Non-hyperbolic Equilibrium Point Research Articles

Prey group defense and hunting cooperation among generalist-predators induce complex dynamics: a mathematical study.

Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense. Additionally, we allow a modification in the attack rate of predators to quantify the hunting cooperation among them. The model admits three boundary equilibria and up to three coexistence equilibrium points. The geometry of the nontrivial prey and predator nullclines and thus the number of coexistence equilibria primarily depends on a specific threshold of the availability of alternative food for predators. We use linear stability analysis to determine the types of hyperbolic equilibrium points and characterize the non-hyperbolic equilibrium points through normal form and center manifold theory. Change in the model parameters leading to the occurrences of a series of local bifurcations from non-hyperbolic equilibrium points, namely, transcritical, saddle-node, Hopf, cusp and Bogdanov-Takens bifurcation; there are also occurrences of global bifurcations such as homoclinic bifurcation and saddle-node bifurcation of limit cycles. We observe two interesting closed 'bubble' form induced by global bifurcations due to change in the strength of hunting cooperation and the availability of alternative food for predators. A three dimensional bifurcation diagram, concerning the original system parameters, captures how the alternation in model formulation induces gradual changes in the bifurcation scenarios. Our model highlights the stabilizing effects of group or gregarious behaviour in both prey and predator, hence supporting the predator-herbivore regulation hypothesis. Additionally, our model highlights the occurrence of "saltatory equilibria" in ecological systems and capture the dynamics observed for lion-herbivore interactions.

Basins of Attraction of Equilibrium Points of Monotone Difference Equations

We investigate the global character of the difference equation of the form $$ x_{n+1} = f(x_n, x_{n-1},\ldots, x_{n-k+1}), \quadn=0,1, \ldots $$ with several equilibrium points, where $f$ is increasing in all its variables. We show that a considerable number of well known difference equations can be embeded into this equation through the iteration process. We also show that a negative feedback condition can be used to determine a part of the basin of attraction of different equilibrium points, and that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points. 2000 Mathematics Subject Classification. 39A10, 39A11

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

In this work, we introduce a simple method for investigating the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.

A New Simple 6D Hyperchaotic System with Hyperbolic Equilibrium and Its Electronic Circuit

This paper represents a sequential design of controlling from a 3D chaotic system to a simple 6D hyperchaotic system by adding three linear controllers via state feedback control and coupling strategies. The new 6D system has a unique non-hyperbolic equilibrium point with four positive Lyapunov exponents (LE). Moreover, some dynamic properties are analyzed which inclined equilibrium and their stability, dissipative, conservative, and multistability. In addition, the corresponding analog electronic circuit is designed and implemented to verify the new simple 6D hyperchaotic system. Finally, the NI Multisim simulation results observed from the digital oscilloscope are consistent with the mathematical simulation based on the Matlab R2021a.

Stability and Bifurcation Analysis of Two-Species Prey–Predator Model Incorporating External Factors

Constant prey refuge with immigration and harvesting in two species would result in significant diversity in the dynamics of a prey–predator population. The phrase “refuge” increases the likelihood of prey population survival in the face of a predator population. Based on these findings, we created and examined a two-species prey–predator system with immigration and harvesting factors, including refuge to only prey population. All ecologically possible equilibrium points are studied for the proposed system. Routh–Hurwitz stability criterion is used for local stability analysis. Global stability of the interior equilibrium point is examined with a suitable Lyapunov function. Local bifurcation of the proposed system, such as saddle-node bifurcation, is analyzed. The conditions for the emergence of this bifurcation at the critical threshold near the nonhyperbolic equilibrium point are established by utilizing Sotomayor’s theorem. The transversality condition is validated for the occurrence of Hopf-bifurcation. The first Lyapunov number is exploited for determining the nature of Hopf bifurcating periodic solution. Finally, numerical simulations are illustrated to validate our theoretical predictions.

The macrodynamics of an endogenous business cycle model of marxist inspiration

This study offers a model that formalizes some of Marx's insights about how capital accumulation generates contradictions that may reproduce never-ending cycles of booms and slumps. The model takes the reserve army of labor as a regulator of the distribution of class-power over the business cycle with a two-sided role: influencing labor productivity, directly through the intensity of labor and indirectly through real wages. The model forms a complex dynamical system capable to yield trajectories for the employment rate, the wage share, and the intensity of labor. Goodwin (1967) model may be considered as a particular case of the model. Complex dynamics may also emerge when we remove some key assumptions and explore and simulate 3-D versions of the system. Though close orbits around non-hyperbolic equilibrium points can be obtained, the possibility of unstable dynamics with increasing amplitudes in the trade cycle and a structural crisis cannot be ruled out.

GLOBAL DYNAMICS OF A PREY–PREDATOR MODEL WITH HOLLING TYPE III FUNCTIONAL RESPONSE IN THE PRESENCE OF HARVESTING

In this paper, we have investigated global dynamics of a two-species food chain model with the Holling type III functional response that includes linear harvesting for the prey and nonlinear harvesting for the predator. The long-time continued existence of both species is discussed using uniform persistence theory. Stability of various equilibrium points is described in terms of model parameters. The local asymptotic stability of non-hyperbolic equilibrium points is determined with the help of center manifold theorem. Global behavior of solutions of the model system when both species are present is determined by considering the global properties of the coexistence equilibrium. Here, we have taken a comprehensive view by considering different bifurcations of co-dimension one and two and have discussed the importance of various model parameters on the system dynamics. The model system shows much more complex and realistic behavior compared to a model system without any harvesting, with constant harvesting or linear-yield harvesting of either or both of the species. Numerical simulations have been conducted to illustrate the theoretical findings.

A New Chaotic System with Only Nonhyperbolic Equilibrium Points: Dynamics and Its Engineering Application

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.

Bifurcation analysis of a predator-prey system with density dependent disease recovery

The center manifold is an invariant manifold that plays a crucial role in the bifurcation analysis of dynamical systems. The center manifold existence theorem assures the local existence of an invariant submanifold of the state space of a dynamical system around a non-hyperbolic equilibrium point. Center manifold theory is essential in the reduction of different bifurcation scenarios to their normal forms. Our study focuses on a predator-prey interactive system with density-dependent growth in predators subject to a contagious disease. The disease is assumed to be horizontally transmitted, and the rate of recovery of the infected predator is assumed to be density-dependent. At the trivial (zero) equilibrium, the center manifold is calculated whose dynamical behaviour is similar to that of the original system. Further, using the center manifolds, the normal form of a Hopf bifurcation point is determined fromwhich the criticality of the system can be deduced. Finally, numerical simulations are performed with biologically plausible parameters to substantiate the analytical findings. Using numerical continuation methods we detect Generalized Hopf and Zero-Hopf bifurcation points. We discuss their normal form coefficients, compute their two-parameter unfoldings and relate these results to the mathematical theory of codimension two bifurcations.

Local and Global Stability of Certain Mixed Monotone Fractional Second Order Difference Equation with Quadratic Terms

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

Stability of Non-Hyperbolic Equilibrium Point for Polynomial System of Differential Equations

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

Dynamical system analysis of self-interacting three-form field cosmological model: stability and bifurcation

The present work deals with Cosmological model of a three-form field, minimally coupled to gravity and interacting with cold dark matter in the background of flat FLRW space-time. By suitable choice of the dimensionless variables, the evolution equations are converted to an autonomous system and cosmological study is done by dynamical system analysis. The critical points are determined and the stability of the (non-hyperbolic) equilibrium points are examined by center manifold Theory. Possible bifurcation scenarios have been examined by the Poincaré index theory to identify possible cosmological phase transition. Also stabilities of the critical points have been analyzed globally using geometric features.

Study of Stability and Bifurcation of Three Species Food Chain Model with Non-monotone Functional Response

In this research article, we consider a tri-trophic food chain model with one prey and two predators such as- prey, intermediate predator and top predator. In this model, the prey and intermediate predator follows non-monotonic functional response; top predator consumes prey and intermediate predator following Holling type I functional response. The positivity, boundedness of solutions of the proposed model and stability conditions of different equilibrium points are discussed here. Then using Center Manifold theorem, the nature of non-hyperbolic type equilibrium points are discussed. After that, different local bifurcations such as- Saddle-node, Transcritical and Hopf bifurcations are studied theoretically as well as numerically by considering half-saturation constant and death rate of intermediate predator as the bifurcation parameters. Finally, the dynamics of the proposed model has been illustrated with the help of some numerical simulations.

Linear and center manifold analysis of FRW cosmological model with variable equation of state in Lyra geometry

The present work deals with the dynamical evolution of flat FRW cosmological model with variable equation of state in Lyra geometry. We investigate the phase-plane analysis of our constructed model. The evolution of cosmological solutions is studied by using dynamical system techniques. Stability and viability of the model are discussed using linear Jacobian stability analysis for hyperbolic equilibrium points and center manifold theory for non-hyperbolic equilibrium points.

Qualitative Stability Analysis of a Non-Hyperbolic Equilibrium Point of a Caputo Fractional System

In this manuscript a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.

Dynamical Analysis of a Novel 4-D Hyper-Chaotic System With One Non-Hyperbolic Equilibrium Point and Application in Secure Communication

In this article, a novel hyper-chaotic system has been introduced and its dynamical properties (i.e., phase plots, time series, lyapunov exponents, bifurcation diagrams, equilibrium points, Poincare sections, etc.) have been studied. Also, the novel chaotic systems have been synchronized using novel synchronization technique multi-switching compound difference synchronization and its application have been shown in the field of secure communication. Numerical simulations have been undertaken to validate the efficacy of the synchronization in secure communication.

5D Hyper-Chaotic System with Multiple Types of Equilibrium Points

A chaotic system with various equilibrium types has rich dynamic behaviors. Its state can switch flexibly among different families of attractors, which is beneficial to the practical applications. So it has been widely concerned in recent years. In this paper, a new 5D hyper-chaotic system is proposed. The important characteristic of the system is that it may have multiple types of equilibrium points by changing system parameters, namely, linear equilibrium point, no equilibrium point, non-hyperbolic unstable equilibrium point and stable hyperbolic-type equilibrium point. Furthermore, there are hyper-chaotic phenomena and multi-stability about the coexistence of multiple chaotic attractors and the coexistence of hyper-chaotic attractors and chaotic attractors in the system. In addition, the system’s complexity is analyzed. It is found that the complexity is close to 1 in the hyper-chaotic state and a pseudo-random sequence generated by the system passes all the statistical tests. Finally, an analog circuit of the system is designed and simulated.

Global Dynamics of Certain Mix Monotone Difference Equation via Center Manifold Theory and Theory of Monotone Maps

We investigate the global dynamics of the following rational difference equation of second order\begin{equation*}x_{n+1}=\frac{Ax_{n}^{2}+Ex_{n-1}}{x_{n}^{2}+f},\quad n=0,1,\ldots ,\end{equation*}where the parameters $A$ and $E$ are positive real numbers and the initial conditions $x_{-1}$ and $x_{0}$ are arbitrary non-negative real numbers such that $x_{-1}+x_{0}>0$. The transition function associated with the right-hand side of this equation is always increasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric values. The unique feature of this equation is that the second iterate of the map associated with this transition function changes from strongly competitive to strongly cooperative. Our main tool for studying the global dynamics of this equation is the theory of monotone maps while the local stability is determined by using center manifold theory in the case of the nonhyperbolic equilibrium point.

Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalized‐type top predator

In this article, we have considered a continuous tritrophic food chain model subject to the Allee effect on the prey growth function with prey-dependent Holling type II functional response between the prey and intermediate predator; Crowley-Martin senses functional response between intermediate predator and top predator, and the top predator is of sexually reproductive type. We have established the positivity and boundedness of the system and the condition of existence of different equilibrium points. The local and global stability of the solutions about the various equilibrium points has been investigated. The center manifold theorem has been used to find the nature of the solution in the neighborhood of the nonhyperbolic equilibrium points and the direction of Hopf bifurcations. Numerical simulation has been carried out to establish the theoretical findings and finally some concluding remarks are given.

A novel 4D chaotic system with nonhyperbolic hyperbolic shape equilibrium points: Analysis, circuit implementation and color image encryption

This paper presents a new 4D chaotic system by extending a 3D system to investigate chaotic systems2 with hidden attractors further and explore their engineering applications in color image encryption. The system has hyperbolic shape equilibriums and the equilibriums are nonhyperbolic which is interesting and rarely mentioned before. The system has been investigated deeply by phase plot, time series, the largest Lyapunov exponent, bifurcation diagram, Poincaré map and 0–1 test. A coexisting hidden attractor has been found in this new system and the circuit simulation model of the system is built by Power Simulation (PSIM). Circuit simulation results are consistent with results of numerical simulations which verify its dynamic behavior further. Then, by combining the generalized Arnold transform with the new chaotic system, a new color image encryption algorithm has been proposed which has large key space and can encrypt either square images or nonsquare images with different size. Finally, the results of encryption experiment and security analysis prove the effectiveness of the algorithm.