Transitional Periodic Patterns and Triple Attractors in a Predator-Prey System with Fear and Refuge

We report the existence of periodic structures in the transitional and chaotic regimes in bi-parameter spaces of a predator-prey system. A model is constructed taking into consideration of two important effects: namely, the prey refuge and fear of predation risk. The fixed points, their existence and stability behaviors are analyzed. The Neimark-Sacker bifurcation in the neighborhood of the interior fixed point is shown selecting refuge strength as a bifurcation parameter. The complex dynamical behaviors are explored in the biparameter space with the help of the largest Lyapunov exponent and isoperiodic diagrams. The period-bubbling transitional patterns, and triple heterogeneous attractors resulting in qualitative unpredictability are identified in the present system. The Wada basin sets for the triple coexisting attractors are found. The study reveals that the oscillations of the populations in certain control parameter regions are highly dependent upon the initial densities of the populations.

The Complex Dynamics of Sharing Platform Competition Game

With the rapid development of Internet technologies and online sharing platforms, sharing economy has become a major trend in economy. The entry of sharing economy leads to profound impacts on incumbent industry. We build a dynamic sharing platform competition model with which agents are bounded rational, and consumer side is heterogeneous. Then, we present the fixed points and the stability conditions of the bifurcation of the dynamic model. We simulate the adjustment speed of sharing platform, sharing platform price, and costs of traditional firm effects on system stability, and we present stable area, bifurcation diagram, the largest Lyapunov exponent, and strange attractor of different parameters, and we give a feedback control method at last. Our main results are as follows: (1) when adjustment speed of sharing platform increases, the system becomes bifurcation, and finally, the system goes into a chaotic state; when the system is stable, price of traditional firm and fee decision of sharing platform are constant. (2) When price of sharing platform increases, sharing platform is more stable while traditional firm is more vulnerable. Suppose the system is in the stable state; when sharing platform price increases, traditional firm price increases, while sharing platform fees decreases. (3) When traditional firm cost is small, the system would be more stable. When the system is stable, with traditional firm cost increasing, traditional firm price increases quicker than sharing platform consumer fee, while sharing platform seller fee decreases. (4) Feedback control can alleviate the chaotic state of system. With feedback control parameter increases, the system becomes more stable.

Chaos Analysis of the Brain Topology in Grey Matter Images for the Recognition of Psychosis

Structural MRI studies in first-episode psychosis (FEP) and in clinical high risk (CHR) patients have consistently shown volumetric abnormalities in frontal, temporal, and cingulate cortex areas. The aim of the present study was to employ chaos analysis in the identification of people with psychosis. Structural MRI were acquired from 73 CHR, 77 FEP and 44 healthy controls (HC). Chaos analysis of the grey matter distribution was performed: first, the distances of each voxel from the center of mass in the grey matter image was calculated. Next, the distances multiplied by the voxel intensity was represented as a spatial-series, which then was analyzed by extracting the Largest-Lyapunov-Exponent (lambda). The lambda brain map depicts how the grey matter topology changes. The classification of a subject’s clinical status was finally predicted by a) comparing the lambda brain maps, which resulted in statistically significant differences in FEP and CHR compared to HC; and b) matching the lambda series with the Morlet wavelet, which resulted in 100% accuracy in distinguishing between FEP and CHR. The proposed framework using spatial-series extraction enhances the classification decision for FEP, CHR and HC subjects, verifies diagnosis-relevant features and may potentially contribute to the identification of structural biomarkers for psychosis.

Hamiltonian energy computation and complex behavior of a small heterogeneous network of three neurons: circuit implementation

Brain functions are sometimes emulated using some analog integrated circuits based on the organizational principle of natural neural networks. Neuromorphic engineering is the research branch devoted to the study and realization of such circuits with striking features. In this contribution, a novel small network of three neurons is introduced and investigated. The model is built from the coupling between two 2D Hindmarsh–Rose neurons through a 2D FitzHugh–Nagumo neuron. Thus, a heterogeneous coupled network is obtained. The biophysical energy released by the network during each electrical activity is evaluated. In addition, nonlinear analysis tools such as two-parameter Lyapunov exponent, bifurcation diagrams, the graph of the largest Lyapunov exponent, phase portraits, time series, as well as the basin of attractions are used to numerically investigate the network. It is found that the model can experience hysteresis justified by the simultaneous existence of three distinct electrical activities using the same set of parameters. Finally, the circuit implementation of the network is addressed in PSPICE to further support the obtained results.

Detachment Detection in Cam Follower System Due to Nonlinear Dynamics Phenomenon

The detachment between the cam and the follower was investigated for different cam speeds (N) and different internal distance of the follower guide from inside (I.D.). The detachment between the cam and the follower were detected using largest Lyapunov exponent parameter, power density function of Fast Fourier Transform (FFT), and Poincare’ maps due to the nonlinear dynamics phenomenon of the follower. The follower displacement and the contact force between the cam and the follower were used in the detection of the detachment heights. Multi-degrees of freedom (spring-damper-mass) systems at the very end of the follower were used to improve the dynamic performance and to reduce the detachment between the cam and the follower. Nonlinear response of the follower displacement was calculated at different cam speeds, different coefficient of restitution, different contact conditions, and different internal distance of the follower guide from inside. SolidWorks program was used in the numerical solution while high speed camera at the foreground of the OPTOTRAK 30/20 equipment was used to catch the follower position. The friction and impact were considered between the cam and the follower and between the follower and its guide. The peak of nonlinear response of the follower displacement was reduced to (15%, 32%, 45%, and 62%) after using multiple degrees of freedom systems.

Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions

We study the chaos and hyperchaos of Rydberg-dressed Bose–Einstein condensates (BECs) in a one-dimensional optical lattice. Due to the long-range, soft-core interaction between the dressed atoms, the dynamics of the BECs are described by the extended Bose-Hubbard model. In the mean-field regime, we analyze the dynamical stability of the BEC by focusing on the ground state and localized state configurations. Lyapunov exponents of the two configurations are calculated by varying the soft-core interaction strength, potential bias, and length of the lattice. Both configurations can have multiple positive Lyapunov exponents, exhibiting hyperchaotic dynamics. We show the dependence of the number of the positive Lyapunov exponents and the largest Lyapunov exponent on the length of the optical lattice. The largest Lyapunov exponent is directly proportional to areas of phase space encompassed by the associated Poincaré sections. We demonstrate that linear and hysteresis quenches of the lattice potential and the dressed interaction lead to distinct dynamics due to the chaos and hyperchaos. Our work is relevant to current research on chaos as well as collective and emergent nonlinear dynamics of BECs with long-range interactions.

Recalculation of error growth models' parameters for the ECMWF forecast system

This article provides a new estimate of error growth models' parameters approximating predictability curves and their differentials, calculated from data of the ECMWF forecast system over the 1986 to 2011 period. Estimates of the largest Lyapunov exponent are also provided, along with model error and the limit value of the predictability curve. The proposed correction is based on the ability of the Lorenz (2005) system to simulate the predictability curve of the ECMWF forecasting system and on comparing the parameters estimated for both these systems, as well as on comparison with the largest Lyapunov exponent (λ=0.35 d−1) and limit value of the predictability curve (E∞=8.2) of the Lorenz system. Parameters are calculated from the quadratic model with and without model error, as well as by the logarithmic, general, and hyperbolic tangent models. The average value of the largest Lyapunov exponent is estimated to be in the < 0.32; 0.41 > d−1 range for the ECMWF forecasting system; limit values of the predictability curves are estimated with lower theoretically derived values, and a new approach for the calculation of model error based on comparison of models is presented.

Strange attractor in earthquake swarms near Valsad (Gujarat), India

Valsad district in south Gujarat near the western coast of the peninsular India experienced earthquake swarms since early February 1986. Seismic monitoring through a network of micro earthquake seismographs showed a well concentrated seismic activity over an area of 7 × 10 km2 with the depth of foci extending from 1 to 15 km. A total number of 21,830 earthquakes were recorded during March 1986 to June 1988. The daily frequency of earthquakes for this period was utilized to examine deterministic chaos through evaluation of dimension of strange attractor and Lyapunov exponent. The low dimension of 2.1 for the strange attractor and positive value of the largest Lyapunov exponent suggest chaotic dynamics in Valsad earthquake swarms with at least 3 parameters for earthquake predictability. The results indicate differences in the characteristics of deterministic chaos in intraplate and interplate regions of India.

Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems’ stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.