Bifurcation Trees Research Articles

Coexisting phenomena and antimonotonic evolution in a memristive Shinriki circuit

This paper is aimed at investigating the dynamical behaviors of the Shinriki circuit with a grounded asymmetric diode-bridge-based memristor through the implicit mapping method, including coexisting phenomena and antimonotonicity. To elucidate these behaviors, the basic properties, including equilibrium points and symmetry, of a Shinriki circuit incorporating a grounded asymmetric diode-bridge-based memristor are first illustrated. The discretization scheme for the circuit is established through the implicit mapping method, and the relatively complete bifurcation trees of each periodic motion by tuning a variable resistor can be analyzed through the Newton-Raphson algorithm. The unique unstable periodic bifurcation routes are also revealed. The evolution of antimonotonicity triggered by period-doubling or saddle-node bifurcations, along with the formed unstable motions, are comprehensively discussed. Furthermore, the simulation results are verified via a Field Programmable Gate Array (FPGA), by which both the asymmetric stable and unstable coexisting periodic attractors can be captured. The results will contribute to explaining the dynamical behaviors of numerous memristive circuits from a novel semi-analytical way, thus opening the possibility of analyzing some phenomena, such as unstable bifurcation routes, that cannot be directly observed via the numerical method.

Periodic motions with impact chatters in an impact Duffing oscillator.

The periodic motions of discontinuous nonlinear dynamical systems are very difficult problems to solve in engineering and physics. Until now, except for numerical studies, one cannot find a better way to solve such problems. In fact, one still has difficulty obtaining periodic motions in continuous nonlinear dynamical systems. In this paper, a method is presented systematically for periodic motions in discontinuous nonlinear dynamical systems. The stability and grazing bifurcations of such periodic motions are studied. Such a method is presented through discussion on a periodically forced, impact Duffing oscillator. Thus, periodic motions with impact chatters in a periodically forced Duffing oscillator with one-sidewall constraint are studied. The analytical conditions for motion grazing at the boundary are developed from discontinuous dynamical systems. The impact Duffing oscillator is discretized to generate subimplicit mappings. With impact, the mapping structures are employed to construct specific impact periodic motions for an impact Duffing oscillator. The bifurcation trees of impact chatter periodic motions are achieved semi-analytically. The grazing and period-doubling bifurcations are obtained, and the grazing bifurcations are for the appearing and disappearance for an impact chatter periodic motion. The impact chatter periodic motions with and without grazing are presented for illustration of impact periodic motion complexity in the impact Duffing oscillator.

Period-1 to Period-4 Motions in a 5D Lorenz System

In this paper, a 5D Lorenz system is discussed. The discrete mappings are developed to solve the periodic motions in the 5D Lorenz system. Then the stability and bifurcations are determined by eigenvalue analysis. A bifurcation tree is presented to demonstrate that the discrete mapping method can provide not only stable orbits but also unstable motions. Finally, trajectory illustrations are given to show bifurcation influences on periodic orbits and homoclinic orbits in the 5D Lorenz system.

Structurally Unstable Synchronization and Border-Collision Bifurcations in the Two-Coupled Izhikevich Neuron Model

This study investigates a structurally unstable synchronization phenomenon observed in the two-coupled Izhikevich neuron model. As the result of varying the system parameter in the region of parameter space close to where the unstable synchronization is observed, we find significant changes in the stability of its periodic motion. We derive a discrete-time dynamical system that is equivalent to the original model and reveal that the unstable synchronization in the continuous-time dynamical system is equivalent to border-collision bifurcations in the corresponding discrete-time system. Furthermore, we propose an objective function that can be used to obtain the parameter set at which the border-collision bifurcation occurs. The proposed objective function is numerically differentiable and can be solved using Newton’s method. We numerically generate a bifurcation diagram in the parameter plane, including the border-collision bifurcation sets. In the diagram, the border-collision bifurcation sets show a novel bifurcation structure that resembles the “strike-slip fault” observed in geology. This structure implies that, before and after the border-collision bifurcation occurs, the stability of the periodic point discontinuously changes in some cases but maintains in other cases. In addition, we demonstrate that a border-collision bifurcation set successively branches at distinct points. This behavior results in a tree-like structure being observed in the border-collision bifurcation diagram; we refer to this structure as a border-collision bifurcation tree. We observe that a periodic point disappears at the border-collision bifurcation in the discrete-time dynamical system and is simultaneously replaced by another periodic point; this phenomenon corresponds to a change in the firing order in the continuous-time dynamical system.

Spikes Adding to Infinity on Period-1 Orbits to Chaos in the Rössler System

In this paper, spikes adding to infinity on bifurcation trees of period-1 orbits to chaos in the Rössler system is studied. The spikes adding on the periodic orbits is completed through a saddle-node bifurcation. With onset of a period-1 orbit, there is 1-spike on such a period-1 orbit, followed by the development from 1-spike to [Formula: see text]-spikes and the period-1 to period-[Formula: see text] orbits have 1-spike to [Formula: see text]-spikes. For a spike bifurcation of a period-1 orbit with [Formula: see text]-spikes ([Formula: see text]), a new spike is added on such a period-1 orbit. Thus, the period-1 orbit has [Formula: see text]-spikes. Such a period-1 to period-[Formula: see text] orbits ([Formula: see text]) have [Formula: see text]-spikes to [Formula: see text]-spikes. The three bifurcation trees of period-1 orbits with [Formula: see text]-spikes ([Formula: see text]) to period-4 orbits with [Formula: see text]-spikes are presented numerically. The phase trajectories and responses of [Formula: see text]-component for period-1 to period-4 orbits with different spikes are given for illustrations of spikes adding on periodic orbits. The spikes adding generating the complexity of period-1 orbits to chaos can be developed.

Period-1 Motions to Twin Spiral Homoclinic Orbits in the Rössler System

Abstract In this paper, period-1 motions to twin spiral homoclinic orbits in the Rössler system are presented. The period-1 motions varying with a system parameter are predicted semi-analytically through an implicit mapping method, and the corresponding stability and bifurcations of the period-1 motions are determined through eigenvalue analysis. The approximate homoclinic orbits are obtained, which can be detected through the periodic motions with the positive and negative infinite large eigenvalues. The two limit ends of the bifurcation diagram of the period-1 motion are at twin spiral homoclinic orbits. For comparison, numerical and analytical results of stable period-1 motion are presented. The approximate spiral homoclinic orbits are demonstrated for a better understanding of complex dynamics of homoclinic orbits. Herein, only initial results on periodic motions to homoclinic orbits are presented for the Rössler system. In fact, the Rössler system has rich complex dynamics existing in other high-dimensional nonlinear systems. Thus, the further studies of bifurcation trees of periodic motions to infinite homoclinic orbits will be completed in sequel.

Bifurcation analysis of a modified FitzHugh-Nagumo neuron with electric field

This paper is devoted to investigating the analytical solutions of the modified FitzHugh-Nagumo (FHN) neuron model with external electric fields through by the discrete mapping method, which can provide a new perspective to the study of complex motions of the neuron. The bifurcation trees of period-1 to period-8 motions and period-3 to period-12 motions of FHN neuron are presented through the implicit discrete mapping method, and the stable and unstable orbits, which cannot study through the traditional numerical method, are calculated. The bifurcations and stability of the periodic orbits are developed via eigenvalues. The spiking firing of neurons can be observed through the discrete nodes in phase portraits and time histories of membrane potential. Moreover, the antimonotonicity of improved FHN model is reported with varying the controlled parameters. The two-parameter diagrams are employed to intuitively explain the generation and evolution of periodic motions. Moreover, the modified FHN system is implemented via hardware circuit, the experiments of which validate the semi-analytical method. The investigation results are useful for the development of artificial intelligence and medicine.

Stability and bifurcations of complex vibrations in a nonlinear brush-seal rotor system.

A brush seal has the advantages of adapting to different vibration conditions and increasing the stability of the nonlinear rotor system. In this research, the stability and bifurcations of complex vibrations in a brush-seal rotor system are studied. An analytical seal force model is obtained through the beam theory and mutual coupling dynamics of the bristles and the rotor. The interaction between the bristles and the rotor is clearly depicted by a geometric map. Periodic and chaotic vibrations as well as the corresponding amplitude-frequency characteristics are first predicted by a numerical bifurcation diagram and 3D waterfalls. Discrete dynamic eigenvalue analysis is adopted for a detailed investigation of the stability and bifurcations of nonlinear vibrations. Jumping, quasi-periodic, and half-frequency vibrations are warned during the speeding up and down process. Four separate nonlinear vibration evolving routes are discovered. Two period-doubling bifurcation trees evolving to chaos are illustrated for the observation of global and independent periodic vibrations. Nonlinear vibration illustrations are presented through displacement orbits as well as harmonic amplitudes and phases. Chaotic vibration and unstable semi-analytical vibration solutions are compared. The obtained results and analysis methods provide new perspectives on nonlinear vibrations in the brush-seal rotor system.

Routes toward chaos in a memristor-based Shinriki circuit.

In this paper, the complex routes to chaos in a memristor-based Shinriki circuit are discussed semi-analytically via the discrete implicit mapping method. The bifurcation trees of period-m (m = 1, 2, 4 and 3, 6) motions with varying system parameters are accurately presented through discrete nodes. The corresponding critical values of bifurcation points are obtained by period-double bifurcation, saddle-node bifurcation, and Neimark bifurcation, which can be determined by the global view of eigenvalues analysis. Unstable periodic orbits are compared with the stable ones obtained by numerical methods that can reveal the process of convergence. The basins of attractors are also employed to analyze the coexistence of asymmetric stable periodic motions. Furthermore, hardware experiments are designed via Field Programmable Gate Array to verify the analysis model. As expected, an evolution of periodic motions is observed in this memristor-based Shinrik's circuit and the experimental results are consistent with that of the calculations through the discrete mapping method.

Nonlinear piezoelectric energy harvesting induced through the Duffing oscillator.

In this paper, nonlinear piezoelectric energy harvesting induced by a Duffing oscillator is studied, and the bifurcation trees of period-1 motions to chaos for such a piezoelectric energy-harvesting system are obtained analytically. Distributed-parameter electromechanical modeling of a piezoelectric energy harvester is presented first, and the electromechanically coupled circuit equation excited by infinitely many vibration modes is developed. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates, and eventually an infinite set of algebraic equations is obtained for the complex modal vibration responses and the complex voltage responses of the energy harvester beam. One single mode case is considered in this paper, and periodic motions with bifurcation trees are obtained through an implicit discrete mapping method. The frequency-amplitude characteristics of periodic motions are obtained for the nonlinear piezoelectric energy-harvesting systems, which provide a better understanding of where and how to achieve the best energy harvesting. This study describes about how the nonlinear oscillator induces piezoelectric energy harvesting through a beam system. The nonlinear piezoelectric energy harvesting is presented through a nonlinear oscillator. Due to the nonlinear oscillator, chaotic piezoelectric energy-harvesting states can get more energy compared to the linear piezoelectric energy-harvesting system.

Analytical predictions of stable and unstable firings to chaos in a Hindmarsh-Rose neuron system.

In this paper, analytical predictions of the firing cascades formed by stable and unstable firings in a Hindmarsh-Rose (HR) neuron system are completed via an implicit mapping method. The semi-analytical firing cascades present the route from periodic firings to chaos. For such cascades, the continuous firing flow of the nonlinear neuron system is discretized to form a special mapping structure for nonlinear firing activities. Stability and bifurcation analysis of the nonlinear firings are performed based on resultant eigenvalues of the global mapping structures. Stable and unstable firing solutions in the bifurcation tree exhibit clear period-doubling firing cascades toward chaos. Bifurcations are predicted accurately on the connections. Phase bifurcation trees are observed, which provide deep cognitions of neuronal firings. Harmonic dynamics of the period-doubling firing cascades are obtained and discussed for a better understanding of the contribution of the harmonics in frequency domains. The route into chaos is illustrated by the firing chains from period-1 to period-16 firings and verified by numerical solutions. The applied methods and obtained results provide new perspectives to the complex firing dynamics of the HR neuron system and present a potential strategy to regulate the firings of neurons.

Period-3 motions to chaos in a periodically forced nonlinear-spring pendulum.

In this paper, the complete bifurcation dynamics of period-3 motions to chaos are obtained semi-analytically through the implicit mapping method. Such an implicit mapping method employs discrete implicit maps to construct mapping structures of periodic motions to determine complex periodic motions. Analytical bifurcation trees of period-3 motions to chaos are determined through nonlinear algebraic equations generated through the discrete implicit maps, and the corresponding stability and bifurcations of periodic motions are achieved through eigenvalue analysis. To study the periodic motion complexity, harmonic amplitudes varying with excitation amplitudes are presented. Once more, significant harmonic terms are involved in periodic motions, and such periodic motions will be more complex. To illustrate periodic motion complexity, numerical and analytical solutions of periodic motions are presented for comparison, and the corresponding harmonic amplitudes and phases are also presented for such periodic motions in the bifurcation trees of period-3 motions to chaos. Similarly, other higher-order periodic motions and bifurcation dynamics for the nonlinear spring pendulum can be determined. The methods and analysis presented herein can be applied for other nonlinear dynamical systems.

On the mechanism of multiscroll chaos generation in coupled non-oscillatory rayleigh-duffing oscillators

We investigate the dynamics of a pair of coupled non oscillatory Rayleigh-Duffing oscillators (RDOs here after). The RDO serves as a model for a class of nonlinear oscillators including microwave Gunn oscillators [Guin et al Comm. in Nonlinear Sci. Numerical Simulat, 2017]. Here, the coupling between the two oscillators is obtained by superimposing to each one’s amplitude a perturbation proportional to the other one. We demonstrate that the coupling induces more equilibrium points and results in extremely complex nonlinear behaviors including multistability (up to six coexisting attractors), multiple Hopf bifurcations, multi-scroll chaos, and coexisting bifurcation trees. These phenomena are studied in detail by utilizing one-parametric bifurcation plots, bi-parametric Lyapunov exponent diagrams, phase space trajectory plots, and basins of attraction as well. Experimental results captured from an Arduino microcontroller-based realization of the coupled RDOs are included to support the observations made through numerical analysis. We would like to point out that the coupling scheme followed in this work may stimulate the research on multiscroll chaos generation based on coupled nonlinear oscillators.

Semi-analytical prediction of the periodic vibration in a sliding bearing–rotor​ system

As for studying the sub-harmonic and super-harmonic periodic motions, eigenvalue dynamics and period-doubling bifurcation tree of the nonlinear rotor system, an implicit discrete mapping method is applied to a nonlinear sliding bearing–rotor system. The nonlinear term of the rotor system is triggered by the oil film forces of the sliding bearing at both ends. In order to accurately reveal the dynamic characteristics of the system, in this paper, a thirteen-parameter polynomial oil film force model is proposed. It considers the effects of displacement and velocity on the oil film force. And, the discrete mapping method used in this paper is a semi-analytical tool for solving equations with high precision. Consequently, continuous eigenvalue diagrams and bifurcation trees of periodic solutions from Period-1 to Period-4 motions of the nonlinear rotor system are obtained. For verification, numerical simulations are completed. Numerical and semi-analytical results are compared in time–history displacement, velocity and trajectories. Furthermore, the active feedback controller is carried out to improve the stability of the system.

36-5-4] 전력전자변환기에 적용된 Chaos PWM 기법의 스펙트럼 확산에 관한 연구

In this study, the power spectrum of the random PWM technique using the Linear Congruential Generator (LCG) and the PWM technique with chaotic logistic, tent and double tent map characteristics were compared with the fixed frequency switching PWM technique. The random technique generates random numbers using LCG, and the chaos number is generated using each chaos map, and a bifurcation tree with a=4.0 and λ=0.99 conditions in which a complete chaos state exists in the ranges of 0 and 1 is used. In order to prove the validity of this study, a DSP-based power electronics converter for driving a three-phase induction motor was manufactured. As a result, the chaos logistic map and tent map were limited in the spread of broadband in the spectrum of power electronics converters. However, the advanced chaos double tent map was similar to the excellent spectrum spread performance of the Random PWM technique by LCG. Harmonic Spread Factor(HSF) of the output voltages of each technique were measured according to the increase in the modulation index, and the chaos double tent map and random technique were very advantageous from the viewpoint of HSF.

Transition from order to chaos in reduced quantum dynamics.

We study a damped kicked top dynamics of a large number of qubits (N→∞) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant r∈[0,1], which plays the role of the single control parameter. In the parameter range for which the classical dynamics is chaotic, while varying r we find the universal period-doubling behavior characteristic to one-dimensional maps: period-2 dynamics starts at r_{1}≈0.3181, while the next bifurcation occurs at r_{2}≈0.5387. In parallel with period-4 oscillations observed for r≤r_{3}≈0.5672, we identify a secondary bifurcation diagram around r≈0.544, responsible for a small-scale chaotic dynamics inside the attractor. The doubling of the principal bifurcation tree continues until r≤r_{∞}∼0.578, which marks the onset of the full scale chaos interrupted by the windows of the oscillatory dynamics corresponding to the Sharkovsky order. Finally, for r=1 the model reduces to the standard undamped chaotic kicked top.

Bifurcation Trees of (1:2)- Asymmetric Periodic Motions with Corresponding Infinite Homoclinic Orbits in the Lorenz System

In this paper, bifurcation dynamics and infinite homoclinic orbits of (1:2)- asymmetric periodic motions in the Lorenz systems are studied through the discrete mapping method. The infinite homoclinic orbits pertaining to the unstable periodic motions on the bifurcation trees of (1:2)-asymmetric periodic motions to chaos are determined. The stability and bifurcations of periodic motions are determined through the eigenvalue analysis. A bifurcation tree of (1:2) asymmetric periodic motions, varying with the Rayleigh number, is presented by discrete nodes, and the corresponding harmonic frequency-amplitude characteristics of periodic motions on the bifurcation tree are discussed through finite Fourier series analysis. A sequence of period-doubling bifurcations is observed on the bifurcation tree of the (1:2)- asymmetric periodic motion to chaos. The scenario of (1:2)- asymmetric period-1, period-2, period-4 motions to chaos scenario is illustrated. Homoclinic orbits are as appearance or vanishing of unstable periodic motions on the bifurcation tree. Illustrations of periodic motions and homoclinic orbits are completed for intuitive demonstrations of the corresponding topological structures. This study extends the initial study of the bifurcation tree from the (1:1)-symmetric periodic motions to asymmetric periodic motions then to chaos in the Lorenz system, and the corresponding infinite homoclinic orbits induced by all unstable periodic motions can be determined. Such results can enrich the understanding of bifurcation dynamics of asymmetric periodic motions to chaos in the Lorenz system.

Period-3 motions to chaos in an inverted pendulum with a periodic base movement

A bifurcation tree of period-3 motions to chaos in an inverted pendulum with a periodic base movement is presented through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are also carried out. Frequency-amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Numerical simulations of periodic motions on the bifurcation are completed. The numerical and analytical results are presented for comparison. Except for period-1 motion to chaos studied before, this study focuses on other periodic motions to chaos existing in the inverted pendulum with periodic base movement. In the earthquake testing, one tests structures from 1hz to 33 hz. However, during such a frequency range, one not only can observe the period-1 motion to chaos, but one can observe period-3 motion to chaos in such an inverted pendulum with a periodic base movement. Thus, in the building design, period-3 motions to chaos should be considered, which have different dynamical behaviors from the period-1 motions to chaos.

On infinite homoclinic orbits induced by unstable periodic orbits in the Lorenz system.

In this paper, infinite homoclinic orbits existing in the Lorenz system are analytically presented. Such homoclinic orbits are induced by unstable periodic orbits on bifurcation trees through period-doubling cascades. Each unstable periodic orbit ends at its corresponding homoclinic orbit. Traditional computational methods cannot obtain homoclinic orbits from the corresponding unstable periodic orbits. This is because unstable periodic orbits in the Lorenz system cannot be achieved in numerical simulations. Herein, the stable and unstable periodic motions to chaos on the period-doubling cascaded bifurcation trees are determined through a discrete mapping method. The corresponding homoclinic orbits induced by the unstable periodic orbits are predicted analytically. A period-doubling bifurcation tree of period-1, period-2, and period-4 motions are generated as an example. The homoclinic orbits relative to unstable period-1, period-2, and period-4 motions are determined. Illustrations of homoclinic orbits and periodic orbits are given. This study presents how to determine infinite homoclinic orbits through unstable periodic orbits in three-dimensional or higher-dimensional nonlinear systems.

On Existence and Bifurcations of Periodic Motions in Discontinuous Dynamical Systems

In this paper, the existence and bifurcations of periodic motions in a discontinuous dynamical system is studied through a discontinuous mechanical model. One can follow the study presented herein to investigate other discontinuous dynamical systems. Such a sampled discontinuous system consists of two subsystems on boundaries and three subsystems in subdomains. From the theory of discontinuous dynamical systems, switchability conditions of a flow at and on the boundaries are developed. From such switchability conditions, grazing motions of a flow at boundaries are discussed, and sliding motions of a flow on boundaries are presented. Based on the motions in each domain and on each boundary, generic mappings are introduced. Using the generic mappings, mapping structures for specific periodic motions are developed. Based on the grazing conditions and appearance and vanishing conditions of sliding motions, parametric dynamics of the existences of the specific periodic motions are presented. In addition, the traditional saddle-node bifurcation, Neimark bifurcations and period-doubling bifurcation are used for parametric dynamics of periodic motions. Bifurcation trees of periodic motions varying with a system parameter are presented first, and phase trajectories of periodic motions are illustrated. The [Formula: see text]-functions are presented for the illustration of the motion switchability at the boundaries and sliding motions on the boundaries. Codimension-2 parametric dynamics of periodic motions are studied and how to develop the 2D parametric maps for specific periodic motions are presented. In the end, periodic motions with grazing are illustrated.