International Journal of Bifurcation and Chaos

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as $Q$-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits.

The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on $Q$-switching lasers with different kinds of saturable absorbers.

In this work, we consider a family of Lotka–Volterra maps $(x^{\prime },y^{\prime })=(x(a-x-y),bxy)$ for $a>1$ and $b>0$ which unfold a map originally proposed by Sharkosky for $a=4$ and $b=1$. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region $y<0$. Some properties and bifurcations are described. The $x$-axis is invariant, on which the map reduces to the logistic. For any $a>1$ an interval of values for $b$ exists for which all the cycles on the $x$-axis are transversely attracting. This invariant set is the source of several kinds of bifurcations. Riddling bifurcations lead to attractors in Milnor sense, not topological but with a stable set of positive measure, which may be the unique attracting set, or coexisting with other topological attractors. The riddling and blowout bifurcations are described related to chaotic intervals on the invariant set, and these global bifurcations have different dynamic results. Chaotic intervals which are not topological attractors may have all the cycles transversely attracting and as Milnor attractors. We show that Milnor attractors may also be related to attracting cycles on the $x$-axis at the bifurcation associated with the transverse and parallel eigenvalues. We show particular examples related to topological attractors with very narrow basins of attraction, when the majority of the trajectories are divergent.

Memory elements, including memristor, memcapacitor, meminductor and second-order memristor, have been widely exploited recently to realize circuit systems for a broad scope of applications. This paper introduces a phasor analysis method for memory elements to help with the understanding of the complex nonlinear phenomena in circuits with memory elements. With the proposed method, all different memory elements could be described in a unified form and the series-connected circuit with memristor, memcapacitor, meminductor and second-order memristor could be simply modeled as one variable ${Ż}_M$. Thus, the phasor vectors provided a way to conveniently calculate the $v$–$i$ relation of different memory elements and to clearly understand the similarities and differences between all memory elements. Then some interesting phenomena were introduced when combining different memory elements. Moreover, a specific ${Ż}_M$ with certain $v$–$i$ relations could be easily obtained with the method. And through the inverse calculation, the specific ${Ż}_M$ could be decomposed to a certain combination of memory elements. Meanwhile, the parameters of ${Ż}_M$ in the phasor domain were analyzed. Furthermore, the frequency characteristic for a $R_M{L}_M{C}_M$ circuit could be easily analyzed with the method and a particular series resonance was introduced.

By introducing an absolute value function for polarity balance, some new examples of chaotic systems with conditional symmetry are constructed that have hidden attractors. Coexisting oscillations along with bifurcations are investigated by numerical simulation and circuit implementation. Such new cases enrich the gallery of hidden chaotic attractors of conditional symmetry that are potentially useful in engineering technology.

With both hunting cooperation and Allee effects in predators, a predator–prey system was modeled as a planar cubic differential system with three parameters. The known work numerically plots the horizontal isocline and the vertical one with appropriately chosen parameter values to show the cases of two, one and no coexisting equilibria. Transitions among those cases with the rise of limit cycle and homoclinic loop were exhibited by numerical simulations. Although it is hard to obtain the explicit expression of coordinates, in this paper, we give the distribution of equilibria qualitatively, discuss all cases of coexisting equilibria, and obtain the Bogdanov–Takens bifurcation diagram to show analytical parameter conditions for those transitions. Our results give analytical conditions for not only the observed saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation but also the transcritical and pitchfork bifurcations at the predator-extinction equilibrium, which were not considered in the known work. Our analytic conditions provide a quantitative instruction to reduce the risk of predator extinction and promote the ecosystem diversity.

This paper investigates the effects of the unsteady nonlinear aerodynamic, plunge/pitch cubic nonlinearities, flap free-play nonlinearity, and coupled nonlinear aeroelasticity on the dynamics of the three-dimensional blade section. The dynamic stall model is developed based on the unsteady Wagner aerodynamics. Coupling the developed nonlinear aerodynamic model and nonlinear elasticity model results in the nonlinear aeroelastic model. The nonlinear aeroelastic equation of motion is converted into a state-space form. The resulting nonlinear state-space equation of motion is simulated by a standard Runge–Kutta algorithm in MATLAB. The proposed model is validated against test data of distinct two- and three-degrees-of-freedom studies and is compared to the ONERA model. Bifurcation diagrams show that there is distinct airspeed, in which the system experiences limit cycle oscillations (LCOs) or chaos. Both hysteresis air loads and structural nonlinearity make the system unstable at airspeed less than linear flutter speed. The nonlinearity of the structure causes supercritical pitchfork bifurcation. Elastic-aerodynamic nonlinearity interaction causes sub-supercritical bifurcation at the lower airspeed and chaotic motion at a higher airspeed. Furthermore, the effects of the initial condition on the response of the nonlinear aero-servo-elastic system are investigated by the Lyapunov exponent method.

In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

This paper proposes a novel chaotic jerk system, which is defined on four domains, separated by codimension-2 discontinuity surfaces. The dynamics of the proposed system are conveniently described and analyzed through a generalization of the Poincaré map which is constructed via an explicit solution of each subsystem. This provides an approach to formulate a robust bifurcation problem as a nonlinear inhomogeneous eigenvalue problem. Also, we establish some criteria for the existence of a period-doubling bifurcation and discuss some of the interesting categories of complex behavior such as multiple period-doubling bifurcations and chaotic behavior when the trajectory undergoes a segment of sliding motion. Our results emphasize that the sharp switches in the behavior are mainly responsible for generating new and unique qualitative behavior of a simple linear system as compared to the nonlinear continuous system.

This paper addresses the issues on the dynamics of nonlinear damping gyros subjected to a quintic nonlinear parametric excitation. The fixed points and their stability are analyzed for the autonomous gyros equation. The number of fixed points of the system varies from one to six. The approximate equation of gyros is considered by expanding the nonlinear restoring force and parametric excitation for the study of the dynamics of gyros. Amplitude and frequency of possible resonances are found by using the multiple scales method. Also obtained are the principal parametric resonance and orders 4 and 6 subharmonic resonances. The stability conditions for each of these resonances are also obtained. Chaotic oscillations, multistability, hysteresis, and coexisting attractors are found using the bifurcation diagrams, the Lyapunov exponents, the phase portraits, the Poincaré section and the time histories. The effects of the damping parameter, the angular spin velocity and the parametric nonlinear excitation are analyzed. Results obtained by using the approximate gyros equation are compared to the dynamics obtained with the exact equation of gyros. The analytical investigations are complemented by numerical simulations.

This paper is devoted to study the spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey model with Michaelis-Menten type harvesting in the prey population. The existence and stability of possible non-negative constant equilibria are investigated. By regarding $\rho$ as a bifurcation parameter, the Hopf bifurcation from the positive constant equilibrium solution is investigated. The necessary and sufficient conditions of Turing instability are explicitly obtained. We show that at the critical value of the bifurcation parameter $d_2$ a Turing bifurcation occurs (i.e. a pattern arises). The conditions for the stability of the pattern are also derived in detail. Moreover, the global steady state bifurcation from the positive constant equilibrium solution is investigated. In particular, the local steady state bifurcation from double zero eigenvalues is also obtained by the techniques of space decomposition and the implicit function theorem. Our results show that Michaelis–Menten type harvesting in our model plays a crucial role in the formation of spatiotemporal dynamics, which is a strong contrast to the case without harvesting.

In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

Based on some essential concepts of fractional calculus and the theorem related to the fractional extension of Lyapunov direct method, we present in this paper a synchronization scheme of fractional-order Lur’e systems. A quadratic Lyapunov function is chosen to derive the synchronization criterion. The derived criterion is a suffcient condition for the asymptotic stability of the error system, formulated in the form of linear matrix inequality (LMI). The controller gain can be achieved by solving the LMI. The proposed scheme is illustrated for fractional-order Chua’s circuits and fractional-order four-cell CNN. Numerical results, which agree well with the proposed theorem, are given to show the effectiveness of this scheme.

This note is concerned with the effect of small $C^1$ perturbations on a discrete dynamical system $(X,f)$, which has heteroclinic repellers. The question to be addressed is whether such perturbed system $(X,g)$ has heteroclinic repellers. It will be shown that if $\parallel f-g{\parallel }_{C^1}$ is small enough, $(X,g)$ has heteroclinic repellers, which implies that it is chaotic in the sense of Devaney. In addition, if $X=R^n$ and $(X,f)$ has regular nondegenerate heteroclinic repellers, then $(X,g)$ has regular nondegenerate heteroclinic repellers, where $g$ is a small Lipschitz perturbation of $f$. Three examples are presented to validate the theoretical conclusions.

A general strategy for suppressing chaos in chaotic Burke–Shaw system using integrative time delay (ITD) control is proposed, as an example. The idea of ITD is that the feedback is integrated over a time interval. Physically, the chaotic system responds to the average information it receives from the feedback. The main feature of integrative is that the stability of the chaotic system occurs over a wider range of the space parameters. Controlling chaotic systems with ITD has not been discussed before as far as we know. Stability and the existence of Hopf bifurcation are studied which demonstrate that the switch stability occurs at critical values of the time delay. Employing the normal form theory and center manifold argument, an explicit formula is derived to determine the stability and the direction of the bifurcating periodic solutions. Numerically, the bifurcation diagram and the eigenvalues of the corresponding characteristic equations are computed to supply a clear interpretation for suppressing chaos via ITD. Furthermore, ITD method is compared with the time delayed feedback (TDF) control numerically. This comparison shows that the stability area with ITD is larger than TDF which demonstrates the feasibility and effectiveness of the ITD. Other examples of chaotic systems can be similarly investigated.

It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.

In this paper, we investigate analytically and numerically the dynamics of a modified Leslie–Gower predator–prey model which is characterized by the reduction of prey growth rate due to the anti-predator behavior. We prove the existence and local/global stability of equilibria of the model, and verify the existence of Hopf bifurcation. In addition, we focus on the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also destabilize the coexistence equilibrium, which are beneficial to the occurrence of limit-cycle-induced oscillation, or prevent the occurrence of limit cycle oscillation and increase the stability of the system.

Based on the dispersion entropy model, combined with multiscale analysis method and fractional order information entropy theory, this paper proposes new models — the generalized fractional order multiscale dispersion entropy (GMDE) and the generalized fractional order refined composite multiscale dispersion entropy (GRCMDE). The new models take the amplitude value information of the sequence itself into consideration, which can make better use of some key information in the sequence and have a higher stability and accuracy. In addition, extending the algorithm to generalized fractional order can make the model better capture the small evolution of the signal data, which is more advantageous for studying the dynamic characteristics of complex systems. This paper verifies the effectiveness of the new models by combining theoretical analysis with empirical research, and further studies the complexity of the financial system and the nature of its multiple time scales. The results show that the proposed GMDE, GRCMDE can better detect the intrinsic nature of financial time series and can distinguish the financial market complexity of different countries.

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their $(n-1)$th input trees are isomorphic, where $n$ is the number of nodes. Here $n-1$ is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in $n$.

Chaotic phenomena may exist in nonlinear evolution equations. In many cases, they are undesirable but can be controlled. In this study, we deal with the chaos control of a three-dimensional chaotic system, reduced from a KdV–Burgers–Kuramoto equation. By adding a single delay feedback term into the chaotic system, we investigate the local stability and occurrence of Hopf bifurcation near the equilibrium point. Some dynamical properties including the direction and stability of bifurcated periodic solutions are presented by using the normal form theory and the center manifold theorem. Numerical simulations are illustrated which agree well with the theoretical results.

In this paper, we report the occurrence of sliding bifurcations admitted by the memristive Murali–Lakshmanan–Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their nonlinear element. The three-segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three subregions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the subregions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the $X$, $Y$, and $Z$ state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The $X$ and $Z$ solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function ($\mathrm{\text{sech}}$) and a hyperbolic secant squared function (${\mathrm{\text{sech}}}^2$), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the $Z$ component and the KdV equation were obtained. Additionally, the ODE for the $X$ component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions.