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Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Bifurcation generally means separation of a structure into two branches; however, in the context of dynamical systems, it refers to period doubling, quadrupling, etc., that lead to the onset of chaos. The bifurcation diagram for the logistic map, which is a fractal, is a plot of the values of the parameter r along the horizontal axis and the possible long-term values of x along the vertical axis (Figure 10.8). The bifurcation diagram shows that period doubling bifurcation leads to a new behaviour with twice the period of the original system and period halving bifurcation leads to a new behaviour with half the period of the original system. Period halving leads from chaos to order, whereas period doubling leads from order to chaos. Logistic map bifurcation diagram is a clear illustration of the link between fractals and chaos. The points of separation of the bifurcation diagram of the logistic map looks like a Cantor set with gaps of varying sizes.
Stability and self-excited vibration of shafts
Published in Zbigniew Osiński, Damping of Vibrations, 2018
The bifurcation diagram is a very important result of the bifurcation analysis since it comprises synthetic information on the behaviour of the system in the neighbourhood of the critical point. A decisive influence on the diagram is exerted by the coefficient ω2, which decides about the role of growth of the self-excited vibration amplitude as well as the orbital stability of the limit cycle. In Fig. 11.5 we can see ω2 and Ω2 in relation to internal friction intensity with the rest of the parameters of the system being fixed.
The logistic map and elements of complex system dynamics
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
The previous paragraphs have highlighted how the behavior of the logistic map strongly depends on the parameter a. This dependence can be systematically explored by varying at small steps the parameter and observing the steady-state behavior obtained. In this way a bifurcation diagram of the system is constructed. The bifurcation diagram, thus, shows how the dynamical behavior of a system changes when a control parameter is varied. Numerical methods are often used to generate bifurcation diagrams. The bifurcation diagram of the logistic map is built in the next example.
An improvement of vibration-driven locomotion module for capsule robots
Published in Mechanics Based Design of Structures and Machines, 2022
Typically, a bifurcation diagram shows the values visited or approached asymptotically of a system as a function of a bifurcation parameter. In this study, Poincaré map of the relative velocity was chosen as the values visited of the system. The excitation frequency, excitation amplitude, mass ratio and cubic stiffness were selected as bifurcation parameters. The plotting process is described as below. For each value of the bifurcation parameter, calculate values of the Poincaré section of the relative velocity, v1-v2. The calculations were run for 300 cycles of the excited force. The first 100 cycles were considered to be transient and thus were omitted. The steady state of the next 200 points of the Poncaré map was collected and plotted on the diagram. After that, the corresponding progression rate of the system is overplot on such bifurcation diagram.
Identification of Chua’s chaotic circuit parameters using penguins search optimisation algorithm
Published in Cyber-Physical Systems, 2022
Fouzia Maamri, Sofiane Bououden, Ilyes Boulkaibet
Generally, in the bifurcation diagram, the trajectory changes from a fixed point until becomes chaotic after successive period doubling [40]. The idea is to force the system to chaotic behaviour by varying randomly the parameter R. The resistor R is used to change the Chua’s circuit trajectory over a range of bifurcation values. In Figure 18, we fix all the parameters of the circuit and we vary the value of the parameter to obtain chaotic behaviour.
Analysis of Fourth-order Chaotic Circuit Based on the Memristor Model for Wireless Communication
Published in IETE Journal of Research, 2023
Sam Thomas, Savarimuthu Prakash, Sankara Malliga Gopalan
The bifurcation diagram indicates the change in the dynamical behaviour of the system as a function of one or more of the parameters. Figure 5 confirms the bifurcation diagram for the proposed system for the circuit values given in Section 2.