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.

An Algorithm for Linearizing the Collatz Convergence

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.

Multistability Control of Hysteresis and Parallel Bifurcation Branches through a Linear Augmentation Scheme

Multistability analysis has received intensive attention in recently, however, its control in systems with more than two coexisting attractors are still to be discovered. This paper reports numerically the multistability control of five disconnected attractors in a self-excited simplified hyperchaotic canonical Chua’s oscillator (hereafter referred to as SHCCO) using a linear augmentation scheme. Such a method is appropriate in the case where system parameters are inaccessible. The five distinct attractors are uncovered through the combination of hysteresis and parallel bifurcation techniques. The effectiveness of the applied control scheme is revealed through the nonlinear dynamical tools including bifurcation diagrams, Lyapunov’s exponent spectrum, phase portraits and a cross section basin of attractions. The results of such numerical investigations revealed that the asymmetric pair of chaotic and periodic attractors which were coexisting with the symmetric periodic one in the SHCCO are progressively annihilated as the coupling parameter is increasing. Monostability is achieved in the system through three main crises. First, the two asymmetric periodic attractors are annihilated through an interior crisis after which only three attractors survive in the system. Then, comes a boundary crisis which leads to the disappearance of the symmetric attractor in the system. Finally, through a symmetry restoring crisis, a unique symmetric attractor is obtained for higher values of the control parameter and the system is now monostable.

Basin of Attraction from Modification Variant of Chebyshev-Halley Methods

The development of Chebyshev-Halley Method for solving nonlinear equation is presented in this paper. Varian of Chebyshev-Halley method by Xiaojian (2008) was modified using Hermite Interpolation. The convergence analysis shows that these methods have sixth-order convergence for   0 and   1 eighth-order convergence for   1 2 . The methods are classified by the order and efficiency index. Here, we considered other criteria, the basin of attractions which are presented for several examples.The development of Chebyshev-Halley Method for solving nonlinear equation is presented in this paper. Varian of Chebyshev-Halley method by Xiaojian (2008) was modified using Hermite Interpolation. The convergence analysis shows that these methods have sixth-order convergence for   0 and   1 eighth-order convergence for   1 2 . The methods are classified by the order and efficiency index. Here, we considered other criteria, the basin of attractions which are presented for several examples.

A Giga-Stable Oscillator with Hidden and Self-Excited Attractors: A Megastable Oscillator Forced by His Twin

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors’ images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters.

Hidden Attractor in a Passive Motion Model of Compass-Gait Robot

Many studies have been done on different aspects of biped robots such as motion, path planning, control and stability. Dynamical properties of biped robot on a sloping surface such as equilibria and their stabilities, bifurcations and basin of attraction are investigated in this paper. Basin of attraction is an important property since it can determine the unseen conditions which affect the attractor of the system with multistabilities. By the help of basin of attractions, the paper claims that the strange attractors of compass-gait robot are hidden.

A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis

This paper proposes a new three-dimensional chaotic flow with one stable equilibrium. Dynamical properties of this system are investigated. The system has a chaotic attractor coexisting with a stable equilibrium. Thus the chaotic attractor is hidden. Basin of attractions shows the tangle of different attractors. Also, some complexity measures of the system such as Lyapunov exponent and entropy will are analyzed. We show that the Kolmogorov-Sinai Entropy shows more accurate results in comparison with Shanon Entropy.

Nonlinear Dynamics of the Hybrid Rotary-Translational Energy Harvester

The analytical modeling and experimental investigation of a nonlinear electromagnetic rotational energy harvester, which can harvest power from rotary and translational excitations, are presented. Some application of energy harvesting such as energy harvesting for tire pressure sensing require an energy harvester which is efficient in generating power from rotational ambient vibrations. The majority of literature on vibration energy harvesting assumes that the ambient excitations are along a single axis. The vibrations from human motion or rotary machines have two components of translational motion as well as a strong rotary motion. The energy harvesting device studied in this paper is a pendulum like device. The base excitations result in rotations of a pendulum. The pendulum is connected to a direct current micro generator. The rotational vibrations of the pendulum generates electricity through the DC generator. Since the energy harvester is responsive to both translational and rotational base excitations, it is called Hybrid Rotary-Translational (HRT) generator. In this paper a small size HRT harvester is introduced and modeled. The model is used to investigate the relation between the frequency and the amplitude of base vibrations on the vibrations and power generation characteristics of the HRT system. For each frequency and amplitude of vibrations the coexistence of multiple solutions and their basin of attractions are investigated. Three types of ambient excitations are studied: rotational, translational along the direction of gravity, and translational normal to the direction of the gravity.