Pub Date : 2024-07-20DOI: 10.1142/s0218127424501220
Qigui Yang, Pengxian Zhu
This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator.
{"title":"Li–Yorke Chaos in Linear Systems with Weak Topology on Hilbert Spaces","authors":"Qigui Yang, Pengxian Zhu","doi":"10.1142/s0218127424501220","DOIUrl":"https://doi.org/10.1142/s0218127424501220","url":null,"abstract":"This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"108 30","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141820483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-20DOI: 10.1142/s021812742450127x
Shengli Chen, Zhiqiang Wu
In general, dynamic systems of higher dimensions or with more complex nonlinearities exhibit more intricate behaviors. Conversely, it is worthwhile to discuss whether a complex phenomenon persists in simpler systems. This paper investigates a single-degree-of-freedom vibration system with a velocity-dependent stiffness affected by additive noise. Although the underlying deterministic system possesses only one stable equilibrium point, under noise actions, it has the potential for a stochastic P-bifurcation to occur. This bifurcation causes the central peak of the joint probability density function to split into two symmetric peaks. At this stage, the behavior of the system resembles the development of two phantom attractors that deviate from the equilibrium point, causing the system’s random states to linger around them for extended periods. The effects of the damping ratio and noise intensity on the phantom attractors are discussed, together with the critical parameter curve associated with the onset of phantom attractors. Moreover, the generation mechanism of phantom attractors is disclosed by investigating the phase trajectories of the underlying conservative system. The distribution law of those critical parameter values is also proven by the stochastic averaging method, which is associated with the most probable amplitude. This study highlights that phantom attractors can manifest in dynamic systems even in the absence of Hopf bifurcation.
一般来说，维度越高或非线性越复杂的动态系统表现出的行为越复杂。反之，在较简单的系统中是否存在复杂现象也值得讨论。本文研究了一个单自由度振动系统，该系统的刚度随速度变化，并受到加性噪声的影响。虽然基本的确定性系统只有一个稳定的平衡点，但在噪声作用下，它有可能发生随机 P 型分岔。这种分岔会导致联合概率密度函数的中心峰分裂成两个对称峰。在这一阶段，系统的行为类似于发展出两个偏离平衡点的幽灵吸引子，导致系统的随机状态在其周围长时间徘徊。本文讨论了阻尼比和噪声强度对幻影吸引子的影响，以及与幻影吸引子出现相关的临界参数曲线。此外，通过研究底层保守系统的相位轨迹，揭示了幻影吸引子的产生机制。还通过随机平均法证明了这些临界参数值的分布规律，并与最可能的振幅相关联。这项研究强调，即使在没有霍普夫分岔的情况下，动态系统中也会出现幻影吸引子。
{"title":"Abnormal Probability Distribution in a Single-Degree-of-Freedom Smooth System with Velocity-Dependent Stiffness","authors":"Shengli Chen, Zhiqiang Wu","doi":"10.1142/s021812742450127x","DOIUrl":"https://doi.org/10.1142/s021812742450127x","url":null,"abstract":"In general, dynamic systems of higher dimensions or with more complex nonlinearities exhibit more intricate behaviors. Conversely, it is worthwhile to discuss whether a complex phenomenon persists in simpler systems. This paper investigates a single-degree-of-freedom vibration system with a velocity-dependent stiffness affected by additive noise. Although the underlying deterministic system possesses only one stable equilibrium point, under noise actions, it has the potential for a stochastic P-bifurcation to occur. This bifurcation causes the central peak of the joint probability density function to split into two symmetric peaks. At this stage, the behavior of the system resembles the development of two phantom attractors that deviate from the equilibrium point, causing the system’s random states to linger around them for extended periods. The effects of the damping ratio and noise intensity on the phantom attractors are discussed, together with the critical parameter curve associated with the onset of phantom attractors. Moreover, the generation mechanism of phantom attractors is disclosed by investigating the phase trajectories of the underlying conservative system. The distribution law of those critical parameter values is also proven by the stochastic averaging method, which is associated with the most probable amplitude. This study highlights that phantom attractors can manifest in dynamic systems even in the absence of Hopf bifurcation.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"111 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141820434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-20DOI: 10.1142/s0218127424501268
C. Lăzureanu
In this paper, using smooth invertible variable transformations and smooth invertible parameter changes, we construct a jerk normal form for the cusp bifurcation of a jerk system with two parameters which displays a nondegenerate fold bifurcation.
{"title":"The Cusp Bifurcation of a Jerk System","authors":"C. Lăzureanu","doi":"10.1142/s0218127424501268","DOIUrl":"https://doi.org/10.1142/s0218127424501268","url":null,"abstract":"In this paper, using smooth invertible variable transformations and smooth invertible parameter changes, we construct a jerk normal form for the cusp bifurcation of a jerk system with two parameters which displays a nondegenerate fold bifurcation.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"107 15","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141820565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-19DOI: 10.1142/s021812742450130x
M. Katsanikas, Stephen Wiggins
In this paper, we extend the notion of periodic orbit-dividing surfaces (PODS) to rotating Hamiltonian systems with two degrees of freedom. First, we present a method that enables us to apply the classical algorithm for the construction of PODS [Pechukas & McLafferty, 1973; Pechukas, 1981; Pollak & Pechukas, 1978; Pollak, 1985] in rotating Hamiltonian systems with two degrees of freedom. Then we study the structure of these surfaces in a rotating quadratic normal-form Hamiltonian system with two degrees of freedom.
{"title":"Periodic Orbit-Dividing Surfaces in Rotating Hamiltonian Systems with Two Degrees of Freedom","authors":"M. Katsanikas, Stephen Wiggins","doi":"10.1142/s021812742450130x","DOIUrl":"https://doi.org/10.1142/s021812742450130x","url":null,"abstract":"In this paper, we extend the notion of periodic orbit-dividing surfaces (PODS) to rotating Hamiltonian systems with two degrees of freedom. First, we present a method that enables us to apply the classical algorithm for the construction of PODS [Pechukas & McLafferty, 1973; Pechukas, 1981; Pollak & Pechukas, 1978; Pollak, 1985] in rotating Hamiltonian systems with two degrees of freedom. Then we study the structure of these surfaces in a rotating quadratic normal-form Hamiltonian system with two degrees of freedom.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"114 48","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141821282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-19DOI: 10.1142/s0218127424501311
F. Montoya, M. Katsanikas, Stephen Wiggins
In prior studies [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored specifically for Hamiltonian systems with three or more degrees of freedom. These approaches, as described in the aforementioned papers, were applied to a quadratic Hamiltonian system in its normal form with three degrees of freedom. Within this framework, we provide a more intricate geometric characterization of this entity within the family of 4D toratopes which elucidates the structure of the dividing surfaces discussed in these works. Our analysis affirmed the nature of this construction as a dividing surface with the property of no-recrossing. These insights were derived from analytical findings tailored to the Hamiltonian system discussed in these publications. In this series of papers, we extend our previous findings to quartic Hamiltonian systems with three degrees of freedom. We establish the no-recrossing property of the PODS for this class of Hamiltonian systems and explore their structural aspects. Additionally, we undertake the computation and examination of the PODS in a coupled scenario of quartic Hamiltonian systems with three degrees of freedom. In the initial paper [Gonzalez Montoya et al., 2024], we employed the first methodology for constructing PODS, while in this paper, we utilize the second methodology for the same purpose.
{"title":"Periodic Orbit Dividing Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II","authors":"F. Montoya, M. Katsanikas, Stephen Wiggins","doi":"10.1142/s0218127424501311","DOIUrl":"https://doi.org/10.1142/s0218127424501311","url":null,"abstract":"In prior studies [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored specifically for Hamiltonian systems with three or more degrees of freedom. These approaches, as described in the aforementioned papers, were applied to a quadratic Hamiltonian system in its normal form with three degrees of freedom. Within this framework, we provide a more intricate geometric characterization of this entity within the family of 4D toratopes which elucidates the structure of the dividing surfaces discussed in these works. Our analysis affirmed the nature of this construction as a dividing surface with the property of no-recrossing. These insights were derived from analytical findings tailored to the Hamiltonian system discussed in these publications. In this series of papers, we extend our previous findings to quartic Hamiltonian systems with three degrees of freedom. We establish the no-recrossing property of the PODS for this class of Hamiltonian systems and explore their structural aspects. Additionally, we undertake the computation and examination of the PODS in a coupled scenario of quartic Hamiltonian systems with three degrees of freedom. In the initial paper [Gonzalez Montoya et al., 2024], we employed the first methodology for constructing PODS, while in this paper, we utilize the second methodology for the same purpose.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"113 35","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141821417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-19DOI: 10.1142/s0218127424501293
Zhongben Gong, Jin Zhou, Meng Huang
Swarmalator model attracts the attention of numerous scholars because of its rich dynamical behaviors. We investigate the reason for the phase transition that occurs in a 2D swarmalator model, and conclude that the limit cycle, a reflection of the coexistence of the forces of maintaining and disrupting orders, results in the various clustering phenomena. Through novel mean-field approximation and using the self-consistency argument method, we prove the clustering conditions and the influence of the number of clusters on the existence of clusters, and provide estimates of the cluster size of the splintered phase wave state and the phase transition threshold between the splintered phase wave state and active phase wave state. Due to the widespread presence of nonlinearity, our study is essential to the analysis of clustering phenomena in real physical models.
{"title":"Approximating the Splitting Point of the Swarmalator Model","authors":"Zhongben Gong, Jin Zhou, Meng Huang","doi":"10.1142/s0218127424501293","DOIUrl":"https://doi.org/10.1142/s0218127424501293","url":null,"abstract":"Swarmalator model attracts the attention of numerous scholars because of its rich dynamical behaviors. We investigate the reason for the phase transition that occurs in a 2D swarmalator model, and conclude that the limit cycle, a reflection of the coexistence of the forces of maintaining and disrupting orders, results in the various clustering phenomena. Through novel mean-field approximation and using the self-consistency argument method, we prove the clustering conditions and the influence of the number of clusters on the existence of clusters, and provide estimates of the cluster size of the splintered phase wave state and the phase transition threshold between the splintered phase wave state and active phase wave state. Due to the widespread presence of nonlinearity, our study is essential to the analysis of clustering phenomena in real physical models.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" March","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141823630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-17DOI: 10.1142/s0218127424501244
Qian Xiang, Yunzhu Shen, Shuangshuang Peng, Mengqiang Liu
In this paper, we present a novel two-dimensional discrete memristor map that is based on a discrete memristor model and a sine–arcsine one-dimensional map. First, an analysis is conducted on the memristor model to understand its characteristics. Then, the model is coupled with the sine–arcsine one-dimensional map to achieve the two-dimensional discrete memristor map. Our investigation reveals the presence of coexisting attractors and hyperchaotic attractors as the bifurcation parameters vary. Numerical simulations show that the discrete memristors effectively enhance the complexity of chaos in the sine–arcsine map. Furthermore, a digital circuit is designed to experimentally verify the new chaotic system. The research results can enrich the theoretical analysis and circuit implementation of chaos.
{"title":"A Two-Dimensional Discrete Memristor Map: Analysis and Implementation","authors":"Qian Xiang, Yunzhu Shen, Shuangshuang Peng, Mengqiang Liu","doi":"10.1142/s0218127424501244","DOIUrl":"https://doi.org/10.1142/s0218127424501244","url":null,"abstract":"In this paper, we present a novel two-dimensional discrete memristor map that is based on a discrete memristor model and a sine–arcsine one-dimensional map. First, an analysis is conducted on the memristor model to understand its characteristics. Then, the model is coupled with the sine–arcsine one-dimensional map to achieve the two-dimensional discrete memristor map. Our investigation reveals the presence of coexisting attractors and hyperchaotic attractors as the bifurcation parameters vary. Numerical simulations show that the discrete memristors effectively enhance the complexity of chaos in the sine–arcsine map. Furthermore, a digital circuit is designed to experimentally verify the new chaotic system. The research results can enrich the theoretical analysis and circuit implementation of chaos.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 40","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141831212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-17DOI: 10.1142/s0218127424300209
Antonio Palacios, Jacinto Tamez, Mani Amani, V. In
Over the past years, we have exploited the bistability features that are commonly found in many individual sensors to develop a network-based [Acebrón et al., 2003; Bulsara et al., 2004; In et al., 2003a; In et al., 2003b; In et al., 2005; In et al., 2006; In et al., 2012; Palacios et al., 2005] approach to modeling, designing, and fabricating extremely sensitive magnetic- and electric-field sensors capable of resolving field changes as low as 150[Formula: see text]pT and 100[Formula: see text]fAmps, respectively. Higher sensitivity is achieved by exploiting the symmetry of the network to create infinite-period bifurcations that render the ensuing oscillations highly sensitive to symmetry-breaking effects from external signals. In this paper, we study the effects of noise on the response of a network-based electric-field sensor as well as the effects of parameter mismatch, which appear naturally due to material imperfections and noise. The results show that Signal-to-Noise Ratio (SNR) increases sharply near the onset of the infinite-period bifurcation, and they increase further as the coupling strength in the network increases while passing the threshold that leads to oscillatory behavior. Overall, the SNR indicates that the negative effects of highly contaminated signals are well-mitigated by the sensitivity response of the system. In addition, computer simulations show the network-based system to be robust enough to mismatches in system parameters, while the deviations from the nominal parameter values form regions where the oscillations persist. Noise has a smoothing effect over the boundaries of these regions.
{"title":"Noise and Parameter Mismatch in a Ring Network for Sensing Extremely Low Electric Fields","authors":"Antonio Palacios, Jacinto Tamez, Mani Amani, V. In","doi":"10.1142/s0218127424300209","DOIUrl":"https://doi.org/10.1142/s0218127424300209","url":null,"abstract":"Over the past years, we have exploited the bistability features that are commonly found in many individual sensors to develop a network-based [Acebrón et al., 2003; Bulsara et al., 2004; In et al., 2003a; In et al., 2003b; In et al., 2005; In et al., 2006; In et al., 2012; Palacios et al., 2005] approach to modeling, designing, and fabricating extremely sensitive magnetic- and electric-field sensors capable of resolving field changes as low as 150[Formula: see text]pT and 100[Formula: see text]fAmps, respectively. Higher sensitivity is achieved by exploiting the symmetry of the network to create infinite-period bifurcations that render the ensuing oscillations highly sensitive to symmetry-breaking effects from external signals. In this paper, we study the effects of noise on the response of a network-based electric-field sensor as well as the effects of parameter mismatch, which appear naturally due to material imperfections and noise. The results show that Signal-to-Noise Ratio (SNR) increases sharply near the onset of the infinite-period bifurcation, and they increase further as the coupling strength in the network increases while passing the threshold that leads to oscillatory behavior. Overall, the SNR indicates that the negative effects of highly contaminated signals are well-mitigated by the sensitivity response of the system. In addition, computer simulations show the network-based system to be robust enough to mismatches in system parameters, while the deviations from the nominal parameter values form regions where the oscillations persist. Noise has a smoothing effect over the boundaries of these regions.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"192 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141828762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-17DOI: 10.1142/s0218127424501153
Demou Luo, Yizhi Qiu
In this study, we discuss the global dynamics of the Holling-II amensalism model for a strong Allee effect of harmful species. We discuss the existence and stabilization of the extinction equilibria, exclusion equilibria, coexistence equilibria, and infinite singularities by analyzing the presence and stabilization of the system characteristics in terms of the possibilities and correspondences in the model when the death rate of the injured species is used as a threshold value. Also, we find that the two equilibrium points in the first quadrant are effective in proving that the model does not have globally stabilizing features and obtain two critical conditions and their corresponding global phase diagrams. Finally, we explore the weak Allee effect of the victim species, and using the analysis from numerical simulations, we recapitulate the analysis and dynamics of the model in equilibrium.
{"title":"Global Dynamics of a Holling-II Amensalism System with a Strong Allee Effect on the Harmful Species","authors":"Demou Luo, Yizhi Qiu","doi":"10.1142/s0218127424501153","DOIUrl":"https://doi.org/10.1142/s0218127424501153","url":null,"abstract":"In this study, we discuss the global dynamics of the Holling-II amensalism model for a strong Allee effect of harmful species. We discuss the existence and stabilization of the extinction equilibria, exclusion equilibria, coexistence equilibria, and infinite singularities by analyzing the presence and stabilization of the system characteristics in terms of the possibilities and correspondences in the model when the death rate of the injured species is used as a threshold value. Also, we find that the two equilibrium points in the first quadrant are effective in proving that the model does not have globally stabilizing features and obtain two critical conditions and their corresponding global phase diagrams. Finally, we explore the weak Allee effect of the victim species, and using the analysis from numerical simulations, we recapitulate the analysis and dynamics of the model in equilibrium.","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":" 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141828957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-11DOI: 10.1142/s0218127424501128
Jean-Pierre Françoise, Daniele Fournier-Prunaret
We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map [Formula: see text]. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map [Formula: see text] and of its inverse [Formula: see text]. We show that the map [Formula: see text] is the product of two involutions. The nature of the fixed points of [Formula: see text] is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping [Formula: see text] is integrable for [Formula: see text] and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for [Formula: see text].
{"title":"Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets","authors":"Jean-Pierre Françoise, Daniele Fournier-Prunaret","doi":"10.1142/s0218127424501128","DOIUrl":"https://doi.org/10.1142/s0218127424501128","url":null,"abstract":"We revisit the Kahan–Hirota–Kimura discretization of a quadratic vector field. The corresponding discrete system is generated by successive iterations of a birational map [Formula: see text]. We include a proof of a formula for the Jacobian of this map. In the following, we essentially focus on the case of the Lotka–Volterra system. We discuss the notion of focal points and prefocal lines of the map [Formula: see text] and of its inverse [Formula: see text]. We show that the map [Formula: see text] is the product of two involutions. The nature of the fixed points of [Formula: see text] is studied. We introduce the notion of asymptotic focal and prefocal sets. We further provide a new proof of the theorem of Sanz-Serna. We show that the mapping [Formula: see text] is integrable for [Formula: see text] and that it preserves a pencil of conics (generic hyperbolas). To conclude, we provide several numerical simulations for [Formula: see text].","PeriodicalId":506426,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":"36 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141658552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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