Period-doubling Bifurcations Research Articles

Chaos, Local Dynamics, Codimension-One and Codimension-Two Bifurcation Analysis of a Discrete Predator–Prey Model with Holling Type I Functional Response

We explore chaos, local dynamics, codimension-one, and codimension-two bifurcations of an asymmetric discrete predator–prey model. More precisely, for all the model’s parameters, it is proved that the model has two boundary fixed points and a trivial fixed point, and also under parametric conditions, it has an interior fixed point. We then constructed the linearized system at these fixed points. We explored the local behavior at equilibria by the linear stability theory. By the series of affine transformations, the center manifold theorem, and bifurcation theory, we investigated the detailed codimensions-one and two bifurcations at equilibria and examined that at boundary fixed points, no flip bifurcation exists. Furthermore, at the interior fixed point, it is proved that the discrete model exhibits codimension-one bifurcations like Neimark–Sacker and flip bifurcations, but fold bifurcation does not exist at this point. Next, for deeper understanding of the complex dynamics of the model, we also studied the codimension-two bifurcation at an interior fixed point and proved that the model exhibits the codimension-two 1:2, 1:3, and 1:4 strong resonances bifurcations. We then investigated the existence of chaos due to the appearance of codimension-one bifurcations like Neimark–Sacker and flip bifurcations by OGY and hybrid control strategies, respectively. The theoretical results are also interpreted biologically. Finally, theoretical findings are confirmed numerically.

Resonance and bifurcations of the discrete SIR model with unfixed incidence rate

In this paper, we investigate a class of discrete SIR epidemic models with unfixed incidence rate, significantly extended the results obtained by predecessors regarding the certain incidence rate. Besides proving the existence of one disease-free fixed point and one endemic fixed point, we provide their complete topological types and stability, which have no explicit expressions because of the unfixed incidence rate. For the endemic fixed point, by the bifurcation theory and normal form method, we identify four non-hyperbolic hypersurfaces in the parameter space and analyse their associated bifurcations: flip bifurcations, Neimark–Sacker bifurcations, and codimension-2 1:2 resonance, a structurally rich phenomenon previously uncharacterized for discrete SIR epidemic models with certain incidence rate. Numerical simulation examples are provided to illuminate the theoretical analysis and demonstrating the influence of uncertain incidence rates.

Two‐Dimensional Hepatitis C Virus Model With Chaos, Stability, and Bifurcations

ABSTRACTIn this paper, we explore the local stability, chaos, and bifurcations of a discrete hepatitis C virus model. More precisely, it is proved that for all model's parameters, model has liver‐free and disease‐free solutions, but it has total‐infection and partial‐infection solutions under certain parametric conditions. Further, we first constructed the linearized system and then local behavior at equilibria of a discrete hepatitis C virus model is explored by the linear stability theory. Next, for deeper understanding the complex dynamics of the discrete hepatitis C virus model, we first identified the bifurcation sets and then detail bifurcation analysis at equilibria is explored by the bifurcation theory. Furthermore, chaos in the hepatitis C virus model is studied due to the appearance of Neimark–Sacker and flip bifurcations by OGY and hybrid control feedback strategies, respectively. Finally, theoretical results are confirmed numerically.

Chaos Control and Bifurcations in a Modified Discrete Prey–Predator Model with Weak Multiple Allee Effect

This study explores the local dynamics of a discrete-time prey–predator model, which has been modified to include a weak multiple Allee effect on the prey population. The investigation encompasses topological categorization, bifurcation analysis, and chaos control. The analysis reveals that the model exhibits flip bifurcation and Neimark–Sacker bifurcation within a narrow vicinity of the coexistence equilibrium point, as demonstrated through bifurcation theory. Furthermore, the direction of these bifurcations is elucidated. To stabilize chaos resulting from the bifurcation, the OGY method is employed. Numerical simulations are conducted to validate the theoretical findings, with maximum Lyapunov exponents indicating the probability of chaos in the model.

Complexity analysis with chaos control: A discretized ratio-dependent Holling-Tanner predator-prey model with Fear effect in prey population.

This study explores a novel two-dimensional discrete-time ratio-dependent Holling-Tanner predator-prey model, incorporating the impact of the Fear effect on the prey population. The study focuses on identifying stationary points and analyzing bifurcations around the positive fixed point, with an emphasis on their biological significance. Our examination of bifurcations at the interior fixed point uncovers a variety of generic bifurcations, including one-parameter bifurcations, period-doubling, and Neimark-Sacker bifurcations. To further understand NS bifurcation, we establish non-degeneracy condition. The system's bifurcating and fluctuating behavior is managed using Ott-Grebogi-Yorke (OGY) control technique. From an ecological perspective, these findings underscore the substantial role of the Fear effect in shaping predator-prey dynamics. The research is extended to a networked context, where interconnected prey-predator populations demonstrate the influence of coupling strength and network structure on the system's dynamics. The theoretical results are validated through numerical simulations, which encompass local dynamical classifications, calculations of maximum Lyapunov exponents, phase portrait analyses, and bifurcation diagrams.

Finding signatures of low-dimensional geometric landscapes in high-dimensional cell fate transitions.

Multicellular organisms develop a wide variety of highly-specialized cell types. The consistency and robustness of developmental cell fate trajectories suggests that complex gene regulatory networks effectively act as low-dimensional cell fate landscapes. A complementary set of works draws on the theory of dynamical systems to argue that cell fate transitions can be categorized into universal decision-making classes. However, the theory connecting geometric landscapes and decision-making classes to high-dimensional gene expression space is still in its infancy. Here, we introduce a phenomenological model that allows us to identify gene expression signatures of decision-making classes from single-cell RNA-sequencing time-series data. Our model combines low-dimensional gradient-like dynamical systems and high-dimensional Hopfield networks to capture the interplay between cell fate, gene expression, and signaling pathways. We apply our model to the maturation of alveolar cells in mouse lungs to show that the transient appearance of a mixed alveolar type 1/type 2 state suggests the triple cusp decision-making class. We also analyze lineage-tracing data on hematopoetic differentiation and show that bipotent neutrophil-monocyte progenitors likely undergo a heteroclinic flip bifurcation. Our results suggest it is possible to identify universal decision-making classes for cell fate transitions directly from data.

Finding signatures of low-dimensional geometric landscapes in high-dimensional cell fate transitions

Multicellular organisms develop a wide variety of highly-specialized cell types. The consistency and robustness of developmental cell fate trajectories suggests that complex gene regulatory networks effectively act as low-dimensional cell fate landscapes. A complementary set of works draws on the theory of dynamical systems to argue that cell fate transitions can be categorized into universal decision-making classes. However, the theory connecting geometric landscapes and decision-making classes to high-dimensional gene expression space is still in its infancy. Here, we introduce a phenomenological model that allows us to identify gene expression signatures of decision-making classes from single-cell RNA-sequencing time-series data. Our model combines low-dimensional gradient-like dynamical systems and high-dimensional Hopfield networks to capture the interplay between cell fate, gene expression, and signaling pathways. We apply our model to the maturation of alveolar cells in mouse lungs to show that the transient appearance of a mixed alveolar type 1/type 2 state suggests the triple cusp decision-making class. We also analyze lineage-tracing data on hematopoetic differentiation and show that bipotent neutrophil-monocyte progenitors likely undergo a heteroclinic flip bifurcation. Our results suggest it is possible to identify universal decision-making classes for cell fate transitions directly from data.

Nonlinear vibration behaviors of HTS levitation system under different load-levitation ratios

High-temperature superconducting (HTS) pinning maglev trains, with their self-stabilizing levitation and low magnetic resistance, are promising candidates for future transportation. This paper investigates the nonlinear vibration behaviors of the single levitation system of the world’s first HTS high-speed maglev train under different load-levitation ratios (LLRs). First, vertical free vibration experiments were conducted under various load conditions, obtaining response curves for stable levitation height, main vibration frequency, stiffness, and damping with various loads. Based on these results, a set of equivalent stiffness and damping parameters for different LLRs was compiled. Second, focusing on three specific LLR conditions, levitation force curves were fitted using experimental data. The system’s stability and nonlinear vibration characteristics were analyzed through theoretical, numerical, and experimental methods. The experimental results show that the stable levitation height decreases nonlinearly as the load increases and stabilizes at 10 mm under rated load conditions. The system’s natural frequency remains nearly constant, showing minimal sensitivity to load changes. Both equivalent stiffness and damping increase with load. The analysis reveals that the HTS levitation system exhibits inherent stability and can recover balance after transient disturbances. It also shows nonlinear dynamics behavior such as period-doubling bifurcation and harmonic resonance, influenced significantly by excitation frequency, amplitude, and LLRs. These phenomena are more pronounced with larger LLR. Under certain conditions, the system exhibits jump phenomena. Resonance primarily occurs in the primary and double-frequency bands. This study enhances the understanding of HTS system dynamics and provides a reference for the dynamic research of HTS pinning maglev trains.

Exploring Dynamic Behavior in a Competition Duopoly Game Based on Corporate Social Responsibility

This study investigates dynamic behaviors within a competition Cournot duopoly framework incorporating consumer surplus, and social welfare through the bounded rationality method. The distinctive aspect of the competition game is the incorporation of discrete difference equations into the players’ optimization problems. Both rivals seek to achieve optimal quantity outcomes by maximizing their respective objective functions. The first firm seeks to enhance the average between consumer surplus and its profit, while the second firm focuses on its profit optimization with a social welfare component. The game map features four fixed points, with one being the Nash equilibrium point at the intersection of marginal objective functions. Our analysis explores equilibrium stability, dynamic complexities, basins of attraction, and the emergence of chaos through double routes via flip bifurcation and Neimark-Sacker bifurcations. We observe that increased adjustment speeds can destabilize the system, leading to a richness of dynamic complexity.

Delayed feedback control and parameter continuation of multistability in a nonsmooth hydraulic rock drill model.

In response to the complex multistable behavior observed in hydraulic rock drills during the drilling process, this study first establishes a four-degree-of-freedom physical model based on dry friction rock mechanics theory. The motion trajectory is classified into three states: non-viscous, impact viscous, and buffer viscous. Using the impact frequency ω as the bifurcation parameter, multistable attractors p0q1 and p1q2 are identified in the system when ω = 9. To control the multistability, a delayed feedback control method is applied, in which the infinite-dimensional delay differential equations are approximated by finite-dimensional ordinary differential equations. The reliability of this approximation is validated through a distance function. When the control gain K = 9 and the delay time τd = 0.35, both attractors p0q1 and p1q2 are successfully converted into a single p0q1 attractor. Next, the pseudo-arclength continuation method and Floquet theory are employed to investigate parameter continuation and parameter domains. The period-doubling bifurcation points PD1 and PD2 divide the parameter space of K and τd into three distinct regions. Crossing these regions induces a supercritical period-doubling bifurcation. For constant K, a smaller τd leads to an increased number of collisions and periodic motions in the system. Simulation results demonstrate that by tuning the delay parameters, the multistability during the drilling process can be effectively controlled, thereby enhancing drilling efficiency and stability. Finally, rock drilling experiments confirm the validity of the model and the presence of multistability. When drilling into rocks with high hardness and brittleness, multistable motions are more likely to occur.

Chaotic dynamics and control in a discrete predation model with Holling II and prey refuge effects.

Understanding predator-prey interactions is crucial for modeling ecological systems. This study investigates a discrete-time predator-prey system with a Holling type II functional response and prey refuge. Linear stability analysis establishes existence conditions and stability criteria for equilibrium points, while bifurcation analysis reveals critical transitions through period-doubling and Neimark-Sacker bifurcations. The center manifold theorem facilitates dimensionality reduction, enabling precise characterization of local dynamics near bifurcation points. Comprehensive numerical simulations-including bifurcation diagrams, phase portraits, maximum Lyapunov exponents, and time series-validate theoretical predictions and uncover complex behavioral regimes. A state feedback control strategy, derived from triangular stability regions, effectively suppresses chaotic fluctuations and stabilizes the system. These results advance fundamental understanding of ecological dynamics while offering practical stabilization techniques, bridging mathematical theory with applied ecological management. The interdisciplinary framework provides actionable insights for maintaining balance in complex ecological systems.

Singular attractors of chaotic systems with line equilibrium

Abstract In this paper three examples of systems of nonlinear ordinary differential equations are considered, in which chaotic attractors coexist with lines of unstable singular points. In modern literature on chaotic dynamics such attractors are usually called as “hidden” attractors, as in the cases of systems that have with a chaotic attractor stable singular points or have no singular points at all. It is shown that the transition to chaos in all considered chaotic systems occurs in accordance with the Feigenbaum-Sharkovsky-Magnitskii bifurcation scenario, as in many other chaotic systems of nonlinear differential equations. In all systems, complete or incomplete subharmonic cascades of bifurcations are realized. So, it is proved that “hidden” attractors of systems with lines of unstable singular points are in fact complex singular attractors as they are understood in the FShM theory and in the cases of systems with stable singular points and systems without singular points. That is any considered “hidden” attractor is bounded non-periodic trajectory in a phase space and is the limit of the sequence of cycles of some cascade of the Feigenbaum period doubling bifurcations.

Bifurcation paths and stability of non-unique solutions of the axisymmetric Lamb–Oseen vortex in a pipe of finite-length

The bifurcation structures and flow evolution of the incompressible axisymmetric Lamb–Oseen swirling flow with uniform axial velocity in a finite-length pipe are investigated by nonlinear dynamical systems analysis and direct numerical simulation. A sequence of high-order transcritical bifurcation and Hopf bifurcation points is found on the columnar base flow branch, corresponding to critical swirl ratios derived from linear stability analysis. The transcritical bifurcation leads to a branch of non-unique solutions that consists of an accelerated and a decelerated sub-branch, while the Hopf bifurcation results in a limit-cycle branch exhibiting a period-doubling cascade leading to chaotic solutions in a certain range of swirl ratio and Reynolds number. Fold bifurcations are also discovered for the first time for this type of flow, where two adjacent decelerated branches are connected via the fold bifurcation, making a natural transition between perturbation modes of varying orders. In addition to the base flow, the study discovers and analyzes four distinct axisymmetric flow states: accelerated flow, decelerated flow, limit cycles, and chaos. These solutions are validated through direct numerical simulation of long-term flow evolution along distinct paths, elucidating the transition from the perturbed base flow to various non-columnar flow and vortex breakdown states. This work uncovers the most complete bifurcation diagram and stability properties that have ever been presented of the viscous axisymmetric Lamb–Oseen vortex.

Relative Equilibria of a 4–Particle Ring Close to the 1 : 2 : 4 Resonance

A 4–particle ring with different masses in nearest-neighbour interaction generalizes the spatially periodic Fermi–Pasta–Ulam chain where all masses are equal. For appropriate mass ratios the system is in 1 : 2 : 4 resonance and the 4–particle ring provides for a versal detuning of the 1 : 2 : 4 resonance. The normal form of the system is not integrable, but can be reduced to two degrees of freedom. We determine the relative equilibria and how these behave under detuning.The reduced phase space consists of a singular part in one degree of freedom and a regular part in two degrees of freedom. On the latter the normal form of the 4–particle ring has at most 4 relative equilibria as these are given by the roots of a single quartic polynomial F in one variable. We find a rich bifurcation scenario, with relative equilibria undergoing Hamiltonian flip bifurcations, centre-saddle bifurcations and Hamiltonian Hopf bifurcations. These bifurcations are both approached from a theoretical point of view for general detuned 1 : 2 : 4 resonances and practically compiled to the set of local bifurcations for the normal form of a 4–particle ring passing through the 1 : 2 : 4 resonance.

Dynamical analysis of discrete Monochamus alternatus-Dastarcus helophoroides model with artificially released and spatial diffusion

In this paper, we consider a discrete-time Monochamus alternatus-Dastarcus helophoroides model with artificially released and spatial diffusion. We prove that the eradication of Monochamus alternatus can be achieved by incorporating sufficient quantities of artificially released Dastarcus helophoroides. If [Formula: see text] is satisfied, then Neimark–Sacker bifurcation and Flip bifurcation occur. In this case, Monochamus alternatus and Dastarcus helophoroides can coexist and the populations varies periodically. When considering spatial diffusion, taking the parameter regulating the growth of Dastarcus helophoroides [Formula: see text] as a bifurcation parameter, we prove that Neimark–Sacker bifurcation and 1:2 Resonance and Turing instability occur. From the ecological point of view, Monochamus alternatus and Dastarcus helophoroides populations generally exhibit complex periodic oscillatory behavior in two-dimensional space.

Bi-stability and period-doubling cascade of frequency combs in exceptional-point lasers

Recent studies have demonstrated that a laser can self-generate frequency combs when tuned near an exceptional point (EP), where two cavity modes coalesce. These EP combs induce periodic modulation of the population inversion in the gain medium, and their repetition rate is independent of the laser cavity’s free spectral range. In this work, we perform a stability analysis that reveals two notable properties of EP combs, bi-stability and a period-doubling cascade. The period-doubling cascade enables halving of the repetition rate while maintaining the comb’s total bandwidth, presenting opportunities for the design of highly compact frequency comb generators.

Nonlinear dynamic analysis of a cam with flat-faced follower system considering the guide rail impact

Abstract A two-degree-of-freedom cam-follower rod system with guide rail clearance is investigated. The dynamic response of a follower rod impact guide subjected to two distinct types of external forces is systematically investigated employing nonlinear dynamics methodologies. A comparative analysis is conducted on the transition dynamics between adjacent motion states 1-0-q and 1-0-(q+1) ( ), with particular focus on their bifurcation characteristics and phase-space evolution. The coexistence properties of the periodic motion 1-0-q with the neighboring motions 1-0-(q+1) and the subharmonic motions 2-0-2(q+1), 2-0-(2q+1), 2-0-2q, etc. occurring in the hysteresis transitive domains due to the difference in initial values are revealed. The evolution mechanism of coexisting attractors in multistable systems is investigated, and it is concluded that the reconstruction of the topology of the attractor domain mainly originates from the saddle-node bifurcation and the grazing bifurcation, while the period-doubling bifurcation does not have a significant effect on the morphology of the attractor domain. It is shown that the impact between the follower rod and the guide rail is positively correlated with the cam rotation speed and preload, and negatively correlated with the contact stiffness and contact damping, i.e., the impact mode shifts from the n-0-q type to the n-p-q ( ) type as the rotation speed and the preload increase, but the shift of the impact mode is reversed with the increase of the contact stiffness and the contact damping. And the conclusion provides a reference to the system for the diagnosis of the big data and the collaborative optimization.

Flip Bifurcation and Numerical Study of Crimean-Congo Hemorrhagic Fever with Sustainable Fractional Approach

Flip Bifurcation and Numerical Study of Crimean-Congo Hemorrhagic Fever with Sustainable Fractional Approach

Feigenbaum universality in subcritical Taylor–Couette flow

Feigenbaum universality is shown to occur in subcritical shear flows. Our testing ground is the counter-rotation regime of the Taylor–Couette flow, where numerical calculations are performed within a small periodic domain. The accurate computation of up to the seventh period-doubling bifurcation, assisted by a purposely defined Poincaré section, has enabled us to reproduce the two Feigenbaum universal constants with unprecedented accuracy in a fluid flow problem. We have further devised a method to predict the bifurcation diagram up to the accumulation point of the cascade based on the detailed inspection of just the first few period-doubling bifurcations. Remarkably, the method is applicable beyond the accumulation point, with predictions remaining valid, in a statistical sense, for the chaotic dynamics that follows.

Sensorless anti-control and synchronization of chaos of brushless DC motor driver

Anti-control and synchronization of period doubling and chaos is a method for bifurcation control. It can be used to detect the occurrence or periodic behavior of a bifurcation at the specified position to meet the requirements of brushless direct current (BLDCM). Antichaotic control can be implemented through the use of an external periodic term or constant. The study of the parametric singularities of brushless direct current (BLDCM) allows the identification of a complex bifurcation structure, namely the limit point (LP), the Hopf (H) and the Bogdanov-Takens (BT) bifurcations, the period-doubling bifurcation and path to chaos . By adjusting the control parameters of the controller, the period doubling bifurcation can be generated or suppressed at the specified position to realize the anti-control of period doubling and chaos bifurcation. Parametric singularities are analyzed using a variety of anti-control signals, including constant voltage, periodic square, sawtooth wave, triangle, etc. The simulation results show that adding constant or periodic factors improves chaos and anti-control effects.