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Journal articles on the topic 'Saddle node bifurcation'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

ROCŞOREANU, CARMEN, NICOLAIE GIURGIŢEANU, and ADELINA GEORGESCU. "CONNECTIONS BETWEEN SADDLES FOR THE FITZHUGH–NAGUMO SYSTEM." International Journal of Bifurcation and Chaos 11, no. 02 (February 2001): 533–40. http://dx.doi.org/10.1142/s0218127401002213.

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By studying the two-dimensional FitzHugh–Nagumo (F–N) dynamical system, points of Bogdanov–Takens bifurcation were detected (Sec. 1). Two of the curves of homoclinic bifurcation emerging from these points intersect each other at a point of double breaking saddle connection bifurcation (Sec. 2). Numerical investigations of the bifurcation curves emerging from this point, in the parameter plane, allowed us to find other types of codimension-one and -two bifurcations concerning the connections between saddles and saddle-nodes, referred to as saddle-node–saddle connection bifurcation and saddle-node–saddle with separatrix connection bifurcation, respectively. The local bifurcation diagrams corresponding to these bifurcations are presented in Sec. 3. An analogy between the bifurcation corresponding to the point of double homoclinic bifurcation and the point of double breaking saddle connection bifurcation is also presented in Sec. 3.

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2

Ueta, Tetsushi, Daisuke Ito, and Kazuyuki Aihara. "Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?" International Journal of Bifurcation and Chaos 25, no. 13 (December 15, 2015): 1550185. http://dx.doi.org/10.1142/s0218127415501850.

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We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

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3

Kuznetsov, Yuri. "Saddle-node bifurcation." Scholarpedia 1, no. 10 (2006): 1859. http://dx.doi.org/10.4249/scholarpedia.1859.

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4

Fang, Ding, Yongxin Zhang, and Wendi Wang. "Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate." Complexity 2020 (May 8, 2020): 1–16. http://dx.doi.org/10.1155/2020/7310347.

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An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

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5

HIZANIDIS, J., R. AUST, and E. SCHÖLL. "DELAY-INDUCED MULTISTABILITY NEAR A GLOBAL BIFURCATION." International Journal of Bifurcation and Chaos 18, no. 06 (June 2008): 1759–65. http://dx.doi.org/10.1142/s0218127408021348.

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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.

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6

Shilnikov, Leonid, and Andrey Shilnikov. "Shilnikov saddle-node bifurcation." Scholarpedia 3, no. 4 (2008): 4789. http://dx.doi.org/10.4249/scholarpedia.4789.

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7

GLENDINNING, PAUL, and COLIN SPARROW. "SHILNIKOV’S SADDLE-NODE BIFURCATION." International Journal of Bifurcation and Chaos 06, no. 06 (June 1996): 1153–60. http://dx.doi.org/10.1142/s0218127496000643.

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In 1969, Shilnikov described a bifurcation for families of ordinary differential equations involving n≥2 trajectories bi-asymptotic to a non-hyperbolic stationary point. At nearby parameter values the system has trajectories in correspondence with the full shift on n symbols. We investigate a simple (piecewise-smooth) example with an infinite number of homoclinic loops. We also present a smooth example which shows how Shilnikov’s mechanism is related to the Lorenz bifurcation by considering the unfolding of a previously unstudied codimension two bifurcation point.

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8

AFRAIMOVICH, V. S., and M. A. SHERESHEVSKY. "THE HAUSDORFF DIMENSION OF ATTRACTORS APPEARING BY SADDLE-NODE BIFURCATIONS." International Journal of Bifurcation and Chaos 01, no. 02 (June 1991): 309–25. http://dx.doi.org/10.1142/s0218127491000233.

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We consider the strange attractors which appear as a result of saddle-node vanishing bifurcations in two-dimensional, smooth dynamical systems. Some estimates and asymptotic formulas for the Hausdorff dimension of such attractors are obtained. The estimates demonstrate a dependence of the dimension growth rate after the bifurcation upon the "pre-bifurcational" picture.

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9

YET, NGUYEN TIEN, DOAN THAI SON, TOBIAS JÄGER, and STEFAN SIEGMUND. "NONAUTONOMOUS SADDLE-NODE BIFURCATIONS IN THE QUASIPERIODICALLY FORCED LOGISTIC MAP." International Journal of Bifurcation and Chaos 21, no. 05 (May 2011): 1427–38. http://dx.doi.org/10.1142/s0218127411029124.

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We provide a local saddle-node bifurcation result for quasiperiodically forced interval maps. As an application, we give a rigorous description of saddle-node bifurcations of 3-periodic graphs in the quasiperiodically forced logistic map with small forcing amplitude.

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10

KIRK, VIVIEN, and EDGAR KNOBLOCH. "A REMARK ON HETEROCLINIC BIFURCATIONS NEAR STEADY STATE/PITCHFORK BIFURCATIONS." International Journal of Bifurcation and Chaos 14, no. 11 (November 2004): 3855–69. http://dx.doi.org/10.1142/s0218127404011752.

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We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit. We show that this phenomenon may occur in an unfolding of the saddle-node/pitchfork normal form equations, although not near the saddle-node/pitchfork instability. By construction and analysis of a return map, we show that the codimension-one bifurcation emerges from a codimension-two point at which there is a heteroclinic bifurcation between two saddle equilibria, one hyperbolic and the other nonhyperbolic. We find the same phenomenon occurs in the normal form equations for the hysteresis/pitchfork bifurcation, in this case arbitrarily close to the instability, and show there are restrictions regarding the way in which such dynamics can occur near pitchfork/pitchfork bifurcations. These conclusions carry over to analogous phenomena in normal forms for steady state/Hopf bifurcations.

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11

García de la Vega, Ignacio, and Ricardo Riaza. "Saddle-Node Bifurcations in Classical and Memristive Circuits." International Journal of Bifurcation and Chaos 26, no. 04 (April 2016): 1650064. http://dx.doi.org/10.1142/s0218127416500644.

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This paper addresses a systematic characterization of saddle-node bifurcations in nonlinear electrical and electronic circuits. Our approach is a circuit-theoretic one, meaning that the bifurcation is analyzed in terms of the devices’ characteristics and the graph-theoretic properties of the digraph underlying the circuit. The analysis is based on a reformulation of independent interest of the saddle-node theorem of Sotomayor for semiexplicit index one differential-algebraic equations (DAEs), which define the natural context to set up nonlinear circuit models. The bifurcation is addressed not only for classical circuits, but also for circuits with memristors. The presence of this device systematically leads to nonisolated equilibria, and in this context the saddle-node bifurcation is shown to yield a bifurcation of manifolds of equilibria; in cases with a single memristor, this phenomenon describes the splitting of a line of equilibria into two, with different stability properties.

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12

Kuznetsov, Yuri. "Saddle-node bifurcation for maps." Scholarpedia 3, no. 4 (2008): 4399. http://dx.doi.org/10.4249/scholarpedia.4399.

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13

Yabuno, Hiroshi, Masahiko Hasegawa, and Manami Ohkuma. "Bifurcation control for a parametrically excited cantilever beam by linear feedback." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226, no. 8 (March 22, 2012): 1987–99. http://dx.doi.org/10.1177/0954406212442603.

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In this article, we propose a bifurcation control method for a parametrically excited cantilever beam by linear feedback. Quadratic damping plays a dominant role in the nonlinear response of the parametrically excited cantilever beam, and two transcritical bifurcations can exist in the frequency–response curve. In the relatively high-amplitude excitation or in sweeping the excitation amplitude, there are two saddle-node bifurcations in addition to the transcritical bifurcations. The discontinuous bifurcation as a saddle-node bifurcation induces jumping phenomena in the sweeps of the excitation amplitude and the excitation frequency. In this article, we focus on the case of the excitation amplitude sweep and propose a control method to avoid the jumping phenomena by bifurcation control, i.e. by shifting the bifurcation set based on the linear feedback. The validity of the control method is experimentally confirmed using a simple apparatus.

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14

Cantin, Guillaume, Nathalie Verdiére, Valentina Lanza, M. A. Aziz-Alaoui, Rodolphe Charrier, Cyrille Bertelle, Damienne Provitolo, and Edwige Dubos-Paillard. "Mathematical Modeling of Human Behaviors During Catastrophic Events: Stability and Bifurcations." International Journal of Bifurcation and Chaos 26, no. 10 (September 2016): 1630025. http://dx.doi.org/10.1142/s0218127416300251.

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The aim of this paper is to present some mathematical results concerning the PCR system (Panic-Control-Reflex), which is a model for human behaviors during catastrophic events. This model has been proposed to better understand and predict human reactions of individuals facing a brutal catastrophe, in a context of an established increase of natural and industrial disasters. After stating some basic properties, that is positiveness, boundedness, and stability of the solutions, we analyze the transitional dynamic. We then focus on the bifurcation that occurs in the system, when one behavioral evolution parameter passes through a critical value. We exhibit a degeneracy case of a saddle-node bifurcation, in a larger context of classical saddle-node bifurcations and saddle-node bifurcations at infinity, and we study the inhibition effect of higher order terms.

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15

Yang, J. H., Miguel A. F. Sanjuán, F. Tian, and H. F. Yang. "Saddle-Node Bifurcation and Vibrational Resonance in a Fractional System with an Asymmetric Bistable Potential." International Journal of Bifurcation and Chaos 25, no. 02 (February 2015): 1550023. http://dx.doi.org/10.1142/s0218127415500236.

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We investigate the saddle-node bifurcation and vibrational resonance in a fractional system that has an asymmetric bistable potential. Due to the asymmetric nature of the potential function, the response and its amplitude closely depend on the potential well where the motion takes place. And consequently for numerical simulations, the initial condition is a key and important factor. To overcome this technical problem, a method is proposed to calculate the bifurcation and response amplitude numerically. The numerical results are in good agreement with the analytical predictions, indicating the validity of the numerical and theoretical analysis. The results show that the fractional-order of the fractional system induces one saddle-node bifurcation, while the asymmetric parameter associated to the asymmetric nature of the potential function induces two saddle-node bifurcations. When the asymmetric parameter vanishes, the saddle-node bifurcation turns into a pitchfork bifurcation. There are three kinds of vibrational resonance existing in the system. The first one is induced by the high-frequency signal. The second one is induced by the fractional-order. The third one is induced by the asymmetric parameter. We believe that the method and the results shown in this paper might be helpful for the analysis of the response problem of nonlinear dynamical systems.

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16

Sheng, Hao, Hsiao-Dong Chiang, and Yan-Feng Jiang. "Local Bifurcations of Electric Distribution Networks with Renewable Energy." International Journal of Bifurcation and Chaos 24, no. 07 (July 2014): 1450102. http://dx.doi.org/10.1142/s0218127414501028.

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Recent years have witnessed a growing trend towards the development and deployment of distributed generation (DG). It is shown that electric distribution networks with DGs can encounter two types of local bifurcations: saddle-node bifurcation and structure-induced bifurcation. The structure-induced bifurcation occurs when a transition between two structures of the distribution network takes place due to limited amount of reactive power supports from renewable energies. The saddle-node bifurcation occurs when the underlying distribution network reaches the limit of its delivery capability. The consequence of structure-induced bifurcation is an immediate instability induced by reactive power limits of renewable energy. It is numerically shown that both types of local bifurcations can occur at both small distribution networks and large-scale distribution networks with DGs. Physical explanations of these two local bifurcations are provided. Studies of local bifurcations in distribution networks provide insights regarding how to design controls to enhance distribution networks with DGs.

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17

Xu, Yeyin, Zhaobo Chen, and Albert C. J. Luo. "Period-1 Motion to Chaos in a Nonlinear Flexible Rotor System." International Journal of Bifurcation and Chaos 30, no. 05 (April 2020): 2050077. http://dx.doi.org/10.1142/s0218127420500777.

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In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

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18

Ai, Wenhuan, Zhongke Shi, and Dawei Liu. "Bifurcation analysis method of nonlinear traffic phenomena." International Journal of Modern Physics C 26, no. 10 (June 24, 2015): 1550111. http://dx.doi.org/10.1142/s0129183115501119.

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A new bifurcation analysis method for analyzing and predicting the complex nonlinear traffic phenomena based on the macroscopic traffic flow model is presented in this paper. This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis. Although the substitution seems to be simple, it can extend the range of the variable to infinity and build a relationship between the traffic congestion and the unstable system in the phase plane. So the problem of traffic flow could be converted into that of system stability. The analysis identifies the types and stabilities of the equilibrium solutions of the new model and gives the overall distribution structure of the nearby equilibrium solutions in the phase plane. Then we deduce the existence conditions of the models Hopf bifurcation and saddle-node bifurcation and find some bifurcations such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles and Bogdanov–Takens bifurcation. Furthermore, the Hopf bifurcation and saddle-node bifurcation are selected as the starting point of density temporal evolution and it will be helpful for improving our understanding of stop-and-go wave and local cluster effects observed in the free-way traffic.

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19

GU, HUAGUANG, HUIMIN ZHANG, CHUNLING WEI, MINGHAO YANG, ZHIQIANG LIU, and WEI REN. "COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION." International Journal of Modern Physics B 25, no. 29 (November 20, 2011): 3977–86. http://dx.doi.org/10.1142/s0217979211101673.

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Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

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20

Liu, Xia, and Yepeng Xing. "Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/478315.

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The bifurcation properties of a predator prey system with refuge and constant harvesting are investigated. The number of the equilibria and the properties of the system will change due to refuge and harvesting, which leads to the occurrence of several kinds bifurcation phenomena, for example, the saddle-node bifurcation, Bogdanov-Takens bifurcation, Hopf bifurcation, backward bifurcation, separatrix connecting a saddle-node and a saddle bifurcation and heteroclinic bifurcation, and so forth. Our main results reveal much richer dynamics of the system compared to the system with no refuge and harvesting.

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21

Xu, Yeyin, and Albert C. J. Luo. "Independent Period-2 Motions to Chaos in a van der Pol–Duffing Oscillator." International Journal of Bifurcation and Chaos 30, no. 15 (December 9, 2020): 2030045. http://dx.doi.org/10.1142/s0218127420300451.

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In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations.

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22

Zhang, Tiansi, and Dianli Zhao. "Bifurcation of an Orbit Homoclinic to a Hyperbolic Saddle of a Vector Field inR4." Discrete Dynamics in Nature and Society 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/571838.

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We perform a bifurcation analysis of an orbit homoclinic to a hyperbolic saddle of a vector field inR4. We give an expression of the gap between returning points in a transverse section by renormalizing system, through which we find the existence of homoclinic-doubling bifurcation in the case1+α>β>ν. Meanwhile, after reparametrizing the parameter, a periodic-doubling bifurcation appears and may be close to a saddle-node bifurcation, if the parameter is varied. These scenarios correspond to the occurrence of chaos. Based on our analysis, bifurcation diagrams of these bifurcations are depicted.

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23

Liu, Ping, Junping Shi, and Yuwen Wang. "A double saddle-node bifurcation theorem." Communications on Pure & Applied Analysis 12, no. 6 (2013): 2923–33. http://dx.doi.org/10.3934/cpaa.2013.12.2923.

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24

Achleitner, Franz, and Peter Szmolyan. "Saddle–node bifurcation of viscous profiles." Physica D: Nonlinear Phenomena 241, no. 20 (October 2012): 1703–17. http://dx.doi.org/10.1016/j.physd.2012.06.008.

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25

Schecter, Stephen. "The Saddle-Node Separatrix-Loop Bifurcation." SIAM Journal on Mathematical Analysis 18, no. 4 (July 1987): 1142–56. http://dx.doi.org/10.1137/0518083.

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26

FABBRI, ROBERTA, RUSSELL JOHNSON, and FRANCESCA MANTELLINI. "A NONAUTONOMOUS SADDLE-NODE BIFURCATION PATTERN." Stochastics and Dynamics 04, no. 03 (September 2004): 335–50. http://dx.doi.org/10.1142/s0219493704001103.

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In this paper we study certain differential equations depending on a small parameter ε which exhibit a bifurcation of saddle-node type as ε passes through zero. We use a classical averaging technique together with methods and results from the modern theory of nonautonomous differential equations.

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27

Esteban, Marina, Enrique Ponce, and Francisco Torres. "Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics." Applied Mathematics and Nonlinear Sciences 2, no. 2 (November 4, 2017): 449–64. http://dx.doi.org/10.21042/amns.2017.2.00036.

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AbstractThis paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from a reduction of symmetric 3D systems with slow-fast dynamics. We concentrate our attention on the saddle dynamics cases, determining the existence of periodic orbits as well as their stability, and possible bifurcations. Dealing with reachable saddles not in the central hysteresis band, we show the existence of subcritical/supercritical heteroclinic bifurcations as well as saddle-node bifurcations of periodic orbits.

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28

Wang, Jinbin, and Lifeng Ma. "Bogdanov–Takens Bifurcation in a Shape Memory Alloy Oscillator with Delayed Feedback." Complexity 2020 (October 24, 2020): 1–10. http://dx.doi.org/10.1155/2020/9132501.

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This work is focused on a shape memory alloy oscillator with delayed feedback. The main attention is to investigate the Bogdanov–Takens (B-T) bifurcation by choosing feedback parameters A 1,2 and time delay τ . The conditions for the occurrence of the B-T bifurcation are derived, and the versal unfolding of the norm forms near the B-T bifurcation point is obtained by using center manifold reduction and normal form. Moreover, it is demonstrated that the system also undergoes different codimension-1 bifurcations, such as saddle-node bifurcation, Hopf bifurcation, and saddle homoclinic bifurcation. Finally, some numerical simulations are given to verify the analytic results.

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29

Salas-Cabrera, R., A. Hernandez-Colin, J. Roman-Flores, and N. Salas-Cabrera. "Bifurcation Analysis of the Wound Rotor Induction Motor." International Journal of Bifurcation and Chaos 25, no. 12 (November 2015): 1550163. http://dx.doi.org/10.1142/s0218127415501631.

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This work deals with the bifurcation phenomena that occur during the open-loop operation of a single-fed three-phase wound rotor induction motor. This paper demonstrates the occurrence of saddle-node bifurcation, hysteresis, supercritical saddle-node bifurcation, cusp and Hopf bifurcation during the individual operation of this electromechanical system. Some experimental results associated with the bifurcation phenomena are presented.

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FREIRE, E., E. PONCE, and J. ROS. "FOLLOWING A SADDLE-NODE OF PERIODIC ORBITS' BIFURCATION CURVE IN CHUA'S CIRCUIT." International Journal of Bifurcation and Chaos 19, no. 02 (February 2009): 487–95. http://dx.doi.org/10.1142/s0218127409023147.

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Starting from previous analytical results assuring the existence of a saddle-node bifurcation curve of periodic orbits for continuous piecewise linear systems, numerical continuation is done to get some primary bifurcation curves for the piecewise linear Chua's oscillator in certain dimensionless parameter plane. The primary period doubling, homoclinic and saddle-node of periodic orbits' bifurcation curves are computed. A Belyakov point is detected in organizing the connection of these curves. In the parametric region between period-doubling, focus-center-limit cycle and homoclinic bifurcation curves, chaotic attractors coexist with stable nontrivial equilibria. The primary saddle-node bifurcation curve plays a leading role in this coexistence phenomenon.

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31

ABED, E. H., H. O. WANG, J. C. ALEXANDER, A. M. A. HAMDAN, and H. C. LEE. "DYNAMIC BIFURCATIONS IN A POWER SYSTEM MODEL EXHIBITING VOLTAGE COLLAPSE." International Journal of Bifurcation and Chaos 03, no. 05 (October 1993): 1169–76. http://dx.doi.org/10.1142/s0218127493000969.

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Dynamic bifurcations, including Hopf and period-doubling bifurcations, are found to occur in a power system dynamic model recently employed in voltage collapse studies. The occurrence of dynamic bifurcations is ascertained in a region of state and parameter space linked with the onset of voltage collapse. The work focuses on a power system model studied by Dobson & Chiang [1989]. The presence of the dynamic bifurcations, and the resulting implications for dynamic behavior, necessitate a re-examination of the role of saddle node bifurcations in the voltage collapse phenomenon. The bifurcation analysis is performed using the reactive power demand at a load bus as the bifurcation parameter. It is determined that the power system model under consideration exhibits two Hopf bifurcations in the vicinity of the saddle node bifurcation. Between the Hopf bifurcations, i.e., in the "Hopf window," period-doubling bifurcations are found to occur. Simulations are given to illustrate the various types of dynamic behaviors associated with voltage collapse for the model. In particular, it is seen that an oscillatory transient may play a role in the collapse.

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32

Beloqui, J., and M. J. Pacifico. "Quasi-transversal saddle-node bifurcation on surfaces." Ergodic Theory and Dynamical Systems 10, no. 1 (March 1990): 63–88. http://dx.doi.org/10.1017/s0143385700005393.

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AbstractIn this paper we give a complete set of invariants (moduli) for mild and strong semilocal equivalence for certain two parameter families of diffeomorphisms on surfaces. These families exhibit a quasi-transversal saddle-connection between a saddle-node and a hyperbolic periodic point.

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33

XIAO, DONGMEI, and SHIGUI RUAN. "CODIMENSION TWO BIFURCATIONS IN A PREDATOR–PREY SYSTEM WITH GROUP DEFENSE." International Journal of Bifurcation and Chaos 11, no. 08 (August 2001): 2123–31. http://dx.doi.org/10.1142/s021812740100336x.

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In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.

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JIN, LIHUA, and YONGZHONG HUO. "BIFURCATION AND STABILITY OF MARTENSITIC TRANSFORMATION DYNAMICS." International Journal of Bifurcation and Chaos 18, no. 12 (December 2008): 3737–52. http://dx.doi.org/10.1142/s021812740802269x.

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In this paper, we have studied the mathematical properties and their physical implications of a system of nonlinear ordinary differential equations with two variables and six parameters, which was proposed to model the martensitic transformation of shape memory alloys. While the system could not have limit cycles, bifurcation to hysteresis was found in load and/or temperature controlled processes with large enough interfacial energies. This agrees qualitatively quite well with the experimental observation and theoretical understandings of the stress-strain-temperature hysteresis of the alloys. A three-dimensional bifurcation diagram was identified. Nonhyperbolic equilibrium points were found as saddle nodes and high order node points. The local behavior was studied and the phase portrait of the system was obtained for the load and temperature parameters. Accordingly, a stable node represents the stable martensitic or austenitic phase, a saddle stands for the unstable phase mixture, and a saddle node corresponds to the beginning or the end of the transformation. Therefore, for a thermal-mechanical loading path, the martensitic transformation process accords with the qualitative changes of the equilibrium points.

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Liu, Yongjian, Chunbiao Li, and Aimin Liu. "Analysis of Geometric Invariants for Three Types of Bifurcations in 2D Differential Systems." International Journal of Bifurcation and Chaos 31, no. 07 (June 15, 2021): 2150105. http://dx.doi.org/10.1142/s0218127421501054.

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Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

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36

Liu, Xia. "Regularization of the Boundary-Saddle-Node Bifurcation." Advances in Mathematical Physics 2018 (2018): 1–10. http://dx.doi.org/10.1155/2018/5094878.

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In this paper we treat a particular class of planar Filippov systems which consist of two smooth systems that are separated by a discontinuity boundary. In such systems one vector field undergoes a saddle-node bifurcation while the other vector field is transversal to the boundary. The boundary-saddle-node (BSN) bifurcation occurs at a critical value when the saddle-node point is located on the discontinuity boundary. We derive a local topological normal form for the BSN bifurcation and study its local dynamics by applying the classical Filippov’s convex method and a novel regularization approach. In fact, by the regularization approach a given Filippov system is approximated by a piecewise-smooth continuous system. Moreover, the regularization process produces a singular perturbation problem where the original discontinuous set becomes a center manifold. Thus, the regularization enables us to make use of the established theories for continuous systems and slow-fast systems to study the local behavior around the BSN bifurcation.

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SHIL'NIKOV, A., G. NICOLIS, and C. NICOLIS. "BIFURCATION AND PREDICTABILITY ANALYSIS OF A LOW-ORDER ATMOSPHERIC CIRCULATION MODEL." International Journal of Bifurcation and Chaos 05, no. 06 (December 1995): 1701–11. http://dx.doi.org/10.1142/s0218127495001253.

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A comprehensive bifurcation analysis of a low-order atmospheric circulation model is carried out. It is shown that the model admits a codimension-2 saddle-node-Hopf bifurcation. The principal mechanisms leading to the appearance of complex dynamics around this bifurcation are described and various routes to chaotic behavior are identified, such as the transition through the period doubling cascade, the breakdown of an invariant torus and homoclinic bifurcations of a saddle-focus. Non-trivial limit sets in the form of a chaotic attractor or a chaotic repeller are found in some parameter ranges. Their presence implies an enhanced unpredictability of the system for parameter values corresponding to the winter season.

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Berthier, F., J. P. Diard, and C. Montella. "Discontinuous impedance near a saddle-node bifurcation." Journal of Electroanalytical Chemistry 410, no. 2 (July 1996): 247–49. http://dx.doi.org/10.1016/0022-0728(96)04699-2.

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TITZ, SVEN, TILL KUHLBRODT, and ULRIKE FEUDEL. "HOMOCLINIC BIFURCATION IN AN OCEAN CIRCULATION BOX MODEL." International Journal of Bifurcation and Chaos 12, no. 04 (April 2002): 869–75. http://dx.doi.org/10.1142/s0218127402004759.

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The qualitative behavior of a conceptual ocean box model is investigated. It is a paradigmatic model of the thermohaline ocean circulation of the Atlantic. In a bifurcation study, the two occurring bifurcations, a saddle-node and a Hopf bifurcation, are computed analytically. Using normal form theory, it is shown that the latter bifurcation is always subcritical. The unstable periodic orbit emerging at the Hopf bifurcation vanishes in a homoclinic bifurcation. The results are interpreted with respect to the stability of the thermohaline circulation.

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SOLIMAN, MOHAMED S., and J. M. T. THOMPSON. "BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION." International Journal of Bifurcation and Chaos 01, no. 01 (March 1991): 107–18. http://dx.doi.org/10.1142/s0218127491000087.

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Heteroclinic and homoclinic connections of saddle cycles play an important role in basin organization. In this study, we outline how these events can lead to an indeterminate jump to resonance from a saddle-node bifurcation. Here, due to the fractal structure of the basins in the vicinity of the saddle-node, we cannot predict to which available attractor the system will jump in the presence of even infinitesimal noise.

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Nakahara*, Hiroyuki, and Kenji Doya. "Near-Saddle-Node Bifurcation Behavior as Dynamics in Working Memory for Goal-Directed Behavior." Neural Computation 10, no. 1 (January 1, 1998): 113–32. http://dx.doi.org/10.1162/089976698300017917.

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In consideration of working memory as a means for goal-directed behavior in nonstationary environments, we argue that the dynamics of working memory should satisfy two opposing demands: long-term maintenance and quick transition. These two characteristics are contradictory within the linear domain. We propose the near-saddle-node bifurcation behavior of a sigmoidal unit with a self-connection as a candidate of the dynamical mechanism that satisfies both of these demands. It is shown in evolutionary programming experiments that the near-saddle-node bifurcation behavior can be found in recurrent networks optimized for a task that requires efficient use of working memory. The result suggests that the near-saddle-node bifurcation behavior may be a functional necessity for survival in nonstationary environments.

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SOTELO HERRERA, Ma DOLORES, JESÚS SAN MARTÍN, and LUCÍA CERRADA. "SADDLE-NODE BIFURCATION CASCADES AND ASSOCIATED TRAVELING WAVES IN WEAKLY COUPLING CML." International Journal of Bifurcation and Chaos 22, no. 07 (July 2012): 1250172. http://dx.doi.org/10.1142/s0218127412501726.

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In this paper, traveling waves are analytically found in weakly coupling CML, with both global and neighbor couplings. The waves are triggered by the individual dynamics of the oscillators belonging to CML: saddle-node bifurcation cascades. Depending on the wave period different results are obtained, since the larger the period, the larger the collapse produced in the boxes of saddle-node bifurcation cascades.

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Krauskopf, Bernd, and Bart E. Oldeman. "Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation." Nonlinearity 19, no. 9 (August 16, 2006): 2149–67. http://dx.doi.org/10.1088/0951-7715/19/9/010.

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TANG, JIASHI, MINGHUA ZHAO, FENG HAN, and WENBIN FU. "SADDLE-NODE BIFURCATION AND ITS CONTROL OF BURGERS–KdV EQUATION." Modern Physics Letters B 24, no. 06 (March 10, 2010): 567–74. http://dx.doi.org/10.1142/s0217984910022627.

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The Burgers–Korteweg-de Vries (KdV) equation had been used as nonlinear modes for acoustic shock waves in dusty plasmas and so on. The variable transformation and the Jacobi elliptic function method was introduced to find the exact solution. In this paper, we will research into the saddle-node bifurcation and its control of the forced Burgers–KdV. By the transformation, PDEs are reduced to ODEs. Analyzing the frequency response function and its unstable region of the trivial steady state, we know that the saddle-node bifurcation which leads to jump and hysteresis may appear in the resonance response. Controllers for bifurcation modification purpose are designed in order to remove or delay the occurrence of jump and hysteresis phenomena. By means of numerical simulations we compare the uncontrolled system with the controlled system and clarify that controllers are adequate for the saddle-node bifurcation control of the forced Burgers–KdV equation.

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CHEN, SHYAN-SHIOU, CHANG-YUAN CHENG, and YI-RU LIN. "APPLICATION OF A TWO-DIMENSIONAL HINDMARSH–ROSE TYPE MODEL FOR BIFURCATION ANALYSIS." International Journal of Bifurcation and Chaos 23, no. 03 (March 2013): 1350055. http://dx.doi.org/10.1142/s0218127413500557.

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In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh–Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: (1) the number and stability of the equilibria are analyzed in detail using a table to demonstrate the matter in which the stability of the equilibrium changes and to determine which two equilibria collapse through the saddle-node bifurcation; (2) the sufficient conditions for an Andronov–Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and (3) we elaborately evaluate the sufficient conditions for the Bogdanov–Takens (BT) and Bautin bifurcations. Several numerical simulations for these conditions are also presented. In particular, two types of bistable behaviors are numerically demonstrated: the BT and Bautin bifurcations. Notably, all of the bifurcation curves in the domain of the remaining parameters are similar when the time scale is large. Additionally, to show the potential for a limit cycle, the existence of a trapping region is demonstrated. These results present a variety of diverse behaviors for this model. The results of this study will be helpful in assessing suitable parameters for fitting the resemblances of experimental observations.

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Li, Xiaodong, Weipeng Zhang, Fengjie Geng, and Jicai Huang. "The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/152518.

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The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

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Szemplińska-Stupnicka, Wanda, and Krzysztof L. Janicki. "Basin Boundary Bifurcations and Boundary Crisis in the Twin-Well Duffing Oscillator: Scenarios Related to the Saddle of the Large Resonant Orbit." International Journal of Bifurcation and Chaos 07, no. 01 (January 1997): 129–46. http://dx.doi.org/10.1142/s0218127497000091.

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Bifurcation phenomena in the twin-well Duffing system are considered for such regions of the system parameters, where the saddle DL associated with the two-well T-periodic attractor (large orbit) undergoes homoclinic bifurcation. In particular, a co-dimension-two bifurcation, the bifurcation defined by the intersection of the homoclinic bifurcation of the large orbit with the saddle-node bifurcation of the nonresonant single-well orbit in the system parameter plane is the main point of interest. In the four subdomains around the co-dimension-two point, the manifold structure of the saddle DL and the basins of attractions are studied numerically. The analysis reveals new phenomena and new features of the system behavior. A complex bifurcational structure is observed that includes boundary crisis of the cross-well chaotic attractor, intermittency (subduction) phenomena, and explosion of the basin of attraction of the large orbit. The analysis also explains why, in one of the subdomains, the two-well T-periodic attractor (large orbit) becomes the unique attractor of the system.

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Tang, Sanyi, Guangyao Tang, and Wenjie Qin. "Codimension-1 Sliding Bifurcations of a Filippov Pest Growth Model with Threshold Policy." International Journal of Bifurcation and Chaos 24, no. 10 (October 2014): 1450122. http://dx.doi.org/10.1142/s0218127414501223.

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A Filippov system is proposed to describe the stage structured nonsmooth pest growth with threshold policy control (TPC). The TPC measure is represented by the total density of both juveniles and adults being chosen as an index for decisions on when to implement chemical control strategies. The proposed Filippov system can have three pieces of sliding segments and three pseudo-equilibria, which result in rich sliding mode bifurcations and local sliding bifurcations including boundary node (boundary focus, or boundary saddle) and tangency bifurcations. As the threshold density varies the model exhibits the interesting global sliding bifurcations sequentially: touching → buckling → crossing → sliding homoclinic orbit to a pseudo-saddle → crossing → touching bifurcations. In particular, bifurcation of a homoclinic orbit to a pseudo-saddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been observed for some parameter sets. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy. One more sliding segment (or pseudo-equilibrium) is induced by the total density of a population guided switching policy, compared to only the juvenile density guided policy, implying that this control policy is more effective in terms of preventing multiple pest outbreaks or causing the density of pests to stabilize at a desired level such as an economic threshold.

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DE, SOMA, PARTHA SHARATHI DUTTA, SOUMITRO BANERJEE, and AKHIL RANJAN ROY. "LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS." International Journal of Bifurcation and Chaos 21, no. 06 (June 2011): 1617–36. http://dx.doi.org/10.1142/s0218127411029318.

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In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

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MAYOL, CATALINA, MARIO A. NATIELLO, and MARTÍN G. ZIMMERMANN. "RESONANCE STRUCTURE IN A WEAKLY DETUNED LASER WITH INJECTED SIGNAL." International Journal of Bifurcation and Chaos 11, no. 10 (October 2001): 2587–605. http://dx.doi.org/10.1142/s0218127401003693.

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We describe the qualitative dynamics and bifurcation set for a laser with injected signal for small cavity detunings. The main organizing center is the Hopf-saddle-node bifurcation from where a secondary Hopf bifurcation of a periodic orbit originates. We show that the laser's stable cw solution existing for low injections, also suffers a secondary Hopf bifurcation. The resonance structure of both tori interact, and homoclinic orbits to the "off" state are found inside each Arnold tongue. The accumulation of all the above resonances towards the Hopf-saddle-node singularity points to the occurrence of a highly degenerate global bifurcation at the codimension-2 point.

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