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![Two limit cycles in an HIV model [18]: when B=D=0.057, A=0.01846287, C=0.11969000: (a) three trajectories with moving directions indicated; and (b) two limit cycles with the inner unstable and outer stable.](publication/299646114/figure/fig1/AS:11431281256785391@1719568565527/Two-limit-cycles-in-an-HIV-model-18-when-BD0057-A001846287-C011969000-a.gif)
Two limit cycles in an HIV model [18]: when B=D=0.057, A=0.01846287, C=0.11969000: (a) three trajectories with moving directions indicated; and (b) two limit cycles with the inner unstable and outer stable.
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![Two limit cycles in an HIV model [18]: when B=D=0.057, A=0.01846287,...](publication/299646114/figure/fig1/AS:11431281256785391@1719568565527/Two-limit-cycles-in-an-HIV-model-18-when-BD0057-A001846287-C011969000-a_Q320.jpg)
In this paper, we study complex dynamical behaviour in biological systems due to multiple limit cycles bifurcation. We use simple epidemic and predator–prey models to show exact routes to new types of bistability, that is, bistability between equilibrium and periodic oscillation, and bistability between two oscillations, which may more realisticall...
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In this paper, the unified behavior for maps with different dimensions is possible from some cases of dynamical systems. This short paper proposes a 2-D noninvertible discrete chaotic map with one bifurcation parameter, and that had only one nonlinear term, and a new 3-D noninvertible discrete chaotic map with twelve bifurcation parameters, and six...
Citations
... Sun et al. (2017), performed a cholera nonlinear transmission model, which is applied to highlight the virus dynamics in China. Yu et al. (Yu and Lin, 2016), proposed the sophisticated behavior in the biological systems using the frequent limit junction cycle of the simple prey-predictor system. ...
... Sun et al. (2017), performed a cholera nonlinear transmission model, which is applied to highlight the virus dynamics in China. Yu et al. (Yu and Lin, 2016), proposed the sophisticated behavior in the biological systems using the frequent limit junction cycle of the simple prey-predictor system. ...
... The change of system parameters has an impact on the dynamic behavior of the oligopoly game model [40,41] . Used the classical chaotic system and biophysical model to simulate the control and detection principle, on the basis of certain theory, white noise can enhance the chaotic synchronization between two chaotic oscillators, and made in-depth research on the application of numerical simulation, in addition, in basic stability in delayed dynamics, by replacing the traditional linearization method with the volume of attraction basin, the proposed an algorithm with cross validation program to numerically estimate the attraction basin in delay dynamics, other delayed Hopfield neural element models with multi-stability and delayed complex networks with synchronous dynamics, at the same time, the complex dynamic phenomena caused by bifurcation of multiple limit cycles are also studied [42][43][44][45][46] . Ma, et al. [47] found that different bifurcation parameters correspond to different chaotic paths in temporary system structure of structural transformation, which has theoretical and practical significance for the macro-control of system. ...
Based on bounded rationality, this paper established a price game model of dual channel supply chain composed of manufacturers and retailers. According to the eigenvalue of Jacobi matrix and Jury criterion, the stability of the equilibrium point is analyzed, and then the dynamic evolution process under the parameters of price adjustment speed and retailer's service input is studied through stability region, bifurcation diagram, maximum Lyapunov exponent diagram and attraction basin. The results show that the system enters chaos through flip and Neimark-Sacker bifurcation, and increase of price adjustment speed and service input value will make the system produce more dynamic behavior. In addition, it can be found that the impact of service input value on itself is much greater than that on manufacturers. Secondly, when adjustment speed is selected as bifurcation parameter, the change curve of sales price is inconsistent, in which the change of retailers mainly remains in periodic state, while manufacturers will gradually enter chaos. Finally, studies the evolution of attraction basin in which three kinds of attractors coexist. In particular, coexistence of boundary attractors and internal attractors increases the complexity of system. Therefore, enterprises need to carefully adjust parameters of the game model to control the stability of system and maintain the long-term stability of market competition.
... The researcher's community investigated the increasing immunization coverage rate and the managing environment to prevent disease spreading. Yu et al. [21] presented the sophisticated behavior based on the biological systems due to the numerous limit cycle bifurcation along with the predictor-prey simple system. Li et al. [22] proposed the SIR nonlinear system using the recovery and incidence rates. ...
The purpose of this study is to present the numerical performances and interpretations of the SEIR nonlinear system based on the Zika virus spreading by using the stochastic neural networks based intelligent computing solver. The epidemic form of the nonlinear system represents the four dynamics of the patients, susceptible patients S(y), exposed patients hospitalized in hospital E(y), infected patients I(y), and recovered patients R(y), i.e., SEIR model. The computing numerical outcomes and performances of the system are examined by using the artificial neural networks (ANNs) and the scaled conjugate gradient (SCG) for the training of the networks, i.e., ANNs-SCG. The correctness of the ANNs-SCG scheme is observed by comparing the proposed and reference solutions for three cases of the SEIR model to solve the nonlinear system based on the Zika virus spreading dynamics through the knacks of ANNs-SCG procedure based on exhaustive experimentations. The outcomes of the ANNs-SCG algorithm are found consistently in good agreement with standard numerical solutions with negligible errors. Moreover, the procedure’s constancy, dependability, and exactness are perceived by using the values of state transitions, error histogram measures, correlation, and regression analysis.
... We emphasize that although limit cycles have been identified in relevant eco-evolutionary research [8], [10], [11], [12], [13], unstable and double limit cycles have not been reported in the relevant studies so far. These behaviors correspond to an interesting dynamic bistable phenomenon which has been discovered in Lotka-Volterra models [14], [15]. More importantly, we also implement analysis revealing how the two limit cycles can coexist through two-parameter bifurcation. ...
... Next, since the trace of the Jacobian is the sum of the eigenvalues, then in view of (15) and (20), one can check that the derivative of the real part of the eigenvalues is ...
... Moreover, for some certain c, two limit cycles, unstable and stable respectively, could coexist, which makes the dynamic behaviors even more complicated. This behavior corresponds to an interesting dynamic bistable phenomenon as discovered in [15], [14]. ...
The fast-slow dynamics of an eco-evolutionary system are studied, where we consider the feedback actions of environmental resources that are classified into those that are self-renewing and those externally supplied. We show although these two types of resources are drastically different, the resulting closed-loop systems bear close resemblances, which include the same equilibria and their stability conditions on the boundary of the phase space, and the similar appearances of equilibria in the interior. After closer examination of specific choices of parameter values, we disclose that the global dynamical behaviors of the two types of closed-loop systems can be fundamentally different in terms of limit cycles: the system with self-renewing resources undergoes a generalized Hopf bifurcation such that one stable limit cycle and one unstable limit cycle can coexist; the system with externally supplied resources can only have the stable limit cycle induced by a supercritical Hopf bifurcation. Finally, the explorative analysis is carried out to show the discovered dynamic behaviors are robust in even larger parameter space.
... The mathematical theory is related to the well-known Hilbert's 16th problem [13] . Very recently, bifurcation of multiple limit cycles has been found in a 3-dimensional physical system [18] , and in 2-dimensional population and disease models [15,29,33] , showing the interesting bistable or even tri-stable phenomenon, which involves equilibria and oscillating motions. ...
... 12 m 4 3 ,where C 0 and C 1 are given in(29) , and the lengthy expressions of v 2 and v 3 are given in Appendix B . Eliminating m 5 from the equations v 1 = v 2 = v 3 = 0 yields two resultants R 1 and R 2 , given below.R 1 = m 3 (m 2 + m 3 )(382993712640 m14 2 − 246 88134 99792 m 13 2 m 3 − 2 , 505 , 129 , 112 , 368 m 4 , 550 , 405 , 074 , 296 m = m 3 (m 2 + m 3 )(115506 8278895746 85029171200 m 28 2 + 4502226 89604 990045276979200 m 27 2 m 3 − 3787528396335697123760 6 682624 m 26 2 m 2 3 + 1397455136412961767746 6 6515008 m 2 2 5 m 3 3 + 284 , 816 , 537 , 546 , 699 , 598 , 961 , 284 , 939 , 088 m 24 2 m 4 3 − 172359285050 6 689778830751030756 m 23 2 m 5 3 − 3670269011319496547801373570 05 m 22 2 m 6 3 + 97890 05276920435124180 050948698 m 21 2 m 7 3 − 596422340 0 0 658757818646 64122257 m 20 2 m 8 3 − 289847370 61956193514958093718206 m 19 2 m 9 3 + 35 , ...
In this paper, we investigate the influence of the effector-regulatory (Teff-Treg) T cell interaction on the T-cell-mediated autoimmune disease dynamics. The simple 3-dimensional Teff-Treg model is derived from the two-step model reduction of an established 5-dimensional model. The reduced 4- and 3-dimensional models preserve the dynamical behaviors in the original 5-dimensional model, which represents the chronic and relapse-remitting autoimmune symptoms. Moreover, we find three co-existing limit cycles in the reduced 3-dimensional model, in which two stable periodic solutions enclose an unstable one. The existence of multiple limit cycles provides a new mechanism to explain varying oscillating amplitudes of lesion grade in multiple sclerosis. The complex multiphase symptom could be caused by a noise-driven Teff population traveling between two coexisting stable periodic solutions. The simulated phase portrait and time history of coexisting limit cycles are given correspondingly.
... Stability and Hopf bifurcation for a recurrent neural network with time delays are investigated in [Yu et al., 2008]. Complex dynamical behavior due to multiple limit cycles bifurcation in biological systems is studied in [Yu & Lin, 2016]. Therefore, the impact of time delay on the performance of the network system often needs to be considered. ...
... Hopf bifurcation analysis has been extensively used to investigate the dynamic characteristics information near a fixed point of a nonlinear system. It is well known that Hopf bifurcations of integer-order systems have been sufficiently studied, and a lot of excellent results have been obtained [Yu & Lin, 2016;Zeng et al., 2016;Xu et al., 2015aXu et al., , 2015bZhao et al., 2015;Xu et al., 2014;Xiao et al., 2013;Shi & Wang, 2013]. Bifurcation control relates to introducing a kind of controller to change the bifurcation characteristics for a nonlinear system, in order to attain certain desired dynamic behaviors, to obtain bifurcation, chaotic dynamic behavior, or to achieve a stable state [Abed & Fu, 1986;Wang & Abed, 1995;Chen et al., 2000;Luo et al., 2003;Yu & Chen, 2004;Liu et al., 2009;Wang & Jian, 2010;Cheng, 2010;Zhao et al., 2011]. ...
In this paper, stability analysis and bifurcation control for a novel incommensurate fractional-order delayed gene regulatory network are investigated. Firstly, the associated characteristic equation is analyzed by taking time delay as a bifurcation parameter, and the conditions of creation for Hopf bifurcation are established. It is demonstrated that the time delay can profoundly affect the dynamics of the proposed system and each order has a significant influence on the creation of bifurcation simultaneously. Then we study the stability and bifurcation behavior of the fractional-order networks by adding a nonlinear feedback controller. Finally, numerical simulations of two examples validate the obtained results.
... Li et al. [47] constructed a multi-group brucellosis model and found out that the best way to contain the disease is to avoid cross infection of animal populations. Moreover, Yu and Lin [48] identified complex dynamical behaviour in epidemiological models and particularly the existence of multiple limit cycle bifurcations using a predictor-prey model. Shi et al. [49] proposed an HIV model with a saturated reverse function to describe the dynamics of infected cells. ...
A worldwide multi-scale interplay among a plethora of factors, ranging from micro-pathogens and individual or population interactions to macro-scale environmental, socio-economic and demographic conditions, entails the development of highly sophisticated mathematical models for robust representation of the contagious disease dynamics that would lead to the improvement of current outbreak control strategies and vaccination and prevention policies. Due to the complexity of the underlying interactions, both deterministic and stochastic epidemiological models are built upon incomplete information regarding the infectious network. Hence, rigorous mathematical epidemiology models can be utilized to combat epidemic outbreaks. We introduce a new spatiotemporal approach (SBDiEM) for modeling, forecasting and nowcasting infectious dynamics, particularly in light of recent efforts to establish a global surveillance network for combating pandemics with the use of artificial intelligence. This model can be adjusted to describe past outbreaks as well as COVID-19. Our novel methodology may have important implications for national health systems, international stakeholders and policy makers.
... The two most frequent dynamic bifurcations detected in biological systems are saddle-node (Tyson and Novak, 2015) and Hopf bifurcations (Sun et al., 2018;Yu and Lin, 2016). Saddle-node bifurcation, in the simplest statement, is when two stable (a node) and unstable (a saddle) steady states collide and destroy each other as a result of parameter change in the system. ...
Many biological processes show switching behaviors in response to parameter changes. Although numerous surveys have been conducted on bifurcations in biological systems, they commonly focus on over-represented parts of signaling cascades, known as motifs, ignoring the multi-motif structure of biological systems and the communication links between these building blocks. In this paper, a method is proposed which partitions molecular interactions to modules based on a control theory point of view. The modules are defined so that downstream effect of one module is a regulator for its neighboring modules. Communication links between these modules are then considered as bifurcation parameters to reveal change in steady state status of each module. As a case-study, we generated a molecular interaction map of signaling molecules during the development of mammalian embryonic kidneys. The whole system was divided to modules, where each module is defined as a group of interacting molecules that result in expression of a vital downstream regulator. Bifurcation analysis was then performed on these modules by considering the communication signals as bifurcation parameters. Two-parameter bifurcation analysis was then performed to assess the effects of simultaneous input signals on each module behavior. In the case where a module had more than two inputs, a series of two parameter bifurcation diagrams were calculated each corresponding to different values of the third parameter. We detected multi-stability for RET which its different activity levels have been suggested to affect cell arrangement in nephric duct, tip and trunk of the bud in embryonic kidney. These results are in agreement with experimental data indicating that cells involved in Embryonic kidney development are bi-potential and they form tip or trunk of the bud based on their RET activity level. Our findings also indicate that Glial cell-derived neurotrophic factor (GDNF), a known potent regulator of kidney development, exerts its fate-determination function on cell placement through destruction of saddle node bifurcation points in RET steady states and confining RET activity level to high activity. In conclusion, embryonic cells usually show a huge decision making potential; the proposed modular modeling of the system in association with bifurcation analysis provides a quantitative holistic view of organ development.
... It is extremely difficult to prove the existence of three limit cycles. Very few articles have been published to discuss the existence of three limit cycles, for example, see [Gonzàlez-Olivares et al., 2011;Yu & Lin, 2016]. ...
In this paper, we consider a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. The main attention is focused on the stability and bifurcation of equilibria when the prey has a linear growth. Coexistence of different species is shown in the food chain, demonstrating bistable phenomenon. Hopf bifurcation is studied to show complex dynamics due to multiple limit cycles bifurcation. In particular, normal form theory is applied to prove that three limit cycles can bifurcate from an equilibrium in the vicinity of a Hopf critical point, yielding a new bistable phenomenon which involves two stable limit cycles.
