Complex dynamic behaviors of a discrete-time predator-prey system

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto''s chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Bifurcations and chaos control in a discrete-time predator-prey system of Leslie type

We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $\R^{2}_+$ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.     

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopfand homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits.interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.     

Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type III functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of $\mathbb{R}_+^2$. Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-$2$, $4$, $8$, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Complex dynamics of a discrete-time predator-prey system with Holling IV functional response

In this paper, a discrete-time predator-prey system with Holling-IV functional response is studied. We first classify the existence of the fixed points of the system, and further investigate their local stabilities. Then the local bifurcation theory for maps is applied to explore the variety of dynamics of the system. Sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation are provided. Numerical results demonstrate that the system may have more complex dynamical behaviors including multiple periodic orbits, quasi-periodic orbits and chaotic behavior. The maximum Lyapunov exponent and sensitivity analysis also confirm the chaotic dynamical behaviors of the system.     

Robust chaos in a credit cycle model defined by a one-dimensional piecewise smooth map

We consider a family of one-dimensional continuous piecewise smooth maps with monotone increasing and monotone decreasing branches. It is associated with a credit cycle model introduced by Matsuyama, under the assumption of the Cobb-Douglas production function. We offer a detailed analysis of the dynamics of this family. In particular, using the skew tent map as a border collision normal form we obtain the conditions of abrupt transition from an attracting fixed point to an attracting cycle or a chaotic attractor (cyclic chaotic intervals). These conditions allow us to describe the bifurcation structure of the parameter space of the map in a neighborhood of the boundary related to the border collision bifurcation of the fixed point. Particular attention is devoted to codimension-two bifurcation points. Moreover, the described bifurcation structure confirms that the chaotic attractors of the considered map are robust, that is, persistent under parameter perturbations.     

Stability and Neimark-Sacker bifurcation of a semi-discrete population model

In this paper, a semi-discrete model is derived for a nonlinear simple population model, and its stability and bifurcation are investigated by invoking a key lemma we present. Our results display that a Neimark-Sacker bifurcation occurs in the positive fixed point of this system under certain parametric conditions. By using the Center Manifold Theorem and bifurcation theory, the stability of invariant closed orbits bifurcated is also obtained. The numerical simulation results not only show the correctness of our theoretical analysis, but also exhibit new and interesting dynamics of this system, which do not exist in its corresponding continuous version.     

Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

Complex dynamics in a two-dimensional noninvertible map

A two-dimensional noninvertible map is investigated. The conditions of existence for pitchfork bifurcation, flip bifurcation and Naimark–Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. Chaotic behavior in the sense of Marotto’s definition of chaos is proven. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including period-34, period-5 orbits, quasi-period orbits, intermittency, boundary crisis as well as chaotic transient. The computation of Lyapunov exponents conforms the dynamical behaviors.     

Bifurcations and chaos in Mira 2 map

In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.     

Bifurcation and chaotic behavior of a discrete-time predator–prey system

The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant ${\mathbb{R}}_{+}^2$. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of ${\mathbb{R}}_{+}^2$ by using a center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits of period 7, 14, 21, 63, 70, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, quasi-periodic orbits and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Generation and suppression of a new hyperchaotic nonlinear model with complex variables

In this paper, we introduce a new hyperchaotic complex Chen model. This hyperchaotic complex system is constructed by adding a complex nonlinear term to the third equation of the chaotic complex Chen system with consideration it’s all variables are complex. The new system is a 6-dimensional continuous real autonomous hyperchaotic system. The properties of this system including invariance, dissipation, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams and hyperchaotic behavior are studied. Different forms of hyperchaotic complex Chen systems are constructed. We suppress the hyperchaotic behavior of our system via passive control method by using one complex controller. The hyperchaotic attractors of the new system are converted to its unstable trivial fixed point and tracked to its unstable non trivial fixed points and periodic orbits. Block diagrams of our system are designed by using Matlab/Simulink after and before the suppression process to ensure the validity of the analytical results.     

Bifurcation analysis in a discrete differential-algebraic predator–prey system

The discrete-time predator–prey biological economic system obtained by Euler method is investigated. Some conditions for the system to undergo flip bifurcation and Neimark–Sacker bifurcation are derived by using new normal form of differential-algebraic system, center mainfold theorem and bifurcation theory. Numerical simulations are given to show the effectiveness of our results and also to exhibit period-doubling bifurcation in orbits of period 2, 4, 8 and chaotic sets. The results obtained here reveal far richer dynamics in discrete differential-algebraic biological economic system. The contents are interesting in mathematics and biology.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Box dimension of Neimark–Sacker bifurcation

In this article we show how a change of a box dimension of orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a non-hyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the centre manifold. We also consider a planar discrete system undergoing a Neimark–Sacker bifurcation. It is shown that box dimension depends on the order of non-degeneracy at the non-hyperbolic fixed point and on the angle–displacement map. As it was expected, we prove that the box dimension is different in the rational and irrational case.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Bifurcation analysis and chaos control in a discrete‐time plant quality and larch budmoth interaction model with Ricker equation

We investigate the dynamics of two‐dimensional discrete‐time model of leaf quality and larch budmoth interaction with Ricker equation. More precisely, the qualitative behavior of larch budmoth model is discussed in which the effect of food source upon the moth population is through intrinsic growth rate. We find the parametric conditions for local asymptotic stability of the unique positive fixed point. It is also proved that under certain parametric conditions, the system undergoes period‐doubling bifurcation with the help of center manifold theory. The parametric conditions for existence and direction of Neimark‐Sacker bifurcation at positive fixed point is investigated with the help of standard mathematical techniques of bifurcation theory. The chaos control in the system is discussed through implementation of hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long‐term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.     

Dynamics of Two Point Vortices in an External Compressible Shear Flow

This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincaré map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.     

HARMONIC AND SUBHARMONIC BIFURCATION IN THE BRUSSEL MODEL WITH PERIODIC FORCE

1.IntroductionABrusselatorisoneofthebestexaminedmodelchemicalreactionswhichconsistsoffourstepsItisshowninFig.1schematicallyandisrepreselltedbythefollowingsetofequationsffevedFebruary6,1995.*~workissupportedbytheNationalNaturalScienceFOundationmanYuan"TermsinChina.ThemodelweadoptistheoneduetoPrigogine,Lefever,andNicolis(Brusselator)t'.Fig.1.'TheschematicdiagramofBrusselmodel(AdditionalcirculararrowsrepreseDttheexistenceofautocatalysis.)Herexandystandfortheconcentrationsofreferencereacta…