Bifurcations Of Dynamical Systems Research Articles

Lévy noise-induced global bifurcation with the compatible cell mapping method

Stochastic bifurcation of dynamical system induced by Lévy noise is investigated in this paper from the global viewpoint, which is demonstrated based on the global attractor obtained with the compatible cell mapping (CCM) method. Compared with the traditional cell mapping method, the efficiency of CCM method is greatly improved by concentrating the computational domain on the covering set of the global attractor rather than the whole state space. The computation time can be further reduced by a simple parallel procedure with the block matrix strategy. With the proposed method, the evolutionary process of global properties has been presented. It shows that the shape and size of stochastic attractors and stochastic saddles vary along the direction of unstable manifold with the parameters of Lévy noise. Stochastic bifurcation occurs when stochastic attractor collides with stochastic saddle. It can be also found that the global bifurcation under Lévy noise is discontinuous because of the jump property. The results are significantly different from those of Gaussian noise, which reveals a distinct global evolutionary mechanism of stochastic bifurcation. Furthermore, it also indicates that the CCM method with block matrix is an effective tool for global analysis.

Multi-parametric optimization for controlling bifurcation structures

Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( > 10 2 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

Predicting saddle-node bifurcations using transient dynamics: a model-free approach

This paper proposes a novel method for predicting the presence of saddle-node bifurcations in dynamical systems. The method exploits the effect that saddle-node bifurcations have on transient dynamics in the surrounding phase space and parameter space, and does not require any information about the steady-state solutions associated with the bifurcation. Specifically, trajectories of a system obtained for parameters close to the saddle-node bifurcation present local minima of the logarithmic decrement trend in the vicinity of the bifurcation. By tracking the logarithmic decrement for these trajectories, the saddle-node bifurcation can be accurately predicted. The method does not strictly require any mathematical model of the system, but only a few time series, making it directly implementable for gray- and black-box models and experimental apparatus. The proposed algorithm is tested on various systems of different natures, including a single-degree-of-freedom system with nonlinear damping, the mass-on-moving-belt, a time-delayed inverted pendulum, and a pitch-and-plunge wing profile. Benefits, limitations, and future perspectives of the method are also discussed. The proposed method has potential applications in various fields, such as engineering, physics, and biology, where the identification of saddle-node bifurcations is crucial for understanding and controlling complex systems.

Investigation of the analytical and numerical solutions with bifurcation analysis for the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili equation

In this work, we investigate the solutions of the (2+1)-dimensional Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation by three powerful analytical methods: the exp _{a} function method, the (frac{G'}{G})-expansion method, and the Sine-Gordon expansion method. This equation describes the nonlinear wave propagation in many applications like waves of evolutionary shallow water, electrical networks, and engineering devices. Moreover, we study the solutions numerically via the finite difference method. We analyze the bifurcation of dynamical system resulting from the BKP equation. Finally, the majority of our solutions are displayed graphically to present the strength of imposed methods.

A physics-informed data-driven approach for forecasting bifurcations in dynamical systems

A physics-informed data-driven approach for forecasting bifurcations in dynamical systems

Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems

The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.

Detecting bifurcations in dynamical systems with CROCKER plots.

Existing tools for bifurcation detection from signals of dynamical systems typically are either limited to a special class of systems or they require carefully chosen input parameters and a significant expertise to interpret the results. Therefore, we describe an alternative method based on persistent homology-a tool from topological data analysis-that utilizes Betti numbers and CROCKER plots. Betti numbers are topological invariants of topological spaces, while the CROCKER plot is a coarsened but easy to visualize data representation of a one-parameter varying family of persistence barcodes. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter collection of differential equations. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use the maximum Lyapunov exponent. We further prove the relationship between the Wasserstein distance to the empty diagram and the norm of the Betti vector, which shows that an even more simplified version of the information has the potential to provide insight into the bifurcation parameter. The results show that our approach reveals more information about the shape of the periodic attractor than standard tools, and it has more favorable computational time in comparison with the Rösenstein algorithm for computing the maximum Lyapunov exponent.

Controlling Bifurcation in Electronic Implementation of Dynamical Systems

In the theoretical and experimental studies of bifurcations in dynamical systems, the adjustments of the parameter values play a key role. The reason is because small variations in these values may result in topological changes in the behavior of the flow of the system. Taking this into account, in this paper, a new design for controlling bifurcation, suitable for electronic implementations of chaotic systems, is presented. The variation of the bifurcation parameter is performed by means of an Arduino UNO micro-controller and a digital controlled potentiometer. In this way, the variation of the electronic components is performed in an automated manner, avoiding the intrinsic problems of a manual variation of the circuit parameters. As a particular example, a scaled Rössler system is considered. One of the advantages of the controlled automated bifurcation is that it is useful for analyzing the robustness of the different limiting behaviors of the system against parameter mismatches.

Experimental detection of Hopf bifurcation in two-dimensional dynamical systems

In this work, we propose a strategy based on an analog active network to detect Hopf bifurcations in two–dimensional dynamical systems described by ordinary differential equations. With the proposed strategy, two parameters of the nonlinear system are established in the analog active network by using external controllable voltage levels in order to explore the dynamical evolution of the system in a fast, easy, and accessible way, making our approach a powerful tool to detect Hopf bifurcations in a two-dimensional space. To demonstrate the proposed strategy’s potential and functionality, we electronically implement the kinetics of a reaction-diffusion model proposed by Barrio et al. in 1999 [1], called BVAM model. Hopf bifurcations are detected by following the changes in the system, going from stationary to periodic solutions when varying the control parameters. Local linear stability analysis is performed to show the quantitative agreement between analytical and experimental bifurcations. We additionally found that global effects are detected by the experimental approach, which cannot be predicted from a local analysis. The proposed strategy opens the way to use analog active networks to detect bifurcations in dynamical systems experimentally.

On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

The problem of the effect of two-frequency quasi-periodic perturbations on systems close to arbitrary nonlinear two-dimensional Hamiltonian ones is studied in the case when the corresponding perturbed autonomous systems have a double limit cycle. Its solution is important both for the theory of synchronization of nonlinear oscillations and for the theory of bifurcations of dynamical systems. In the case of commensurability of the natural frequency of the unperturbed system with frequencies of quasi-periodic perturbation, resonance occurs. Averaged systems are derived that make it possible to ascertain the structure of the resonance zone, that is, to describe the behavior of solutions in the neighborhood of individual resonance levels. The study of these systems allows determining possible bifurcations arising when the resonance level deviates from the level of the unperturbed system, which generates a double limit cycle in a perturbed autonomous system. The theoretical results obtained are applied in the study of a two-frequency quasi-periodic perturbed pendulum-type equation and are illustrated by numerical computations.

Parameter identification on Abelian integrals to achieve Chebyshev property

<p style='text-indent:20px;'>Chebyshev criterion is a powerful tool in the study of limit cycle bifurcations in dynamical systems based on Abelian integrals, but it is difficult when the Abelian integrals involve parameters. In this paper, we consider the Abelian integrals on the periodic annuli of a Hamiltonian with one parameter, arising from the generalized Liénard system, and identify the parameter values such that the Abelian integrals have Chebyshev property. In particular, the bounds on the number of zeros of the Abelian integrals are derived for different parameter intervals. The main mathematical tools are transformations and polynomial boundary theory, which overcome the difficulties in symbolic computations and analysis, arising from large parametric-semi-algebraic systems.

Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. In an experimental setting, this transition could lead to incorrect data or damage to the entire experiment. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here, we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.

Method of Topological Roughness in Tasks of Research and Control of Synergetic Systems

The method of a research of roughness of dynamic systems based on a concept of roughness according to Andronov—Pontryagin and named with "method of topological roughness" is considered. The concepts "roughness" and "bifurcation" of dynamic systems formulated at a dawn of formation of the scientific direction of mathematics — topology, are given by the great French scientist A. Poincare. Also the concept of roughness according to Andronov—Pontryagin is formulated and conditions of accessibility of required roughness of a dynamic system are defined. The definitions of concepts of the maximum roughness and minimum non roughness of dynamic systems entered by the author earlier are given. The corresponding theorems of necessary and sufficient conditions of accessibility of the maximum roughness and the minimum non roughness and also emergence of bifurcations of topological structures of dynamic systems, which were proved in the fundamental works of the author given in the list of references are formulated. At the same time it is claimed that sets of rough and not rough systems make roughnesses of a set, continuous on an indicator. As a roughness indicator in a method the number of conditionality of a matrix of reduction to a diagonal (quasidiagonal) type of a matrix of Jacobi in special points of phase space of a system is used. The method allows to controling roughness of control systems on the basis of the theorem formulated with use of the matrix equation of Sylvester and proved in works of the author which is also provided in this work. The main stages of researches of roughness and bifurcations of systems by means of the considered method are formulated in the form of the corresponding algorithm. In work questions of synergetic systems and chaos (strange attractors) in them, founders of science of synergetics — H. Haken, I. Prigozhin are briefly stated. The method can be used for researches of roughness and bifurcations of dynamic systems and also synergetic systems and chaos of the different physical nature. In works of the author the method is approved on examples of many synergetic systems, such as Lorenz attractors and R ö ssler, Belousov-Zhabotinsky’s systems, Chua, "predator-prey", Henon, Hopf’s bifurcations, etc. In this work of a possibility of a method are illustrated on examples of synergetic system Chua and also a technical system in the form of the nonlinear servomechanism.

Method of topological roughness of dynamic systems: applications to synergetic systems

The paper presents a method of dynamic system roughness research, based on Andronov-Pontryagin concept of roughness (method of topological roughness). Andronov-Pontryagin concept of roughness has been formulated. Reachability conditions of dynamic system required roughness are defined. Concept definition for maximum roughness and minimum non-roughness of dynamic systems is given. Theorems on necessary and sufficient conditions of eachability of maximum roughness and minimum non-roughness and occurrence of bifurcations of dynamic system topological structures are formulated. It is claimed that the sets of rough and non-rough systems are continuous in terms of the set roughness. The condition number of the matrix of bringing to the diagonal (quasi-diagonal) view of the Jacobi matrix at special points of the system phase space is used as an indicator of roughness. The method gives the possibility to control the roughness of control systems based on a theorem formulated using Sylvester’s matrix equation. The basic concepts on synergetics and synergetic systems are presented. The method can be used for studies of roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. The method is tested on the examples of many synergetic systems: Lorenz and Rössler, Belousov-Zhabotinsky, Chua, “predator-prey”, Henon, and Hopf bifurcation. The main provisions of the topological roughness method are given. The possibilities of the method are illustrated by examples of Belousov-Zhabotinsky and Chua synergetic systems.

Bifurcation analysis of dissipative rogue wave in electron-positron-ion plasma with relativistic ions and superthermal electrons

Abstract In this manuscript, a modified nonlinear Schrodinger equation (MNLSE) is derived for an unmagnetized collisionless three components plasma containing superthermal electrons, Boltzmann distribution of positrons and relativistic ions. By bifurcation of dynamical system, we determined the stable and unstable regions and predicted the kinds of solutions of MNLSE. This solutions reveal dark soliton in heteroclinic areas and rogue wave in unstable regions. A novel form of rogue wave is obtained also the effects of viscosity, superthermal electron κ, ratio of electron to positron temperature, ratio of ion to electron temperature and the density of positron are illustrated with some graphics in two-dimensional and three-dimensional. The derived results have numerous applications in plasma physics and it could be much importance in predicting and enriching rogue wave happening in dissipation plasma physics.

Hopf bifurcations in dynamical systems

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

Bifurcation Formulas and Algorithms of Constructing Central Manifolds of Discrete Dynamical Systems

Ones of the main questions in theory of local bifurcations and its applications are questions about direction of bifurcations (sub- or supercriticality) and on stability of the solutions arising in neighborhood of a nonhyperbolic equilibrium point or cycle dynamic system. We consider problems of local bifurcations in dynamical systems with discrete time. New features are proposed to orientation of bifurcations and properties stability of bifurcation solutions for problems on basic scenarios of bifurcations. We also propose new algorithms for constructing central manifolds of the corresponding problems, allowing to obtain new bifurcation formulas, in particular, formulas to calculate Lyapunov quantities. Proposed algorithms and formulas are based on the common operator method the study of problems on local bifurcations and allow under the new conditions effective qualitative analysis of bifurcations in terms of the initial equations.

Early-warning signals Early-warning signals for bifurcation in dynamical systems

Early-warning signals Early-warning signals for bifurcation in dynamical systems

Complexity Analysis of Prefabrication Contractors’ Dynamic Price Competition in Mega Projects with Different Competition Strategies

This paper considers a repeated duopoly game of prefabrication contractors in mega infrastructure projects and assumes the contractors exhibit bounded rationality. Based on the theory of bifurcation of dynamical systems, a dynamic price competition model is constructed considering different competition strategies. Accordingly, the stability of the equilibrium point of the system is discussed considering different initial market capacities, and numerical simulation is performed. The results show the system has a unique equilibrium solution when initial capacity is high and the parameters meet certain conditions. The contractors’ price adjustment strategy has an important influence on system stability. However, an overly aggressive competition strategy is not conducive to system stability. Moreover, the system is sensitive to initial parameter values.

Bifurcation in Transmission Networks Under Variation of Link Capacity

Much research attention has been devoted to the investigation of how the structure of a network affects its intended performance. However, conclusions drawn from the previous studies are often inconsistent and even contradictory. In order to identify the causes of these diverse results and to explore the impact of network topology on performance, we apply the concept of bifurcation in dynamical systems and consider the effect of varying a crucial parameter for networks of different structures. In this paper, we study transmission networks and identify the capacity setting as the parameter. Upon varying this parameter, the behavioral change of the network is observed. Specifically, we focus on communication networks and power grids, and study the improvement or degradation of robustness of such networks under variation of link capacity. We observe that the effect of increasing link capacity on robustness differs for different networks, and a bifurcation point exists in some cases which divides regions of opposite robustness behavior. Our work demonstrates that capacity settings play a crucially important role in determining how network structure affects the intended performance of transmission networks, and clarifies the previous reported contradictory results.