Topic
Bifurcation diagram
About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.
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TL;DR: In this paper, the chaos induced in a new type of phase locked loop (PLL) having a second-order loop filter is investigated, where the system under consideration is modeled as a third-order autonomous system with sinusoidal phase detector characteristics.
Abstract: The chaos induced in a new type of phase locked loop (PLL) having a second-order loop filter is investigated. The system under consideration is modeled as a third-order autonomous system with sinusoidal phase detector characteristics. The modern of nonlinear theory such as bifurcation and chaos is applied to a third-order of PLL. A method is developed to quantitatively study the type of bifurcations that occur in this type of PLL’s. The study showed that PLL experiencing a Hopf bifurcation point as well as chaotic behaviour. The method of multiple scales is used to find the normal form near the Hopf bifurcation point. The point is found to be supercritical one.
41 citations
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TL;DR: In this article, the stability points in nonlinear elasticity may be classified into limit points and bifurcation points, and path-tracing, pin-pointing and path switching strategies are described.
40 citations
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TL;DR: In this paper, a chaos neuron model represented by the fractional differential equation as a novel artificial neuron model is presented, which has an ability of a chaotic response which depends on the differential order and delay term in the dynamics.
Abstract: This paper presents a chaos neuron model represented by the fractional differential equation as a novel artificial neuron model. This model has an ability of a chaotic response which depends on the differential order and delay term in the dynamics. There is also observed a burst response in a certain parameter of chaotic region. We investigate basic characteristics of the model by some observations such as time-sequential data, bifurcation diagram for the differential order, and Lyapunov exponent analysis to the fractional differential system including delay. The result of the Lyapunov analysis confirms the existence of chaos on the presently proposed neuron model.
40 citations
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TL;DR: In this article, the existence of multiple solutions for a class of second order impulsive equations was investigated using bifurcation techniques, and multiple solutions were found for the same class of impulsive problems.
40 citations
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TL;DR: Numerically different cases of bistability between steady, periodic, and quasi-periodic regimes are studied and the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bIfurcation diagrams of the original laser equations and the slow time amplitude equation.
Abstract: Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities are starting from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.
40 citations