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Primer on Spike Processing and Excitability
Published in Paul R. Prucnal, Bhavin J. Shastri, Malvin Carl Teich, Neuromorphic Photonics, 2017
Paul R. Prucnal, Bhavin J. Shastri, Malvin Carl Teich
Some dynamical systems have periodic solutions (X(t) = X(t + T), for a given T > 0). This periodic orbit is called a limit cycle if the system is attracted back to it upon small perturbation. We are interested in systems where these periodic excursions are fast and represent action potentials or spikes. In these dynamical systems, a saddle-node bifurcation can occur on a limit cycle; they are referred to as saddle-node on limit cycle (SNLC) bifurcation (Fig. 2.6). When this happens, such a local bifurcation has effect on the global behavior of the system: if the system is at the right side of a saddle point, the system travels along the path of the limit cycle and returns to the attractor node (see Fig. 2.6). The qualitative behavior of the system thus changes from periodic to excitable[18].
A Note on Continuation Algorithms for Periodic Orbits
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
In parallel with theoretical studies, problems of location, dependence on parameters and branching of periodic trajectories of dynamical systems have been extensively inves-tigated numerically for some time. Several computer codes have been developed for this purpose [DK,HK,KKLN,K] some of which are distributed commercially at present. In most of the underlying algorithms the orbit is represented by one or several of its points. A substantial part of the algorithm consists in continuation: some of the parameters of the system are changed in small steps and, knowing a point on a periodic orbit for some value of the parameter (or, a sequence of such points for several values of the parameter) a nearby point on a periodic orbit for the next value of the parameter is found.
Strange attractors and continuous-time chaotic systems
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
In this section we discuss an approach to do this. In particular, the methodology is in the form of a conjecture, introduced by Roberto Genesio and Alberto Tesi in 1991 [35], and applicable to Lur’e systems. Quoting their words the conjecture says that “a Lur’e feedback system presents a chaotic behavior when a predicted limit cycle and an equilibrium point of certain characteristics interact between themselves with a suitable filtering effect along the system.” In fact, chaotic motion is seen as the result of a perturbation of a periodic orbit due to some other property of the system. Clearly, this represents a basic mechanism leading to chaos and not excluding that more complex chaotic motions can be obtained by the interaction of other characteristics of the system.
Effect of noise on residence times of a heteroclinic cycle
Published in Dynamical Systems, 2023
Valerie Jeong, Claire Postlethwaite
This paper is organized as follows. In Section 2, we construct a Poincaré map for (2) and obtain expressions for the residence time and fixed point. The fixed point corresponds to the periodic orbit in the system. We present a two-parameter bifurcation analysis of the periodic orbit, with one parameter associated with the resonance bifurcation and the other being the constant perturbation. This analysis is analogous to that done by Chow and his collaborators [10] for a homoclinic orbit. In Section 3, we use an Ornstein–Uhlenbeck process that captures the dynamics of (1), for which an analytical solution for the stationary distribution is available. To obtain the mean residence time prediction, we integrate the product of the probability distribution and the residence time expression from Section 2. It is not possible to find an analytic solution for this prediction, and we numerically integrate the expression using a quadrature method in MATLAB. The plots show an increase in the mean residence time prediction for the relevant noise level range, which agrees with the numerical simulations. We further show that it is the nonlinearity in the residence time expression that leads to the increase in the mean residence time, by evaluating the mean residence time predictions using a polynomial approximation for the residence time. In Section 4, we compare the mean residence time predictions from Section 3 to numerical simulations. Further numerical simulations suggest linear relationships between the parameters and the noise levels at which the mean residence times reach maxima. We conclude in Section 5.
Transient and steady-state nonlinear vibrations of FGM truncated conical shell subjected to blast loads and transverse periodic load using post-difference method
Published in Mechanics of Advanced Materials and Structures, 2023
G. Kai, S. W. Yang, W. Zhang, X. J. Gu, W. S. Ma
To analyze the chaotic and periodic vibrations of the FGM truncated conical shell, the bifurcation diagrams, time histories as well as phase portraits are obtained by using Runge-Kutta algorithm. Figure 13 depicts the bifurcation diagram of the transverse excitation versus the amplitude of the first-order mode for the FGM truncated conical shell when the transverse excitation increases from to From this picture, four chaotic regions can be found. It is shown that the first chaotic region is detected after a stable periodic orbit. The second chaotic region is located in The third and fourth chaotic regions occur when is close to and respectively. Based on Figure 13, it is observed that the vibration law of the FGM truncated conical shell is presented when the transverse excitations increase from to periodic chaotic periodic chaotic periodic chaotic periodic chaotic vibrations. In addition, it is observed that the chaotic vibration is caused by the period-doubling bifurcation, as shown in Figure 13.