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 effect of fractional order of damping on the dynamic behaviors of the motion of the fractionally damped Duffing equation is examined. And the authors show that the size of the attractor trends to enlarge when fractional ordering α increases.
Abstract: Vibration phenomena of the fractionally damped systems have attracted increasing attentions in recent years. In this paper, dynamics of the fractionally damped Duffing equation is examined. The fractionally damped Duffing equation is transformed into a set of fractional integral equations solved by a predictor–corrector method. The effect of fractional order of damping on the dynamic behaviors of the motion is the main subject of the study. In this work, bifurcation of the parameter-dependent system is drawn numerically. The time evolutions of the nonlinear dynamical system responses are also described in phase portraits and the Poincare map technique. In addition, the occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. Results obtained from this study illustrates that the fractional order of damping has a significant effect on the dynamic behaviors of the motion. The size of the attractor trends to enlarge when fractional order α increases. Regular motions (including period-3 motion) and chaotic motions are examined. Moreover, a period doubling route to chaos is also found. Many period-3 windows are also observed in bifurcation diagram.
95 citations
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TL;DR: A control-based setup in combination with Newton iterations ensures that the periodic orbit can be continued even when it is unstable, as demonstrated with the continuation of initially stable rotations of a vertically forced pendulum experiment through a fold bifurcation to find the unstable part of the branch.
Abstract: We present a continuation method that enables one to track or continue branches of periodic orbits directly in an experiment when a parameter is changed. A control-based setup in combination with Newton iterations ensures that the periodic orbit can be continued even when it is unstable. This is demonstrated with the continuation of initially stable rotations of a vertically forced pendulum experiment through a fold bifurcation to find the unstable part of the branch.
95 citations
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TL;DR: An analytical method is derived to investigate the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type and determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems.
Abstract: In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.
95 citations
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TL;DR: In this article, a bifurcation diagram of the most dangerous eigenvalues as functions of the Reynolds number was determined, together with the most important eigenvalue functions.
Abstract: The flow produced in an enclosed cylinder of height-to-radius ratio of two by the counter-rotation of the top and bottom disks is numerically investigated. When the Reynolds number based on cylinder radius and disk rotation is increased, the axisymmetric basic state loses stability and different complex flows appear successively: steady states with an azimuthal wavenumber of 1; travelling waves; near-heteroclinic cycles; and steady states with an azimuthal wavenumber of 2. This scenario is understood in a dynamical system context as being due to a nearly codimension-two bifurcation in the presence of O(2) symmetry. A bifurcation diagram is determined, together with the most dangerous eigenvalues as functions of the Reynolds number. Two distinct types of near-heteroclinic cycles are observed, with either two or four bursts per cycle. The physical mechanism for the primary instability could be the Kelvin-Helmholtz instability of the equatorial azimuthal shear layer of the basic state
95 citations
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TL;DR: In this paper, the authors introduce a method for tracking nonlinear oscillations and their bifurcations in nonlinear dynamical systems, which does not require a mathematical model of the dynamical system nor the ability to set its initial conditions.
Abstract: We introduce a method for tracking nonlinear oscillations and their bifurcations in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system nor the ability to set its initial conditions. Instead it relies on feedback stabilizability, which makes the approach applicable in an experiment. This is demonstrated with a proof-of-concept computer experiment of the classical autonomous dry-friction oscillator, where we use a fixed time step simulation and include noise to mimic experimental limitations. For this system we track in one parameter a family of unstable nonlinear oscillations that forms the boundary between the basins of attraction of a stable equilibrium and a stable stick-slip oscillation. Furthermore, we track in two parameters the curves of Hopf bifurcation and grazing-sliding bifurcation that form the boundary of the bistability region.
95 citations