International Journal of Bifurcation and Chaos最新文献

{"title":"Spatial Dynamics of a Competitive and Cooperative Model with Multiple Delay Effects: Turing Patterns and Hopf Bifurcation","authors":"Yu Mu, Wing-Cheong Lo","doi":"10.1142/s0218127424500627","DOIUrl":"https://doi.org/10.1142/s0218127424500627","url":null,"abstract":"<p>Competing populations within an ecosystem often release chemicals during the interactions and diffusion processes. These chemicals can have diverse effects on competitors, ranging from inhibition to stimulation of species’ growth. This work constructs a competition model that incorporates stimulatory substances, spatial effects, and multiple time lags to investigate the combined impact of these phenomena on competitors’ growth. When the stimulation rate from the produced chemicals falls within a suitable threshold interval, all species within the system can coexist. Under the species’ coexistence, their diffusive phenomenon leads to a spatially heterogeneous distribution, resulting in patchy structures (Turing patterns) within their habitat. As the parameter values exceed their thresholds, species begin to exhibit spatially periodic oscillations (spatial Hopf bifurcation). The presence of multiple delays and competitors’ diffusion contributes to spatially complex and heterogeneous behaviors (Turing–Hopf bifurcation). The results help us understand the underlying mechanisms behind these heterogeneous behaviors and enable us to mitigate their negative impact on species’ growth and harvest. Numerical simulations are used to measure the dynamics of competitors under different parameter conditions.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exploring Iterated Implicit Function Systems: Existence and Properties of Attractors","authors":"Zhong Dai, Shutang Liu","doi":"10.1142/s0218127424500597","DOIUrl":"https://doi.org/10.1142/s0218127424500597","url":null,"abstract":"<p>This paper investigates a type of iterated implicit function systems composed of equations <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>c</mi></math></span><span></span>, where <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is a continuous function, and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi></math></span><span></span> is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>c</mi></math></span><span></span> containing the implicit function or being <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>α</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span>-contractive about <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>y</mi></math></span><span></span>. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computation of Normal Form and Unfolding of Codimension-3 Zero-Hopf–Hopf Bifurcation","authors":"Xin Xu, Xiaofang Zhang, Qinsheng Bi","doi":"10.1142/s0218127424500639","DOIUrl":"https://doi.org/10.1142/s0218127424500639","url":null,"abstract":"<p>The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language <i>Maple</i> is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Emergence Behavior Versus Physical-Like Behavior","authors":"Xiaobo Hou, Wanshan Lin, Xueting Tian, Xutong Zhao","doi":"10.1142/s0218127424500548","DOIUrl":"https://doi.org/10.1142/s0218127424500548","url":null,"abstract":"<p>In this paper, we study the dynamical complexity of points with emergence behavior but without weak face behavior, especially for points without physical-like behavior in certain dynamical systems such as transitive Anosov systems. We use the tools of saturated sets to prove that these points show strong dynamical complexity in the sense of entropy, density and distributional chaos. We obtain some observations of those results related to irregular sets and level sets. These results strengthen the previous results of [Catsigeras <i>et al</i>., 2019; Hou <i>et al</i>., 2023].</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spatiotemporal Dynamics of a General Two-Species System with Taxis Term","authors":"Wenjie Zuo, Yongli Song","doi":"10.1142/s021812742450055x","DOIUrl":"https://doi.org/10.1142/s021812742450055x","url":null,"abstract":"<p>In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Periodic Orbit Dividing Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – I","authors":"Francisco Gonzalez Montoya, Matthaios Katsanikas, Stephen Wiggins","doi":"10.1142/s0218127424300118","DOIUrl":"https://doi.org/10.1142/s0218127424300118","url":null,"abstract":"<p>In prior work [Katsanikas &amp; Wiggins, 2021a, 2021b, 2023c, 2023d], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored for Hamiltonian systems possessing three or more degrees of freedom. The initial approach, outlined in [Katsanikas &amp; Wiggins, 2021a, 2023c], was applied to a quadratic Hamiltonian system in normal form having three degrees of freedom. Within this context, we provided a more intricate geometric characterization of this object within the family of 4D toratopes that describe the structure of the dividing surfaces discussed in these papers. Our analysis confirmed the nature of this construction as a dividing surface with the no-recrossing property. All these findings were derived from analytical results specific to the case of the Hamiltonian system discussed in these papers. In this paper, we extend our results for quartic Hamiltonian systems with three degrees of freedom. We prove for this class of Hamiltonian systems the no-recrossing property of the PODS and we investigate the structure of these surfaces. In addition, we compute and study the PODS in a coupled case of quartic Hamiltonian systems with three degrees of freedom.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Crossing Limit Cycles in Planar Piecewise Linear Systems Separated by a Nonregular Line with Node–Node Type Critical Points","authors":"Liping Sun, Zhengdong Du","doi":"10.1142/s0218127424500494","DOIUrl":"https://doi.org/10.1142/s0218127424500494","url":null,"abstract":"<p>In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy=\"false\">)</mo></math></span><span></span>, which are the positive <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span>- and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>y</mi></math></span><span></span>-axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Controlled Quasi-Latitudinal Solutions for Ultra-Fast Spin-Torque Magnetization Switching","authors":"Alessandro Fortunati, Massimiliano d’Aquino, Claudio Serpico","doi":"10.1142/s0218127424500561","DOIUrl":"https://doi.org/10.1142/s0218127424500561","url":null,"abstract":"<p>The aim of this paper is to present a novel class of time-dependent controls to realize ultra-fast magnetization switching in nanomagnets driven by spin-torques produced by spin-polarized electric currents. Magnetization dynamics in such complex systems is governed by the Landau–Lifshitz–Slonczewski equation which describes the precessional motion of (dimensionless) magnetization vector on the unit-sphere. The relevant case of nanoparticles with uniaxial anisotropy having in-plane easy and intermediate axes as well as out-of-plane hard axis is considered. By exploiting the characteristic smallness of damping and spin-torque intensity, the complexity of the magnetic system’s dynamic is dealt with by employing tools borrowed from Hamiltonian Perturbation Theory. More precisely, the aforementioned controls are constructed via suitable perturbative tools in a way to realize approximate <i>latitudinal solutions</i> (i.e. motions on a sphere in which the out-of-plane magnetization component stays constant) with the effect to fast “switch” the system from one stationary state to another. The possibility to keep a (“small”) bounded value of the out-of-plane coordinate throughout this process of “transfer” turns out to be advantageous in the applications as it sensibly reduces the post-switching relaxation oscillations that may cause the failure of switching in real samples. Further relevant quantitative results on the behavior of the solutions during the pre- and post-switching stages (termed “expulsion” and “attraction”, respectively) are given as a by-product. A selection of validating numerical experiments is presented alongside the corresponding theoretical results.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiobjective Optimization of Chaotic Image Encryption Based on ABC Algorithm and DNA Coding","authors":"Jinwei Yu, Wei Xie, Langwen Zhang","doi":"10.1142/s0218127424500573","DOIUrl":"https://doi.org/10.1142/s0218127424500573","url":null,"abstract":"<p>As digital communication and storage continue to expand, the protection of image privacy information becomes increasingly critical. To safeguard sensitive visual information from unauthorized access, this paper proposes a novel image encryption scheme that integrates multiobjective Artificial Bee Colony (ABC) optimization algorithm and DNA coding. Multiple evaluation metrics including correlation relationship, Number of Pixel Change Rate (NPCR), Unified Average Changing Intensity (UACI), and information entropy are collaboratively optimized by the ABC algorithm. The proposed method begins with the application of the SHA-256 algorithm to generate keys and random sequences using chaotic systems. These sequences are then employed for shuffling, DNA coding, decoding, and diffusion, generating initial encrypted images. Subsequently, the encrypted images serve as individuals within the ABC algorithm to determine optimal parameters of the chaotic systems and the best ciphertext image. Simulation experiments demonstrate that the ciphertext images achieved excellent results in information entropy, pixel correlation coefficient, NPCR, and UACI. The integration of the multiobjective ABC optimization algorithm with DNA coding in our proposed image encryption scheme results in heightened security, as evidenced by superior performance in various metrics.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Attractor Merging and Amplitude Control of Hyperchaos in a Self-Reproducing Memristive Map","authors":"Yongxin Li, Chunbiao Li, Qing Zhong, Tengfei Lei, Sicong Liu","doi":"10.1142/s0218127424500500","DOIUrl":"https://doi.org/10.1142/s0218127424500500","url":null,"abstract":"<p>Memristor-type feedback provides a unique passage for chaos produced with easy control. In this work, a novel memristive map with amplitude control and coexisting hyperchaotic attractors is designed, in which two nonbifurcation parameters are extracted for partial amplitude control crossing the origin and total amplitude control. Attractor splitting and attractor merging are captured, which lead to five regimes of attractor self-reproducing. In this case, double-cavity attractors extend in the direction of 0<span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow></mrow><mrow><mo stretchy=\"false\">∘</mo></mrow></msup></math></span><span></span>, 45<span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow></mrow><mrow><mo stretchy=\"false\">∘</mo></mrow></msup></math></span><span></span>, 90<span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow></mrow><mrow><mo stretchy=\"false\">∘</mo></mrow></msup></math></span><span></span> in negative <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span>-axis, 90<span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow></mrow><mrow><mo stretchy=\"false\">∘</mo></mrow></msup></math></span><span></span> in positive <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span>-axis, 135<span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow></mrow><mrow><mo stretchy=\"false\">∘</mo></mrow></msup></math></span><span></span> in <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span>-axis forming a specific wall of self-reproduction. Furthermore, the FPGA-based hardware implementation is carried out showing consistent results with numerical simulation. Finally, a high-security chaotic encryption scheme is proposed for the orthogonal frequency division multiplexing transmission system, where the power division multiplexing technique and two-dimensional region joint encryption are effectively utilized. The security of the transmitted information is improved by the various coexisting attractors and nonbifurcation amplitude controllers.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}