Hacınlıyan, A. | Kandıran, E.

Conference Object | 2019 | Springer Proceedings in Complexity , pp.149 - 158

Lyapunov exponents characterize the rate of approach or recession of nearby trajectories in a dynamical system defined by differential equations or maps. They are usually taken as indicators of chaotic behavior. The density of orbits in the state space or equivalently, the Poincare map is usually taken as another such indicator. Although these indicators usually give correct results, there are instances in which they can lead to confusing or misleading information. For instance, a system of three linear differential equations can have three positive eigenvalues ?i leading to a solution ?it. The Wolf-Benettin algorithm [4] would repo . . .rt three positive Lyapunov exponents, in spite of the fact that the system is not chaotic. Another example is the Khomeriki model [1] or even the usual Bloch equations that would report a spectrum of all negative Lyapunov exponents but produce completely full state space plots, if the AC field is sufficiently strong. We will consider the class of systems proposed by Sprott [3] consisting of three-dimensional ODE’s with at most two quadratic nonlinearities as examples. Many of them obey two scenarios one of which is Lorenz model like behavior where an unstable linearized fixed point is surrounded by two stable fixed points so that the unstable fixed-point leads to a throw and catch behavior. The other is Rössler-like behavior whereas the system moves away from a weakly unstable linearized fixed point, nonlinear terms return it to equilibrium with a spiral out catch in mechanism. Since the presence of an attractor may involve structural stability, these two mechanisms are expected to produce different spatial extents for the attractor. Although Lyapunov exponents indicate time dependent behavior, spatial extent would complement this as a spatial measure of localization, thus complementing the Lyapunov exponents that characterize horizon of predictability. Direct numerical simulation and where feasible, the normal form approach will be used to investigate selected examples of the three degree of freedom systems. © Springer Nature Switzerland AG 2019 Daha fazlası Daha az

Hacınlıyan, A. | Kandıran, E.

Conference Object | 2019 | Springer Proceedings in Complexity , pp.65 - 75

Many models of physical systems involving electronic circuit elements [6], population dynamics [5] involve evolution equations with discontinuities. The key to understand such systems is to hope that the discontinuity does not adversely affect the integration process. There are also three variable chaotic dynamical system examples, such as the Sprott systems for deriving jerky dynamics that have also become of interest [10]. In order to calculate dynamical invariants in chaotic systems such as characteristic exponents and fractal dimensions we often need to find the Jacobian; this often requires attempting to differentiate discontin . . .uous functions. Therefore finding a suitable continuous approximation to the discontinuities becomes important. In previous communications, two example systems had been used with two parametrizations for approximating discontinuous functions with continuous ones, one of which is the same as that used in the literature. In this work, we will use further examples to optimize the parameters of the continuous approximation to discontinuities using different examples in order to test the degree of applicability of this approach. Where possible, the invariants calculated by this method will be compared to the corresponding invariants calculated from its time series. © Springer Nature Switzerland AG 2019 Daha fazlası Daha az

Aybar, I.K.

Conference Object | 2012 | CHAOS 2012 - 5th Chaotic Modeling and Simulation International Conference, Proceedings , pp.271 - 276

A relation between the optimal solution of the optimization problem and the stability and bifurcation properties of the corresponding dynamical system is suggested in this work. There exists a relation between the optimal solution of an optimization problem and an equilibrium point of a dynamical system. In this sense stability properties, Lyapunov exponents and bifurcations of the resulting dynamical systems can be studied. © 2012 5th Chaotic Modeling and Simulation International Conference All Rights Reserved

Ozbal, S. | Südor, H.C. | Keskin, A.U.

Conference Object | 2018 | 2018 Medical Technologies National Congress, TIPTEKNO 2018 , pp.271 - 276

This paper analyzes and reports the dynamical behavior of a jerk system used in biomedical modelling applications. This system employs a hyperbolic tangent function as a single source of nonlinearity. The system trajectories, Poincaré maps and Lyapunov exponents and spectrum, as well as Bifurcation diagram are presented to verify the chaotic dissipative behavior of the system, generating a double-scroll chaotic attractor. © 2018 IEEE.

Aybar, O.O. | Hacinliyan, A.S. | Ayba, I.K. | Koseyan, K. | Deruni, B.

Conference Object | 2012 | CHAOS 2012 - 5th Chaotic Modeling and Simulation International Conference, Proceedings , pp.43 - 49

A motivation for looking at chaos in the classical realizations of the Yang-Mills or Yang Mills augmented by Higgs equations is the importance of this system in the initial (in)stability at big bang, since in the initial stages all interactions were of the same strength and were based on non abelian gauge theories, of which the SU(2) Yang Mills is a first example. In this study we consider the following two particle effective Hamiltonian suggested by Biro, Matinyan and Müller: H=px2+py2-b2y2/2/2 -x2y2+1+1/2a2x2+b2y2). © 2012 5th Chaotic Modeling and Simulation International Conference All Rights Reserved

Alan, N. | Aybar, I.K. | Aybar, O.O. | Hacinliyan, A.S.

Conference Object | 2012 | CHAOS 2012 - 5th Chaotic Modeling and Simulation International Conference, Proceedings , pp.19 - 25

In this study, the International market gold prices over the last 31 years were analyzed for trends by five different methods, linear trend analysis, ARMA analysis, Rescaled range analysis, attractor reconstruction and maximal Lyapunov Exponent, detrended fluctuation analysis. Unfortunately not all methods give consistent results. The linear analysis reveals three regions with different trends. This is not supported by the rescaled range or detrended fluctuation analysis results. The maximal Lyapunov exponent calculation reveals chaotic behavior. The detrended fluctuation analysis reveals behavior close to brown noise. This is not c . . .orroborated by the rescaled range analysis, which indicates anti persistent behavior. The ARMA model implies first differencing that indicates a strong underlying linear trend. Combining these results, one probable explanation is that the strong linear trend, (also corroborated by ARMA analysis) affects the rescaled range calculation, because of its dependence on extreme values. The detrended fluctuation analysis removes this trend and reveals brown noise. This is consistent with a maximal positive Lyapunov exponent. Hence, we have a linear trend plus brown noise and neither of these two effects is dominant. © 2012 5th Chaotic Modeling and Simulation International Conference All Rights Reserve Daha fazlası Daha az