Recent advances in bifurcation analysis: Theory, methods, applications and beyond…

Editorial

13 April 2022

Editorial: Recent Advances in Bifurcation Analysis: Theory, Methods, Applications and Beyond…

Pablo Aguirre

and

Víctor F. Breña-Medina

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Editors

Víctor F. Breña-Medina

Instituto Tecnológico Autónomo de México

Pablo Aguirre

Federico Santa María Technical University

Impact

The schematic diagram of the number of positive roots of Equation (5) in the (u, Q)-space for (A, M) = (1/10, −1/10) fixed and with T = 1 − A + M, L = A(M + 1) − Q − M, Δ = (u* − T(A, M))2 − 4(u*(u* − T(A, M)) − L(A, M, Q)) and u* is determined by the solution(s) of (5). At Q=Q±:=(73±5)/200, that is, at Q− ≈ 0.35381966 and Q+ ≈ 0.376180, two positive equilibria collapse. For Q < Q− or Q > Q+, Equation (5) has only one positive root (light grey regions). Hence, system (4) has only one positive equilibrium P = (u*, u*). For Q− < Q < Q+, Equation (5) has three positive roots (dark grey region) and system (4) thus has three positive equilibria. (A) Q < Q−. (B) Q = Q−. (C) Q− < Q < Q+. (D) Q = Q+. (E) Q > Q+.

Original Research

26 January 2022

Stability Analysis of a Modified Leslie–Gower Predation Model With Weak Allee Effect in the Prey

Claudio Arancibia-Ibarra

, 1 more and

Peter van Heijster

In this manuscript, we study a Leslie–Gower predator-prey model with a hyperbolic functional response and weak Allee effect. The results reveal that the model supports coexistence and oscillation of both predator and prey populations. We also identify regions in the parameter space in which different kinds of bifurcations, such as saddle-node bifurcations, Hopf bifurcations and Bogdanov–Takens bifurcations.

5,933 views

8 citations

Original Research

18 March 2022

Classification of Codimension-1 Singular Bifurcations in Low-Dimensional DAEs

Ivan Ovsyannikov

and

Haibo Ruan

3,720 views

0 citations

(A) Dependence of system behaviors on the internal parameter. The bifurcation diagram is shown as a function of the internal system parameter a. The red and blue dots indicate negative and positive x (0) cases (top part), respectively, and the attractor-merging condition F (fmax, min)− xd is shown (middle part) (b =10, xd =0). The orbit, map functions, and attractor-merging condition F (fmax, min)− xd (indicated by magenta and green dots, respectively) are also shown (bottom part). (B) Attractor-merging and separating achieved by “reduced region of orbit” (RRO) feedback signals in a cubic map (b = 10, σ =0.6). The orbit, map functions, and attractor-merging condition F (fmax, min)+ Ku(fmax, min)− xd (indicated by the magenta and green dots, respectively) under the attractor-separating condition (a =2.75, K =0) (left-hand side in the upper section) and those under the attractor-merging condition induced by the RRO feedback signal with a negative strength (K = −0.2) (right-hand side in the upper part) are shown. The orbit, map functions, and attractor-merging condition F (fmax, min) + Ku(fmax, min)− xd under the attractor-merging condition of a = 2.86, K =0 (left-hand side in the bottom part), as well as those under the attractor-separating condition induced by an RRO feedback signal with a positive strength (K = 0.2) (right-hand side in the bottom part) are shown. (C) Ability of the signal response measured by the correlation coefficient maxτC(τ) as a function of the attractor-merging condition F (fmax, min) + Ku(fmax, min)− xd in the case where attractor-merging is induced by the RRO feedback signal (a = 2.82, 2.825,2.83) (left side), and where attractor-separating is induced by the RRO feedback signal (a =2.85, 2.855,2.86) (right side). Solid line and shaded area represent the mean and standard deviation, respectively, among 10 trials with different initial x (0).

Perspective

30 November 2021

Recent Trends of Controlling Chaotic Resonance and Future Perspectives

Sou Nobukawa

, 4 more and

Tetsuya Takahashi

Stochastic resonance is a phenomenon in which the effects of additive noise strengthen the signal response against weak input signals in non-linear systems with a specific barrier or threshold. Recently, several studies on stochastic resonance have been conducted considering various engineering applications. In addition to additive stochastic noise, deterministic chaos causes a phenomenon similar to the stochastic resonance, which is known as chaotic resonance. The signal response of the chaotic resonance is maximized around the attractor-merging bifurcation for the emergence of chaos-chaos intermittency. Previous studies have shown that the sensitivity of chaotic resonance is higher than that of stochastic resonance. However, the engineering applications of chaotic resonance are limited. There are two possible reasons for this. First, the stochastic noise required to induce stochastic resonance can be easily controlled from outside of the stochastic resonance system. Conversely, in chaotic resonance, the attractor-merging bifurcation must be induced via the adjustment of internal system parameters. In many cases, achieving this adjustment from outside the system is difficult, particularly in biological systems. Second, chaotic resonance degrades owing to the influence of noise, which is generally inevitable in real-world systems. Herein, we introduce the findings of previous studies concerning chaotic resonance over the past decade and summarize the recent findings and conceivable approaches for the reduced region of orbit feedback method to address the aforementioned difficulties.

8,013 views

8 citations

Original Research

29 July 2021

Entrainment Dynamics Organised by Global Manifolds in a Circadian Pacemaker Model

Jennifer L. Creaser

, 1 more and

Kyle C. A. Wedgwood

(A–C): Simulations of the FJK model Eqs 1–5 for N=12, τc=24.2 with I=50. (D–F): Simulations of the 2D version of the FJK model using (Eq. 8) in place of (Eq. 4). (A) + (D): Time series of A (black), C (blue), and n (red). The shaded regions indicate where f(t)=0, whereas f(t)=1 in the unshaded regions. (B) + (E): Stable entrained limit cycle. The dark portion of the orbit corresponds to the interval in which f(t) = 0. The marker shows the point along the orbit where f changes from 0 to 1. (C) + (F): Projection of panels (B, E), respectively, into the (A,C) plane.

Circadian rhythms are established by the entrainment of our intrinsic body clock to periodic forcing signals provided by the external environment, primarily variation in light intensity across the day/night cycle. Loss of entrainment can cause a multitude of physiological difficulties associated with misalignment of circadian rhythms, including insomnia, excessive daytime sleepiness, gastrointestinal disturbances, and general malaise. This can occur after travel to different time zones, known as jet lag; when changing shift work patterns; or if the period of an individual’s body clock is too far from the 24 h period of environmental cycles. We consider the loss of entrainment and the dynamics of re-entrainment in a two-dimensional variant of the Forger-Jewett-Kronauer model of the human circadian pacemaker forced by a 24 h light/dark cycle. We explore the loss of entrainment by continuing bifurcations of one-to-one entrained orbits under variation of forcing parameters and the intrinsic clock period. We show that the severity of the loss of entrainment is dependent on the type of bifurcation inducing the change of stability of the entrained orbit, which is in turn dependent on the environmental light intensity. We further show that for certain perturbations, the model predicts counter-intuitive rapid re-entrainment if the light intensity is sufficiently high. We explain this phenomenon via computation of invariant manifolds of fixed points of a 24 h stroboscopic map and show how the manifolds organise re-entrainment times following transitions between day and night shift work.

3,390 views

9 citations