International Journal of Bifurcation and Chaos最新文献

{"title":"Design of Higher-Dimensional Switching Chaos Generators by Constructing a Closed Hyper-Polyhedron","authors":"Changchun Sun","doi":"10.1142/s0218127424501372","DOIUrl":"https://doi.org/10.1142/s0218127424501372","url":null,"abstract":"A novel and unified design approach on higher-dimensional switching chaos generators is derived in this paper. The whole [Formula: see text]-dimensional linear space is divided into two parts by a closed hyper-polyhedron. Two higher-dimensional linear systems with the simplest structures as switching chaos generators are designed successfully to generate chaos. State matrix of the first linear system is Hurwitz stable. State matrix of the second linear system is not Hurwitz stable. Chaotic dynamical behaviors take place due to switching two systems. The switching trajectories go through the boundary of the closed hyper-polyhedron endlessly. Moreover, the size of the hyper-polyhedron can determine and control the amplitude of the chaotic signals. Specific numerical examples on four-dimensional, five-dimensional and six-dimensional switching chaos generators are employed, respectively, to illustrate the effectiveness of the novel and advanced approach presented in this paper. The proposed approach can also be applied to designing other switching chaos generators with the higher dimension beyond six.","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141924690","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":"How Does the Fractional Derivative Change the Complexity of the Caputo Standard Fractional Map","authors":"Ugne Orinaite, Inga Telksniene, Tadas Telksnys, Minvydas Ragulskis","doi":"10.1142/s0218127424500858","DOIUrl":"https://doi.org/10.1142/s0218127424500858","url":null,"abstract":"<p>The impact of power-law memory on the dynamics of the Caputo standard fractional map is investigated in this paper. The definition of a complexity measure for the Caputo standard fractional map is introduced. This measure evaluates both the average algebraic complexity of a trajectory and the distribution of the trajectories of different types in the phase space of the system. The interplay between the small-scale spatial chaos and the large-scale spatial behavior is observed and measured during the transition of the Caputo standard fractional map from the circle map to the classical standard map. It is demonstrated that the impact of the fractional derivative on the complexity of the fractional system is not straightforward and is predetermined by the physical properties governing the dynamics of that system.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259786","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":"Slow–Fast Dynamics of a Piecewise-Smooth Leslie–Gower Model with Holling Type-I Functional Response and Weak Allee Effect","authors":"Xiao Wu, Feng Xie","doi":"10.1142/s021812742450086x","DOIUrl":"https://doi.org/10.1142/s021812742450086x","url":null,"abstract":"<p>The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator’s food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254632","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":"Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map","authors":"Àngel Jorba, Joan Carles Tatjer, Yuan Zhang","doi":"10.1142/s0218127424500846","DOIUrl":"https://doi.org/10.1142/s0218127424500846","url":null,"abstract":"<p>We study a family of one-dimensional quasi-periodically forced maps <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>F</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>𝜃</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>𝜃</mi><mo stretchy=\"false\">)</mo><mo>,</mo><mi>𝜃</mi><mo stretchy=\"false\">+</mo><mi>ω</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, where <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span> is real, <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜃</mi></math></span><span></span> is an angle, and <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span> is an irrational frequency, such that <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>𝜃</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is a real piecewise-linear map with respect to <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span> of certain kind. The family depends on two real parameters, <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>&gt;</mo><mn>0</mn></math></span><span></span> and <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>&gt;</mo><mn>0</mn></math></span><span></span>. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>&lt;</mo><mn>1</mn></math></span><span></span> and any <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>,</mo></math></span><span></span> there is only one continuous invariant curve. For <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi><mo>&gt;</mo><mn>1</mn><mo>,</mo></math></span><span></span> there exists a smooth map <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo></math></span><span></span> such that: (a) For <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>&lt;</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>f</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span><span></span> has two continuous","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153496","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":"Global Stabilization of a Bounded Controlled Lorenz System","authors":"Héctor Martínez Pérez, Julio Solís-Daun","doi":"10.1142/s0218127424500895","DOIUrl":"https://doi.org/10.1142/s0218127424500895","url":null,"abstract":"<p>In this work, we present a method for the <i>Global Asymptotic Stabilization</i> (GAS) of an affine control chaotic Lorenz system, via <i>admissible</i> (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the <i>control Lyapunov function</i> (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are <i>point-dissipative</i>, i.e. there is an explicit <i>absorbing ball</i><span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">ℬ</mi></math></span><span></span> given by the level set of a certain Lyapunov function, <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>V</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. However, since the minimum point of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>V</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> does not coincide with any rest point of Lorenz system, we apply <i>a modified</i> solution to the “uniting CLF problem” (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu &amp; Prieur, 2010]) in order to obtain a CLF <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of “any” rest point of this system via bounded and <i>regular</i> feedback controls by using the proposed CLF method, which also contains the following controllers: (i) <i>damping controls</i> outside <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">ℬ</mi></math></span><span></span>, so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) <i>feedback controls</i> inside <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">ℬ</mi></math></span><span></span> that stabilize the control system at “any” desired rest point of the (unforced) Lorenz system.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153499","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":"Canard Cycles and Their Cyclicity of a Fast–Slow Leslie–Gower Predator–Prey Model with Allee Effect","authors":"Tianyu Shi, Zhenshu Wen","doi":"10.1142/s0218127424500913","DOIUrl":"https://doi.org/10.1142/s0218127424500913","url":null,"abstract":"<p>We study canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey system with Allee effect. More specifically, we find necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its location (or their locations), and then we further completely determine its (or their) dynamics under explicit conditions. Besides, by geometric singular perturbation theory and the slow–fast normal form, we find explicit sufficient conditions to characterize singular Hopf bifurcation and canard explosion of the system. Additionally, the cyclicity of canard cycles is completely solved, and of particular interest is that we show the existence and uniqueness of a canard cycle, whose cyclicity is at most two, under corresponding precise explicit conditions.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153584","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":"Studying the Upper Bounds of the Numbers of Zeros of Abelian Integrals by the Law of Polynomials","authors":"Lijun Hong, Jinling Liu, Xiaochun Hong","doi":"10.1142/s0218127424500810","DOIUrl":"https://doi.org/10.1142/s0218127424500810","url":null,"abstract":"<p>For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>, the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153640","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":"Qualitative Properties of a Physically Extended Six-Dimensional Lorenz System","authors":"Fuchen Zhang, Ping Zhou, Fei Xu","doi":"10.1142/s0218127424500834","DOIUrl":"https://doi.org/10.1142/s0218127424500834","url":null,"abstract":"<p>In this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon <i>et al</i>., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov’s direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy=\"false\">)</mo></math></span><span></span> of this system is globally exponentially stable.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153542","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":"Analysis and Synchronization of the Chen System with Fractional Derivative","authors":"Chuntao Yin, Yufei Zhao, Xianghong Li, Yongjun Shen","doi":"10.1142/s0218127424500883","DOIUrl":"https://doi.org/10.1142/s0218127424500883","url":null,"abstract":"<p>In this paper, we study the dynamic behaviors and chaos synchronization of the Chen system described by Caputo–Hadamard fractional derivative. First, the existence and uniqueness of a solution to the Chen system with Caputo–Hadamard derivative are proved by qualitative analysis. Further, the stability of equilibria of the considered system is analyzed with the aid of Routh–Hurwitz criteria. Meanwhile, the bifurcation condition of the Caputo–Hadamard Chen system is compared with the integer-order Chen system, where the differences between the two systems are demonstrated numerically. In the study of chaos synchronization of the drive–response Chen systems with Caputo–Hadamard derivative, two control schemes are developed: three nonlinear controllers and single linear controller. The feasibility of two control schemes is verified, and the synchronization performances of these two schemes are compared by numerical simulations. Based on this, the influence of the fractional-order on chaos synchronization performance is illustrated as well.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153585","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":"A Concise 4D Conservative Chaotic System with Wide Parameter Range, Offset Boosting Behavior and High Initial Sensitivity","authors":"Baoqing Lu, Juan Du, Jiulong Du, Zeyang Zhao","doi":"10.1142/s0218127424500809","DOIUrl":"https://doi.org/10.1142/s0218127424500809","url":null,"abstract":"<p>In this paper, we present a concise four-dimensional (4D) conservative chaotic system with a wide parameter range. Since there are no terms higher than first order, the circuit does not contain multipliers, resulting in a simple circuit implementation. The nonlinear dynamic characteristics, such as phase diagrams, equilibrium points, divergence, Poincaré cross-sections, Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension, are analyzed in detail, which illustrates the conservativity. Besides, the system exhibits different offset boosting behaviors. Through offset boosting, the system can propagate along a line, convert signal polarity, control variable amplitude, generate coexisting attractors, and even induce changes in its state. Specially, we realize the transition from a single-vortex attractor to a multivortex one by some changes in the initial values. Furthermore, the Pearson correlation coefficient is used to demonstrate the higher initial value sensitivity of the system. Finally, the system is implemented through Multisim simulation and analog circuit separately, and their consistency validates the system effectively.</p>","PeriodicalId":50337,"journal":{"name":"International Journal of Bifurcation and Chaos","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153686","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}