zero-Hopf Bifurcation Research Articles

First Integrals and a Zero-Hopf Bifurcation of the Four-Dimensional Lotka-Volterra Systems

In this paper, the integrability and a zero-Hopf bifurcation of the four-dimensional Lotka-Volterra systems are studied. The requirements for this kind of system's integrability and a line of singularities with two zero eigenvalues are provided. We identify the parameters that lead to a zero-Hopf equilibrium point at each point along the line of singularities. We show that there is only one parameter that displays such equilibria. The first-order averaging method is also employed, although this method will not give any information about the bifurcate periodic solutions that bifurcate from the zero-Hopf equilibria.

Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations.

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

Limit cycle bifurcation from a zero-Hopf equilibrium for a class of 3-dimensional Kolmogorov systems

A zero-Hopf equilibrium point p of a 3-dimensional autonomous differential system in R3 is an equilibrium point such that the eigenvalues of the linear part of the system at p are 0 and ±ωi with ω≠0. A zero-Hopf bifurcation takes place when from a zero-Hopf equilibrium bifurcate some small-amplitude limit cycles moving the parameters of the system.Polynomial Kolmogorov systems are the natural extension of the polynomial Lotka–Volterra systems of degree 2 to higher degree. The Kolmogorov systems have been studied intensively due to their applications for modeling many natural phenomena.In this paper we characterize, up to first order in the averaging theory, the eight distinct zero-Hopf bifurcations which can exhibit the class of 3-dimensional Kolmogorov systems of degree 3, providing an explicit approximation of the bifurcated small-amplitude limit cycles, together with information about their kind of stability.From each one of this eight different zero-Hopf bifurcations emerges one or two limit cycle using the averaging theory of first order. Moreover we provide an explicit example of each one of these eight zero-Hopf bifurcations.

Hopf and Zero-Hopf Bifurcation Analysis for a Chaotic System

Hopf and Zero-Hopf Bifurcation Analysis for a Chaotic System

Periodic orbits and non-existence of $$C^1$$ first integrals for analytic differential systems exhibiting a zero-Hopf bifurcation in $$\mathbb {R}^4$$

Periodic orbits and non-existence of $$C^1$$ first integrals for analytic differential systems exhibiting a zero-Hopf bifurcation in $$\mathbb {R}^4$$

Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior.

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.

Zero Hopf Bifurcation and Chaotic Behavior in a Leslie Tritrophic Model

Zero Hopf Bifurcation and Chaotic Behavior in a Leslie Tritrophic Model

The dud canard: Existence of strong canard cycles in [formula omitted

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R3 through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove – in the analytic case only – that for all 0<ϵ≪1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to O(1) cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in R2, but in contrast to the planar case, the family of periodic orbits in R3 is not explosive. For this reason, we have chosen to call the phenomena in R3, the “dud canard”. The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in R2, we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit ϵ=0. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic.

Zero-Hopf bifurcation and ultimate boundness of an asymmetrical hyperchaotic Lorenz system

The complex dynamics of a newly proposed 4D hyperchaotic Lorenz-type system are studied in this paper. The sufficient conditions for the emergence and stability of periodic solutions at bifurcation points are derived using averaging theory. The ultimate boundedness of the proposed system is also established.

The zero-Hopf bifurcations of a new hyperchaotic system

Abstract The aim of this work is to study the existence of zero-Hopf bifurcations of a new hyperchaotic system, using the averaging theory of dynamical systems of second order. Furthermore, at most one periodic orbit can bifurcate from the origin of coordinates.

Complex Dynamics of Rössler–Nikolov–Clodong O Hyperchaotic System: Analysis and Computations

This paper discusses the analysis and computations of chaos–hyperchaos (or vice versa) transition in Rössler–Nikolov–Clodong O (RNC-O) hyperchaotic system. Our work is motivated by our previous analysis of hyperchaotic transitional regimes of RNC-O system and the results recently obtained from another researchers. The analysis and numerical simulations show that chaos–hyperchaos transition in RNC-O system is coupled to change in the equilibria type as one large hyperchaotic attractor occurs. Moreover, we show that for this system, a zero-Hopf bifurcation is not possible. We also consider the cases when the divergence of the system is a constant and detected two families of exact solutions.

BISTABILITY, GENERALIZED AND ZERO-HOPF BIFURCATIONS IN A PEST CONTROL MODEL WITH FARMING AWARENESS

Introduction of biopesticides and predators in the crop fields has a crucial impact on the pest management as the biopesticides induce infection among the pest population and the infected pests are easy to be captured by the predators. In this paper, we investigate a mathematical model for a pest management system with farming awareness-based inclusion of predators and biopesticides. The level of awareness among the farmers is assumed to grow at a constant rate besides its growth due to an increase in the pest population in the crop field. The proposed model is analyzed mathematically for the feasibility and stability of the equilibrium points; existence of Hopf bifurcation together with direction and stability of the bifurcating periodic solutions are discussed; sufficient conditions are derived for which the system does not exhibit any periodic solution. Our simulation results show that the activeness of farmers for employing biopesticides in the crop field potentially suppresses the oscillations and brings forth stability in the system. Further, we observe that the system showcases the appearances of Generalized Hopf bifurcation, Zero-Hopf bifurcation, transcritical bifurcation and bistability. Overall, our findings manifest that by imparting knowledge among the farmers regarding the use of biopesticides and predators in the crop fields, the pest population can be effectively controlled.

Pattern Formation in a Reaction-Diffusion BAM Neural Network With Time Delay: (k1, k2) Mode Hopf-Zero Bifurcation Case.

This article investigates the joint effects of connection weight and time delay on pattern formation for a delayed reaction-diffusion BAM neural network (RDBAMNN) with Neumann boundary conditions by using the (k1,k2) mode Hopf-zero bifurcation. First, the conditions for k1 mode zero bifurcation are obtained by choosing connection weight as the bifurcation parameter. It is found that the connection weight has a great impact on the properties of steady state. With connection weight increasing, the homogeneous steady state becomes inhomogeneous, which means that the connection weight can affect the spatial stability of steady state. Then, the specified conditions for the k2 mode Hopf bifurcation and the (k1,k2) mode Hopf-zero bifurcation are established. By using the center manifold, the third-order normal form of the Hopf-zero bifurcation is obtained. Through the analysis of the normal form, the bifurcation diagrams on two parameters' planes (connection weight and time delay) are obtained, which contains six areas. Some interesting spatial patterns are found in these areas: a homogeneous periodic solution, a homogeneous steady state, two inhomogeneous steady state, and two inhomogeneous periodic solutions.

Combined influences of environmental enrichment and harvesting mediate rich dynamics in an intraguild predation fishery system

Natural resources and its enrichment activities are often being associated with the paradoxical effects in ecology such as in predator–prey and fishery models. Several studies have shown that an increase in resources can cause the “Paradox of Enrichment” i.e., the destabilization of certain species population as the system is enriched. This situation is rather contradictory to our logical intuition. Parallel with this view, harvesting effort and the intensity of enrichment activities are among crucial factors in shaping the sustainability of fishery activities and natural resources. This is because an extreme harvesting effort could lead to resource depletion and threaten vulnerable marine ecosystems. In this work, we investigate the joint effects of enrichment and harvesting activities in an intraguild predator–prey fishery model that incorporates the Michaelis–Menten type of harvesting of predatory fishes. We study the existence and stability of the steady states in this system analytically and conduct numerical simulations, together with bifurcation analysis to verify our analytical findings. Our co-dimension one bifurcation analysis shows that the system can exhibit local bifurcations such as transcritical, saddle–node, subcritical, and supercritical Hopf bifurcations. Co-dimension two bifurcation analysis further demonstrates the existence of global dynamics through the appearance of Generalized Hopf (GH) and Zero-Hopf (ZH) bifurcations, which occur due to the interactions of distinct local bifurcations in this ecological system. The limit point bifurcation of cycles that emanates from the GH point has a bounded parameter space region in which tristability phenomenon (between stable steady states and stable limit cycles) occurs. In a scenario where the resources are enriched in the presence of harvesting activity, bistability emerges with the long-term outcomes lead to the exclusion of either prey or predator population, depending on the initial conditions of species. Based on these findings, we propose another interpretation to the “Paradox of Enrichment” as postulated in ecological theories: this predator–prey fishery system can be destabilized through an alternative stable states phenomenon, which results in the extinction of certain species as the ecosystem is further enriched.

Bursting dynamics and the zero-Hopf bifurcation of simple jerk system

We characterize the zero-Hopf bifurcation at a singular point of a parameter jerk system. By employing the second order averaging theory, we demonstrate that up to three periodic orbits generated as disturbance parameters tend to zero. Again, both the bifurcation mechanism and bursting dynamics of the 3D jerk system with external periodic excitation are systematically explored. While an order difference exists between the frequency of external excitation and the average frequency of the system, the system exhibits bursting oscillations. The mechanisms of different bursting oscillations are investigated by means of the equilibrium point curve and the transformed phase portraits. As the amplitudes of the excitation change, the system displays “delayed symmetric pitchfork/point, delayed symmetric pitchfork/supHopf, and delayed pitchfork/supHopf/homoclinic connection bursting”. Finally, an analog circuit is designed to verify the complex bursting phenomena of the system.

Slow–Fast Dynamics in a Non-smooth Vector Field with Zero-Hopf Bifurcation

Slow–Fast Dynamics in a Non-smooth Vector Field with Zero-Hopf Bifurcation

Zero-Hopf bifurcation in the general Van der Pol-Duffing equation

In this work we study the invariant sets which emerge from the zero-Hopf bifurcations that general Van der Pol-Duffing equations can exhibit. We provide sufficient conditions for the simultaneous bifurcation of three periodic solutions and two invariant torus from the origin of this system. We use recent results related to the averaging method in order to analytically obtain our results. We also provide numerical examples for all the analytical results that we provide.

Algebraic Analysis of Zero-Hopf Bifurcation in a Chua System

This article first studies the stability conditions of a Chua system depending on six parameters. After, using the averaging method, as well as the methods of the Gröbner basis and real solution classification, we provide sufficient conditions for the existence of three limit cycles bifurcating from a zero-Hopf equilibrium of the Chua system. As we know, this last phenomena is first found. Some examples are presented to verify the established results.

Zero-Hopf bifurcation of periodic orbits in the generalized Rössler system

We apply a technique of Llibre based on the averaging method to a Rössler-type three-dimensional quadratic system and we prove the existence of a periodic orbit.

Chaotic RG flow in tensor models

We study a bi-antisymmetric tensor quantum field theory with $O(N_1)\times O(N_2)$ symmetry. Working in $4-\epsilon$ dimensions we calculate the beta functions up to second order in the coupling constants and analyze in detail the Renormalization Group (RG) flow and its fixed points. We allow $N_1$ and $N_2$ to assume general real values and treat them as bifurcation parameters. In studying the behavior of the model in the space of $N_1$ and $N_2$ we find a point where a zero-Hopf bifurcation occurs. In the vicinity of this point, we provide analytical and numerical evidence for the existence of Shilnikov homoclinic orbits, which induce chaotic behavior in the RG flow of the model. As a simple warm-up example for the study of chaotic RG flows, we also review the non-hermitian Ising chain and show how for special complex values of the coupling constant, its RG transformations are chaotic and equivalent to the Bernoulli map.