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The purpose of this paper is to survey literature on macroeconomic nonlinear dynamics.

The paper identifies five influential types of models where the possible generation of endogenous cycles and chaotic motion arises. First, the frameworks that make use of the one‐hump logistic type equation; second, the models inspired on the growth literature of the 1940s; third, intertemporal utility maximization problems with increasing returns; fourth, models that can be represented as piecewise dynamic maps; and, fifth, bounded rationality – heterogeneous expectations setups.

The attention will be mainly focused on the theme of business cycles; an interpretation of the deterministic real business cycle model with increasing returns is proposed and a graphical analysis of the underlying system shows that strange attractors are observable for specific sets of parameter values.

The study of endogenous cycles in macroeconomic literature has important implications for policy: if fluctuations are due to deterministic reasons this may imply that by manipulating policy parameters governments may be able to change the qualitative nature of the economy's dynamics.

The paper gives a comprehensive view of nonlinear dynamics in macroeconomics. It shows that various relevant subjects might be addressed in this kind of models, e.g. economic growth, asset pricing, business cycles, consumption decisions, among others.

Computer simulations were done extensively in order to study non‐linear dynamics of laser‐plasma interaction in InSb semiconductor. We constructed the modified Duffing kind of non‐linear semiconductor plasma oscillator equation. Collision frequency is found to be dominant parameter to influence the bifurcation, chaos, hysteresis and bistable effects of plasma wave. Small windows of higher period cascade above the critical value of laser parameter (α₁α₂) in the chaos region are observed. Laser‐plasma exhibits too much chaotic regime at lower value of laser driving frequency (δ). Hysteresis and bistable regions of plasma wave are presented and the conditions for their occurence are identified. The unstable regions completely merge at higher value of effective collision frequency (γ).

The purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control.

The authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019’s dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem.

The occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization.

As the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited.

In this paper we report on a mathematical pattern that we call bios, and its generation by recursions of bipolar feedback. Bios is a newly found form of organization, that resembles chaos in its aperiodic pattern and its extreme sensitivity to initial conditions, but has additional properties (diversification, novelty, nonrandom complexity, life‐limited patterning, 1/f power spectrum) found in natural creative processes, and absent in chaos. The process equation A_t+1=A_t+g_tsin(A_t) generates convergence to π, a cascade of bifurcations, chaos, bios and infinitation, as the value of the feedback gain g_t increases. In the complex plane, series generated by orthogonal process equations display fractal organic patterns.

It is often observed in practice that the essential behavior of mathematical models involving many variables can be captured by a much smaller model involving only a few variables. Further, the simpler model very often displays oscillatory behavior of some sort, especially when critical problem parameters are varied in certain ranges. This paper attempts to supply arguments from the theory of dynamical systems for why oscillatory behavior is so frequently observed and to show how such behavior emerges as a natural consequence of focusing attention upon so‐called “essential” variables in the process of model simplification. The relationship of model simplification and oscillatory behavior is shown to be inextricably intertwined with the problems of bifurcation and catastrophe in that the oscillations emerge when critical system parameters, i.e. those retained in the simple model, pass through critical regions. The importance of the simplification, oscillation and bifurcation pattern is demonstrated here by consideration of several examples from the environmental, economic and urban areas.

The purpose of this paper is to develop two types of simple jerk circuits and to carry out their dynamical analyses using a unified mathematical model.

Two types of simple jerk circuits only involve a nonlinear resistive feedback channel composited by a nonlinear device and an inverter. The nonlinear device is implemented through parallelly connecting two diode-switch-based series branches. According to the classifications of switch states and circuit types, a unified mathematical model is established for these two types of simple jerk circuits, and the origin symmetry and scale proportionality along with the origin equilibrium stability are thereby discussed. The coexisting bifurcation behaviors in the two types of simple jerk systems are revealed by bifurcation plots, and the origin symmetry and scale proportionality are effectively demonstrated by phase plots and attraction basins. Moreover, hardware experimental measurements are performed, from which the captured results well validate the numerical simulations.

Two types of simple jerk circuits are unified through parallelly connecting two diode-switch-based series branches and a unified mathematical model with six kinds of nonlinearities is established. Especially, the origin symmetry and scale proportionality for the two types of simple jerk systems are discussed quantitatively. These jerk circuits are all simple and inexpensive, easy to be physically implemented, which are helpful to explore chaos-based engineering applications.

Unlike previous works, the significant values are that through unifying these two types of simple jerk systems, a unified mathematical model with six kinds of nonlinearities is established, upon which symmetrically scaled coexisting behaviors are numerically disclosed and experimentally demonstrated.

This study aims to improve the complexity of chaotic systems and the security accuracy of information encrypted transmission. Applying five-dimensional memristive Hopfield neural network (5D-HNN) to secure communication will greatly improve the confidentiality of signal transmission and greatly enhance the anticracking ability of the system.

Chaos masking: Chaos masking is the process of superimposing a message signal directly into a chaotic signal and masking the signal using the randomness of the chaotic output. Synchronous coupling: The coupled synchronization method first replicates the drive system to get the response system, and then adds the appropriate coupling term between the drive The synchronization error and the coupling term of the system will eventually converge to zero with time. The synchronization error and coupling term of the system will eventually converge to zero over time.

A 5D memristive neural network is obtained based on the original four-dimensional memristive neural network through the feedback control method. The system has five equations and contains infinite balance points. Compared with other systems, the 5D-HNN has rich dynamic behaviors, and the most unique feature is that it has multistable characteristics. First, its dissipation property, equilibrium point stability, bifurcation graph and Lyapunov exponent spectrum are analyzed to verify its chaotic state, and the system characteristics are more complex. Different dynamic characteristics can be obtained by adjusting the parameter k.

A new 5D memristive HNN is proposed and used in the secure communication

As the transmission component of a locomotive, the traction gear pair system has a direct effect on the stability and reliability of the whole machine. This paper aims to provide a detailed dynamic analysis for the traction system under internal and external excitations by numerical simulation.

A non-linear dynamic model of locomotive traction gear pair system is proposed, where the comprehensive time-varying meshing stiffness is obtained through the Ishikawa formula method and verified by the energy method, and then the sliding friction excitation is analyzed based on the location of the contact line. Meantime, the adhesion torque is constructed as a function of the adhesion-slip feature between wheelset and rail. Through Runge–Kutta numerical method, the system responses are studied with varying bifurcation parameters consisting of exciting frequency, load fluctuation, gear backlash, error fluctuation and friction coefficient. The dynamic behaviors of the system are analyzed and discussed from bifurcation diagram, time history, spectrum plot, phase portrait, Poincaré map and three-dimensional frequency spectrum.

The analysis results reveal that as control parameters vary the system experiences complex transition among a diverse range of motion states such as one-periodic, multi-periodic and chaotic motions. Specifically, the significant difference in system bifurcation characteristics can be observed under different adhesion conditions. The suitable gear backlash and error fluctuation can avoid the chaotic motion, and thus, reduce the vibration amplitude of the system. Similarly, the increasing friction coefficient can also suppress the unstable state and improve the stability of the system.

The numerical results may provide a systemic understanding of dynamic characteristics and present some available information to design and optimize the transmission performance of the locomotive traction system.

To reconcile theoretical discrepancies between discrete emotion, dimensional emotion (positive vs. negative affect), and the circumplex model, we propose the bifurcation model of affect structure (BMAS). Based on complexity theory, this model explores how emotion as an adaptive complex system reacts to affective events through negative and positive feedback loops, resulting in self-organizing oscillation and transformations between three states: equilibrium emotion, discrete positive and negative emotion in the near-equilibrium state, and chaotic emotion. We argue that the BMAS is superior to the extant models in revealing the dynamic connections between emotions and the intensity of affective events in organizational settings.

The purpose of this study is to analyze the short-term development pattern and long-term development trend of the digital supply chain.

This study uses the combination of short-term game and long-term evolutionary game theory.

Findings of this study suggest that irrational decisions can make the evolutionary path of the digital supply chain complex and unpredictable.

This study proposes an evolutionary game model for the digital supply chain that can provide good guidance for the digitalization process of enterprises.