Crisis Bifurcation Research Articles

ИННОВАЦИОННЫЕ МЕТОДЫ, ПОВЫШАЮЩИЕ ЭФФЕКТИВНОСТЬ БЕРЕЖЛИВОГО ПРОИЗВОДСТВА В РОССИЙСКОЙ ПРОМЫШЛЕННОСТИ

The article is devoted to the study of the features of the application of the lean manufacturing concept in Russian industry. Particular attention is paid to the possibility of increasing labor productivity using traditional and innovative methods of continuous improvement of business processes in industry. The purpose of this article is to study the concepts of lean production as a factor in increasing labor productivity of Russian companies and to develop innovative methods that increase the efficiency of lean production. As part of the study, an assessment was made of the impact of the implementation of the National Project “Labor Productivity” on the growth of industrial efficiency, the dynamics of the productivity index from the moment the measures began state support. The regions leading in productivity growth and the regions of the Central Black Sea Region are considered. The results of the study show that in general, for the type of economic activity “Manufacturing”, the growth rate of the labor productivity index is negative during the implementation of the national project; there is only a jump in 2020. It is suggested that this is not related to the implementation of the project, but to the general crisis bifurcation situation in economics.
 The article shows that the experience of those regions that have achieved success despite the general socio-economic instability in the country and industry deserves a closer look. Summarizing best practices, the article systematizes the main methods of lean production, paying special attention to innovative methods that can be used not only in industry, but also in all types of economic activity. A survey was conducted of the opinions of specialists representing a number of industrial enterprises of the Central Black Sea Region involved in the implementation of the National Project “Labor Productivity”, this made it possible to generalize productive methods of lean production and identify the main problems in applying the concept of lean production in Russian industry.

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Crises and chaotic scattering in hydrodynamic pilot-wave experiments.

Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.

Extreme multistability and phase synchronization in a heterogeneous bi-neuron Rulkov network with memristive electromagnetic induction.

Memristive electromagnetic induction effect has been widely explored in bi-neuron network with homogeneous neurons, but rarely in bi-neuron network with heterogeneous ones. This paper builds a bi-neuron network by coupling heterogeneous Rulkov neurons with memristor and investigates the memristive electromagnetic induction effect. Theoretical analysis discloses that the bi-neuron network possesses a line equilibrium state and its stability depends on the memristor coupling strength and initial condition. That is, the stability of the line equilibrium state has a transition between unstable saddle-focus and stable node-focus via Hopf bifurcation. By employing parameters located in the stable node-focus region, dynamical behaviors related to the memristor coupling strength and initial conditions are revealed by Julia- and MATLAB-based multiple numerical tools. Numerical results demonstrate that the proposed heterogeneous bi-neuron Rulkov network can generate point attractor, period, chaos, chaos crisis, and period-doubling bifurcation. Note that extreme multistability are disclosed with respect to initial conditions of memristor and gated ion concentration. Coexisting infinitely multiple firing patterns of periodic firing patterns with different periodicities and chaotic firing patterns for different memristor initial conditions are demonstrated by phase portrait and time-domain waveform. Besides, the phase synchronization related to the memristor coupling strength and its initial condition is explored, which suggests that the two heterogeneous neurons become phase synchronization with large memristor coupling strength and initial condition. This also reflects that the plasticity of memristor synapse enables adaptive regulation in keeping energy balance between the neurons. What's more, MCU-based hardware experiments are executed to further confirm the numerical simulations.

Society Ladenness Model: System-Parametric Analysis

The system-parametric method developed by A. Uyemov allows creating a model of the contemporary bifurcational reality. There are many terms adopted from other sciences which are often used in postmodern philosophy. Bifurcation is one of such terms adopted from synergetics. In this article emphasis is placed on the contemporary time which is bifurcational in its essence. Using the key thesis of the systems method which states that any object can be presented as a system, we propose to create a model of social ladenness. The mere fact that ladenness exists, makes the system of crisis (bifurcation) nature. The social ladenness is defined using system descriptors. This definition contains the concept, the structure, and the substrate of the object and analysis of their elements. Consistence and hierarchical arrangement of the descriptors is the social ladenness model under discussion. The elements identified in the social ladenness model are the economic, political, technical, ecological, religious, creative and bifurcation impact. We identified some markers of human existence in the community system. The coronavirus pandemic is used as an example to demonstrate the way the bifurcation society system model can be built in order to find solutions to overcome the social and individual crises.

Dynamic analysis of a flexible rotor supported by ball bearings with damping rings based on FEM and lumped mass theory

A dynamic model of a flexible rotor supported by ball bearings with rubber damping rings was proposed by combining the finite element and the mass-centralized method. In the proposed model, the rotor was built with the Timoshenko beam element, while the supports and bearing outer rings were modelled by the mass-centralized method. Meanwhile, the influences of the rotor’s gravity, unbalanced force and nonlinear bearing force were considered. The governing equations were solved by precise integration and the Runge-Kutta hybrid numerical algorithm. To verify the correctness of the modelling method, theoretical and experimental analysis is carried out by a rotor-bearing test platform, where the error rate between the theoretical and experimental studies is less than 10%. Besides that, the influence of the rubber damping ring on the dynamic properties of the rotor-bearing coupling system is also analyzed. The conclusions obtained are in agreement with the real-world deployment. On this basis, the bifurcation and chaos behaviors of the coupling system were carried out with rotational speed and rubber damping ring’s stiffness. The results reveal that as rotational speed increases, the system enters into chaos by routes of crisis, quasi-periodic and intermittent bifurcation. However, the paths of crisis, quasi-periodic bifurcation, and Hopf bifurcation to chaos were detected under the parameter of rubber damping ring’s stiffness. Additionally, the bearing gap affects the rotor system’s dynamic characteristics. Moreover, the excessive bearing gap will make the system’s periodic motion change into chaos, and the rubber damping ring’s stiffness has a substantial impact on the system motion.

Transition to doubly diffusive chaos

Doubly diffusive convection driven by horizontal gradients of temperature and salinity is studied in a three-dimensional enclosure of square horizontal cross section and large aspect ratio. Previous studies focused on the primary instability and revealed the formation of subcritical branches of spatially localized states. These states lose stability because of their twist instability, thereby precluding the presence of any related stable steady states beyond the primary bifurcation and giving rise to spontaneous temporal complexity for supercritical parameter values. This paper investigates the emergence of this behavior. In particular, chaos is shown to be produced at a crisis bifurcation located close to the primary bifurcation. The critical exponent related to this crisis bifurcation is computed and explains the unusually abrupt transition. The construction of a low-dimensional model highlights that only a few requirements are necessary for this type of transition to occur. As a consequence, it is believed to be observable in many other systems.

Tumor Stabilization Induced by T-Cell Recruitment Fluctuations

The influence of random fluctuations on the recruitment of effector cells towards a tumor is studied by means of a stochastic mathematical model. Aggressively growing tumors are confronted against varying intensities of the cell-mediated immune response for which chaotic and periodic oscillations coexist together with stable tumor dynamics. A thorough parametric analysis of the noise-induced transition from this oscillatory regime to complete tumor dominance is carried out. A hysteresis phenomenon is uncovered, which stabilizes the tumor at its carrying capacity and drives the healthy and the immune cell populations to their extinction. Furthermore, it is shown that near a crisis bifurcation, such transitions occur under weak noise intensities. Finally, the corresponding noise-induced chaos-order transformation is analyzed and discussed in detail.

Unexpected Behaviors in a Single Mesh Josephson Junction Based Self-Reproducing Autonomous System

In the literature, existing Josephson junction based oscillators are mostly driven by external sources. Knowing the different limits of the external driven systems, we propose in this work a new autonomous one that exhibits the unusual and striking multiple phenomena among which coexist the multiple hidden attractors in self-reproducing process under the effect of initial conditions. The eight-term autonomous chaotic system has a single nonlinearity of sinusoidal type acting on only one of the state variables. A priori, the simplicity of the system does not predict the richness of its dynamics. We also find that a limit cycle attractor widens to a parameter controlling coexisting multiple-scroll attractors through the splitting and the inverse splitting of periods. Multiple types of bifurcations are found including period-doubling and period-splitting (antimonotonicity) sequences to chaos, crisis and Hopf type bifurcation. To the best of our knowledge, some of these interesting phenomena have not yet been reported in similar class of autonomous Josephson junction based circuits. Moreover, analytical investigations based on the Hopf theory analysis lead to the expressions that determine the direction of appearance of the Hopf bifurcation, confirming the existence and determining the stability of bifurcating periodic solutions. To observe this latter bifurcation and to illustrate the theoretical analysis, numerical simulations are performed. Chaos can be easily controlled by the frequency of the linear oscillator, the superconducting junction current, as well as the gain of the amplifier or circuit component values. The circuit and Field Programmable Gate Arrays (FPGA)-based implementation of the system are presented as well.

Noise‐induced transformations in corporate dynamics of coupled chaotic oscillators

Stochastic effects in corporate dynamics of two symmetrically coupled chaotic oscillators are studied. As a functional unit, we use a discrete system defined by the logistic map. It is shown that increasing coupling parameter changes corporate deterministic dynamics and causes transitions “chaos‐order‐chaos.” We study an order window between Neimark–Sacker and crisis bifurcation points and show how this window is contracted and eliminated by increasing noise. For parametric analysis of these stochastic phenomena in coupled system, we apply the method of confidence domains based on the mathematical theory of stochastic sensitivity.

Stochastic deformations of coupling-induced oscillatory regimes in a system of two logistic maps

We consider a system of two coupled identical logistic maps, which in isolation demonstrate stable equilibrium modes. Under increasing coupling strength, this system exhibits transitions from initial equilibrium regime to synchronized oscillatory behavior with complex modes, both regular (periodic or quasiperiodic) and chaotic. Moreover, the coupled subsystem can demonstrate synchronization with anti-phase oscillations in tonic or burst form. A novelty of this paper is related to analysis of noise-induced deformations of these corporative oscillatory regimes. For this model, the following constructive stochastic effects are studied: (i) noise-induced temporal destruction of the anti-phase synchronization with transition from tonic to burst oscillations, (ii) noise-induced destruction of 2-torus with the transition from quasiperiodic oscillations to the bursting, (iii) stochastic transformations of burst oscillations from regular to chaotic. An important role of transients and “riddled” basins in the study of stochastic shifts of crisis bifurcation points is discussed. For the analytical investigation of these phenomena, a method based on the stochastic sensitivity of attractors (discrete cycles and tori) and confidence domains is applied.

Combined impacts of the Allee effect, delay and stochasticity: Persistence analysis

We study a combined influence of the Allee effect, delay and stochasticity on the base of the phenomenological Hassell mathematical model of population dynamics. This bistable dynamical model possesses a wide variety of regimes, both regular and chaotic. In the persistence zone, these regimes coexist with the trivial equilibrium that corresponds to the extinction of the population. It is shown that borders of the persistence zone are defined by the crisis and saddle-node bifurcation points. Noise-induced transitions from the persistence to the extinction are studied both numerically and analytically. Using numerical modeling, we have found that the persistence zone can decrease and even disappear under the influence of random noise. For the theoretical study of this phenomenon, we apply the stochastic sensitivity analysis and Mahalanobis metrics.

A direct transition to chaos in hysteretic systems with focus dynamics.

A specific transition to chaos is detected in the study of periodic orbits of hysteretic systems with symmetry and dynamics of focus type. The corresponding bifurcation is rigorously justified by resorting to the analysis of transition maps, whose mathematical expressions are adequately derived. It is shown that, depending on a parameter related to the location of equilibria, such transition maps can pass from being a smooth function to a discontinuous, piecewise-smooth function. We deal with the intermediate situation for which the transition map is continuous but nonsmooth. Using a second parameter, and previous known results on chaotic maps, we show in a rigorous way the existence of a chaos boundary crisis bifurcation, where the transition from a configuration without periodic orbits to another with bounded chaotic solutions occurs.

Stochastic variability and transitions to chaos in a hierarchical three-species population model

A variability of the dynamic behavior in stochastically forced multi-species population models is studied. We address how noise can generate complex oscillatory regimes with transitions between attractors and order-chaos transformations. For the parametric analysis of noise-induced transitions, we utilize a semi-analytical technique based on the stochastic sensitivity analysis of attractors and confidence domains method. This approach is used in the study of the fairly realistic three-species population model describing the interaction of prey, predator and top predator. We consider in detail the parametric zone where the system is monostable with excitable limit cycle, or bistable with coexisting limit cycle and chaotic attractor. These zones are separated by the crisis bifurcation point. Noise-induced transitions between regular and chaotic attractors in the bistability zone are analysed by the confidence ellipses method. In the monostability zone, a mechanism of the transition from regular periodic to multimodal chaotic oscillations is studied.

Stochastic Sensitivity Analysis of Noise-Induced Extinction in the Ricker Model with Delay and Allee Effect.

A susceptibility of population systems to the random noise is studied on the base of the conceptual Ricker-type model taking into account the delay and Allee effect. This two-dimensional discrete model exhibits the persistence in the form of equilibria, discrete cycles, closed invariant curves, and chaotic attractors. It is shown how the Allee effect constrains the persistence zones with borders defined by crisis bifurcations. We study the role of random noise on the contraction and destruction of these zones. This phenomenon of the noise-induced extinction is investigated with the help of direct numerical simulations and semi-analytical approach based on the stochastic sensitivity functions. Stochastic transitions from the persistence regimes to the extinction are studied by the analysis of the mutual arrangement of the basins of attraction and confidence domains.

Crises, noise, and tipping in the Hassell population model.

We consider a problem of the analysis of the noise-induced tipping in population systems. To study this phenomenon, we use Hassell-type system with Allee effect as a conceptual model. A mathematical investigation of the tipping is connected with the analysis of the crisis bifurcations, both boundary and interior. In the parametric study of the abrupt changes in dynamics related to the noise-induced extinction and transition from order to chaos, the stochastic sensitivity function technique and confidence domains are used. The effectiveness of the suggested approach to detect early warnings of critical stochastic transitions is demonstrated.

Solitary states for coupled oscillators with inertia.

Networks of identical oscillators with inertia can display remarkable spatiotemporal patterns in which one or a few oscillators split off from the main synchronized cluster and oscillate with different averaged frequency. Such "solitary states" are impossible for the classical Kuramoto model with sinusoidal coupling. However, if inertia is introduced, these states represent a solid part of the system dynamics, where each solitary state is characterized by the number of isolated oscillators and their disposition in space. We present system parameter regions for the existence of solitary states in the case of local, non-local, and global network couplings and show that they preserve in both thermodynamic and conservative limits. We give evidence that solitary states arise in a homoclinic bifurcation of a saddle-type synchronized state and die eventually in a crisis bifurcation after essential variation of the parameters.

Stochastic sensitivity of regular and multi-band chaotic attractors in discrete systems with parametric noise

We consider a response of nonlinear dynamic systems on random disturbances. A stochastic sensitivity of regular and chaotic attractors of discrete systems with parametric noise is studied. Cases of equilibria, cycles, one- and multi-band chaotic attractors are considered, and explicit parametric formulas for the stochastic sensitivity of these attractors are derived. We give a constructive application of this theory to the analysis of the dispersion of random states near chaotic attractors on the example of the logistic map. It is shown how this technique can be effectively used in the analysis of noise-induced crisis bifurcation of merging chaotic bands.

Analysis of Noise-Induced Chaos-Order Transitions in Rulkov Model Near Crisis Bifurcations

Noise-induced transitions from chaos to order in nonlinear systems with crisis bifurcations are studied. In this study, a discrete-time Rulkov system is used as a conceptual model of the neuronal activity. We investigate probabilistic mechanisms of noise-induced transitions from chaotic spiking to quiescence in zones of crisis bifurcations. To analyze these transitions parametrically, we apply a mathematical technique based on the stochastic sensitivity functions and confidence domains. A stochastic phenomenon of the shifts of crisis bifurcation points and the expansion of the order window under increasing noise is discussed and analyzed. Using our analytical approach, we construct a parametric description of chaotic and regular regimes for the randomly forced Rulkov model.

Analysis of chaotic saddles in a nonlinear vibro-impact system

In this paper, a computational investigation of chaotic saddles in a nonlinear vibro-impact system is presented. For a classical Duffing vibro-impact oscillator, we employ the bisection procedure and an improved stagger-and-step method to present evidence of visual chaotic saddles on the fractal basin boundary and in the internal basin, respectively. The results show that the period saddles play an important role in the evolution of chaotic saddle. The dynamics mechanics of three types of bifurcation such as saddle-node bifurcation, chaotic saddle crisis bifurcation and interior chaotic crisis bifurcation are discussed. The results reveal that the period saddle created at saddle-node bifurcation is responsible for the switch of the internal chaotic saddle to the boundary chaotic saddle. At chaotic saddle crisis bifurcation, a large chaotic saddle can divide into two different chaotic saddle connected by a period saddle. The intersection points between stable and unstable manifolds of this period saddle supply access for chaotic orbits from one chaotic saddle to another and eventually induce the coupling of these two chaotic saddle. Interior chaotic crisis bifurcation is associated with the intersection of stable and unstable manifolds of the period saddle connecting two chaotic invariant sets. In addition, the gaps in chaotic saddle is responsible for the fractal structure.