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![The periodic orbit (3) for the values α=ω=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\omega =1$$\end{document}, β=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =0$$\end{document}, γ=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =2$$\end{document} and ε=1/100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =1/100$$\end{document}](publication/311945663/figure/fig1/AS:941803117219900@1601554769290/The-periodic-orbit-3-for-the-values-ao1documentclass12ptminimal.gif)
The periodic orbit (3) for the values α=ω=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\omega =1$$\end{document}, β=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =0$$\end{document}, γ=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =2$$\end{document} and ε=1/100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =1/100$$\end{document}
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In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation \({\dddot{x}} + a {\ddot{x}} + b {\dot{x}} + c x - x^2 = 0\), called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a f...
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The discretized approach for the space variable will be used in this paper to transform nonlinear parabolic partial differential equation into system of ordinary differential equations, then using the backstepping transformation approach to stabilize and solve the obtained system of nonlinear ordinary differential equations, based on evaluating the...
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... The next goal is to determine the nature of bifurcation (such as elementary saddle-node, transcritical or pitchfork) that one obtains in Theorem 6. For this analysis, the following theorem is recalled: 31]). Consider the one-parameter family . ...
Understanding of the glucose risk factors-mediated mechanism in human breast cancer remains challenging. In this perception, for the first time, we proposed a complex nonlinear dynamical model that may provide a basic insight into the mechanism of breast cancer for the patient with existing glucose risk factors. The impact of glucose risk factors on the cancer cells’ population is evaluated using the formulated analytical model. The dynamical features of the cancer cells are described by a system of ordinary differential equations. Furthermore, the Routh–Hurwitz stability criterion is used to analyze the dynamical equilibrium of the cells’ population. The occurrence of zero bifurcation as well as two and three of the Jacobian matrix are obtained based on the sums of principal minors of order one. The glucose risk factors are exploited as the bifurcation parameters (acted as necessary and sufficient conditions) to detect the Hopf bifurcation. The presence of excess glucose in the body is found to affect negatively the breast cancer cells’ dynamics, stimulating chaos in the normal and tumor cells and thus drastically deteriorating the efficiency of the human immune system. The theoretical results are validated using the numerical simulations. It is concluded that the present findings may be beneficial for the future breast cancer therapeutic drug delivery and cure.
... In [6] J. Llibre studied the zero-Hopf bifurcation using averaging theory of first order, and applied it to the system (1). This method can be applied to any differential system in R n , n ≥ 3. Recently, it was successfully applied for other interesting models (see for example [1,2,3,4,5,7,8] and references therein). ...
We apply a technique of Llibre based on the averaging method to a Rössler-type three-dimensional quadratic system and we prove the existence of a periodic orbit.
... In [6] J. Llibre studied the zero-Hopf bifurcation using averaging theory of first order, and applied it to the system (1). This method can be applied to any differential system in R n , n ≥ 3. Recently, it was successfully applied for other interesting models (see for example [1,2,3,4,5,7,8] and references therein). ...
... In [1], the authors analyzed the presence of zero-Hopf bifurcations for the ordinary differential equation of order three ... ...
The purpose of the present paper is to study the presence of bifurcations of zero-Hopf type at a generalized Genesio differential equation. More precisely, by transforming such differential equation in a first-order differential system in the three-dimensional space R3, we are able to prove the existence of a zero-Hopf bifurcation from which periodic trajectories appear close to the equilibrium point located at the origin when the parameters a and c are zero and b is positive.
... In the present paper, we shall study the periodic orbits of this class of third-order differential equations, by using the averaging theory of dynamical systems, which is a completely different approach to the ones used in the mentioned papers. In [11][12][13] averaging theory was applied to order three systems too. ...
The aim of the present work is to study the necessary and sufficient conditions for the existence of periodic solutions for a class of third order differential equations by using the averaging theory. Moreover, we use the symmetry of the Monodromy matrix to study the stability of these solutions.
... This motivates us to derive a set of bifurcation formulae for the general family of 3-D quadratic systems (includes the quadratic family in [29]) independently, not based on the projection method. By the way, Cardin and Llibre [32] studied the transcritical and Zero-Hopf bifurcations of Genesio-Tesi system. ...
The purpose of this paper is to propose some formulae for Hopf bifurcation analysis, and investigate applications to a chaotic system. We perform a substantial simplification for the classical Hopf bifurcation formulae. Our results can be extended to multi-dimensional quadratic systems. As an application, we consider the Genesio-Tesi system. Finally, the dynamics of the system are analyzed by bifurcation diagrams, Lyapunov exponents, phase portraits and Poincaré maps. We show that the system can generate chaos via a Hopf bifurcation and period doubling cascade as the control parameter varies. Some other bifurcations can be observed, which includes saddle-node bifurcations, interior crises and boundary crises.
... This motivates us to derive a set of bifurcation formulae for the general family of 3-D quadratic systems ( includes the quadratic family in [9]) independently, not based on the projection method. By the way, Cardin and Llibre [6] studied the transcritical and Zero-Hopf bifurcations of Genesio-Tesi system. ...
... In the following discussion, we'll always choose a = 1, b = 7, the initial conditions x(0) = y(0) = z(0) = 0.5 and vary c. According to Section 3, a supercritical Hopf bifurcation occurs at c = 7 , see Fig. 2, which are bifurcation diagrams of the successive maxima of x, as a function of c in the ranges [6,12] and [12, 15.5]. It reveals the following route to chaos: stable equilibrium, limit cycle and period doubling of the limit cycles. ...
The purpose of this paper is to propose some formulae for Hopf bifurcation analysis, and consider the application to a chaotic system.
We perform a substantial simplification for the classical Hopf bifurcation formulae. The obtained final forms are concise and general for 3-D quadratic systems. The results can be extended for multi-dimensional quadratic systems. As an application, we consider the Genesio-Tesi system.
We get detailed information about the Hopf bifurcation: the direction of bifurcation, the period, amplitude and stability of bifurcating limit cycles. Finally, in this paper, the dynamics of the system are analyzed by bifurcation diagrams, Lyapunov exponents, phase portraits and Poincar\'{e} maps. We show that the system can generate chaos via a Hopf bifurcation and period doubling cascade as the control parameter varies. Some other bifurcations can be observed, which includes saddle-node bifurcations, interior crises and boundary crises.
... where the constants a i 1 ···i n , b i 1 ···i n , c i 1 ···i n ,k , a, b and c k are real, ab = 0 and P, Q and R k are the remainder terms in the Taylor series. We refer the reader to [1,2,10,11,14] and the references therein for details on the study of limit cycles and averaging theory. ...
... Then for |ε| > 0 sufficiently small, there exists an isolated T -periodic solution x(t, ε) of system (2) such that ...
For a \(C^{m+1}\) differential system on \(\mathbb {R}^n\), we study the limit cycles that can bifurcate from a zero–Hopf singularity, i.e., from a singularity with eigenvalues \(\pm ~bi\) and \(n-2\) zeros for \(n\ge 3\). If the singularity is at the origin and the Taylor expansion of the differential system (without taking into account the linear terms) starts with terms of order m, then \(\ell \) limit cycles can bifurcate from the origin with \(\ell \in \{0,1,\ldots , 2^{n-3}\}\) for \(m=2\) [see Llibre and Zhang (Pac J Math 240:321–341, 2009)], with \(\ell \in \{ 0,1,\ldots ,3^{n-2}\}\) for \(m=3\), with \(\ell \le 6^{n-2}\) for \(m=4\), and with \(\ell \le 4\cdot 5^{n-2}\) for \(m=5\). Moreover, \(\ell \in \{0,1,2\}\) for \(m=4\) and \(n=3\), and \(\ell \in \{0,1,2,3,4,5\}\) for \(m=5\) and \(n=3\). In particular, the maximum number of limit cycles bifurcating from the zero–Hopf singularity grows up exponentially with n for \(m=2,3\).
... In the meantime, due to the great potential applications in technological fields, such as secure communications, nonlinear circuits, control and synchronization, biological networks, complex dynamics of three-dimensional dynamical systems has become a focal topic in theories and applications of nonlinear sciences [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. ...
This paper analyzes complex dynamics of the generalized Langford system (GLS) with five parameters. First, some important local dynamics such as the Hopf bifurcations and the stabilities of hyperbolic and zero/double-zero equilibrium are investigated using normal form theory, center manifold theory and bifurcation theory. Besides, an accurate expression of a periodic orbit and some approximate expressions of bifurcating limit cycles by Hopf bifurcation are obtained. Second, by using averaging theory, the zero-Hopf bifurcation at the origin is analyzed and also the stability of the bifurcating limit cycle is obtained. Of particular interest is that one numerically finds an annulus in three-dimensional space which appears nearby the bifurcating limit cycle under proper conditions. If an amplitude system of GLS has a center, its suspension may lead the GLS to exhibit such annulus. Third, it is proved rigorously that there exist two heteroclinic cycles and the coexistence of such two heteroclinic cycles and a periodic orbit under some conditions. This implies system has no chaos in the sense Shil’nikov heteroclinic criterion. Finally, by further numerical observation, it is shown that three different types of attractors exist simultaneously, such as two kinds of periodic orbits, periodic orbit and invariant torus.
