points are considered. In an enlarged system of equations, a period doubling bifurcation can be considered as a Z2 symmetry breaking bifurcation so that the standard theory for Z2 × Z2 mode interactions can be used. However, a Lyapunov-Schmidt reduction is required to reduce the problem to two dimensions before applying the theory. In this case, only one of the two coordinate axes is invariant under the flow. However, we show that there is another flow invariant curve which is locally quadratic. An example showing all these features is then considered.

This research deals with the derivation and dynamical analysis of a discrete-time evolutionary Ricker population model. The model is built using Evolutionary Game Theory and takes into consideration the effect of immigration. The positive fixed point’s existence and local asymptotic stability are examined. Further, it is shown that the evolutionary model experiences Neimark–Sacker bifurcation (NSB) and period doubling bifurcation (PDB) in a small neighborhood of the positive fixed point under certain conditions. To make the chaotic behavior predictable and stable, three different chaos control strategies are applied. Detailed numerical simulations are carried out to not only verify our theoretical results but also exhibit the rich dynamics of the derived system.

An inertial shaker as a vibratory system with impact is considered. By means of differential equations, periodicity and matching conditions, the theoretical solution of periodic n−1 impacting motion can be obtained and the Poincaré map is established. Dynamics of the system are studied with special attention to interaction of Hopf and period doubling bifurcations corresponding to a codimension-2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses −1 simultaneously for the Jacobi matrix. The four-dimensional map can be reduced to a three-dimensional normal form by the center manifold theorem and the theory of normal forms. The two-parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated. It is proved that there exist curve doubling bifurcation (a torus doubling bifurcation), Hopf bifurcation of 2–2 fixed points as well as period doubling bifurcation and Hopf bifurcation of 1–1 fixed points near the critical point. Numerical results indicate that the vibro-impact system presents complicated and interesting curve doubling bifurcation and Hopf bifurcation as the two controlling parameters vary.

Transition to chaos through a cascade of period doublings of the primary synchronization mode is discovered in steady approaching flow around a forced inline oscillating cylinder near a plane boundary at a Reynolds number of 175. The transition occurs well within the otherwise synchronized region (known as the Arnold tongue) in the frequency and amplitude space of the oscillating cylinder, creating two parameter strips of desynchronized flows within the Arnold tongue. Five orders of period doublings from mode to mode are revealed by progressively increasing the frequency resolution in the simulation. The ratio of frequency intervals of two successive period-doubling modes asymptotes towards the first Feigenbaum constant, reaching a value of 4.52 at mode of . Additional three-dimensional simulations demonstrate the existence of period doubling with a regular spanwise flow structure similar to regular mode B of steady flow around an isolated cylinder. Although transition to chaos through cascades of period doublings is primarily reported for the primary synchronization mode, it is also observed for other synchronization modes (Tang et al., J. Fluid Mech., vol. 832, 2017, pp. 146–169), where and are integers with a non-reducible , such as . The physical mechanisms responsible for the present period-doubling bifurcations and transition to chaos through cascades of period doublings are ascribed to the interaction of asymmetric vortex shedding from the cylinder (due to a geometric asymmetry) and the boundary layer developed on the plane boundary, through specifically designed numerical tests.

In this paper routes to chaotic oscillation in power systems are deeply studied. Using a three-bus simple system, three routes which may cause chaos in power systems are presented, illustrated and discussed. They are the route of cascading period doubling bifurcation (PDB), torus bifurcation route and directly initiated by large disturbance route. PDB is caused by a real Floquet multiplier (FM) moving counter to the real axis and going out of the unit circle from a point (-1,0) in the complex plane. The route of cascading PDB is a typical route to chaos and has been studied deeply in many nonlinear systems. In this paper, we give a full bifurcation diagram of PDB and discuss some of its detail structure. Torus bifurcation (TB) is also a typical route to chaos. TB is caused by a couple of conjugate Floquet multipliers (FM) going out of the unit circle with a nonzero imaginary part in the complex plane. Chaos caused by TB has some interesting features, such as self-organizing phenomenon, coexistence of divergent subspace and chaotic subspace. These features are helpful to deeply understand various modes of power system instability. The last route, which is directly initiated by a large disturbance, is reported and studied for the first time to the authors' knowledge. All studies told us that chaos in power systems is in fact caused by some kind of external disturbances.