International Journal of Bifurcation and Chaos

The nonlinear dynamical properties of the recently proposed $T\pm 𝜖$ feedback control for suppressing cardiac alternans is investigated in detail for the discrete cardiac restitution map model and the continuous Luo–Rudy I ionic model. It is shown that cardiac alternans, induced by fast pacing, can be dramatically reduced by small changes of fixed magnitude ($𝜖$) under feedback control when $𝜖$ exceeds a critical threshold. Remarkably, the suppression is achieved by utilizing the induced chaotic dynamics. Detailed information on the nature and mechanism of the control, and properties of the attractors are analyzed and discussed in the light of nonlinear dynamics.

Homeostasis is an important and common biological phenomenon wherein an output variable does not change very much as an input parameter is varied over an interval. It can be studied by restricting attention to homeostasis points — points where the output variable has a vanishing derivative with respect to the input parameter. In a feedforward network, if a node has a homeostasis point then downstream nodes will inherit it. This is the case except when the downstream node has a bifurcation point coinciding with the homeostasis point. We apply singularity theory to study the behavior of the downstream node near these homeostasis-bifurcation points. The unfoldings of low codimension homeostasis-bifurcation points are found. In the case of steady-state bifurcation, the behavior includes multiple homeostatic plateaus separated by hysteretic switches. In the case of Hopf bifurcation, the downstream node may have limit cycles with a wide range of near-constant amplitudes and periods. Homeostasis-bifurcation is therefore a mechanism by which binary, switch-like responses or stable clock rhythms could arise in biological systems.

Generalized periodic discharges (GPD) are generalized waveforms that recur with a relatively uniform morphology and duration observed in EEG recordings of many types of metabolic encephalopathy, which are often referred to as epileptiform. In this paper, we try to link these spatiotemporal electrocortical activities and significant attributes of cortex based on Liley’s mean field model, and seek possible generation mechanisms of GPD rhythms from several factors. To these ends, the dynamical properties of simulated EEG consistent with neurophysiological features of human cortex are investigated, among which GPD patterns are our focus. Firstly, with different value sets of model parameters, we reproduce some typical simulation waveforms which are analogous to mammalian normal or abnormal brain activities detected by EEG. Or in other words, we put more emphasis on brain waveforms of GPD states, normal states, and low firing rate states in our numerical simulations, and mode transitions among different firing states are our main interests. Secondly, through analysis of maximum Lyapunov exponents and frequency spectrum, we give several mode transitions by varying synaptic connections between excitatory and inhibitory populations, which support the conjecture that selective changes of synaptic connections can trigger GPD states, such as in excitatory (AMPA receptors) and inhibitory neurotransmitters (GABA receptors). Thirdly, we stress the importance of time delay on neural population connections and find that they are free to transfer among different firing modes with appropriate time delays. Furthermore, considering the effects of external inputs to cerebral cortex, we verify that stimulation can lead to good controls on GPD patterns, including facilitation and elimination. Finally, we show that more dynamical rhythms can be produced when taking into account the cortico-cortical connections. These modeling results are expected to shed light into the pathophysiological mechanisms of GPD modes from a theoretical viewpoint.

This paper focuses on the classification of the bifurcation modes of a Duffing system under the combined excitations of constant force and harmonic excitation. The Harmonic Balance method combined with the arc-length continuation is used to obtain the periodic solutions of the system, and the Floquet theory is employed to analyze the stability of the corresponding solutions. Accordingly, the frequency-response curves affected respectively by the constant force and the magnitude of the harmonic excitation are analyzed to show the basic dynamical properties of the system. Afterwards, the bifurcation investigations are carried out with the aid of the two-state variable singularity method. It is derived that there are a total of six different types of bifurcation modes due to the effects of the constant force and the magnitude of the harmonic excitation. At last, the effects of the nonlinearity parameter and the damping ratio on the bifurcation modes of the system are also discussed. The results obtained in this paper extend the findings in reference that the system can have markedly three types of frequency-response curves: with only one solution, or with maximum three or five solutions for a certain excitation frequency, and contribute to a better understanding of the significant influence of the constant force.

In this paper, a new four-dimensional chaotic flow is proposed. The system has a cyclic symmetry in its structure and shows a complicated, chaotic attractor. The dynamical properties of the system are investigated. The system shows multistability in an interval of its parameter. Fractional order model of the proposed system is discussed in various fractional orders. Bifurcation analysis of the fractional order system shows that it has a kind of multistability like the integer order system, which is a very rare phenomenon. Circuit realization of the proposed system is also carried out to show that it is usable for engineering applications.

The topological entropy of finite representations of the Hénon map is studied. Efficient methods to compute the topological entropy of finite representations of maps are presented. Accurate finite representations of the Hénon map and its iterates are constructed and the topological entropy of these representations is calculated. The relation between the topological entropy of the Hénon map and the topological entropy of its finite representations is discussed.

In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.

In this study, we investigate an enhanced architecture of multicarrier differential chaos-shift keying (MC-DCSK) scheme using quadrature modulation (QM), namely QMC-DCSK. The use of quadrature modulation aims at doubling the data rate over a defined bandwidth and hence improve bandwidth efficiency of the system. In the proposed scheme, the chaotic spreading sequence is transmitted on a predefined frequency while each of the remaining frequencies is phase-shifted at a $90^{\circ }$ angle in order to produce two quadrature subcarriers located at the same frequency. These subcarriers are modulated by the product of the chaotic spreading sequence and the corresponding bit substreams in parallel. Noncoherent demodulation is carried out in the receiver in order to recover the original data based on the sign of correlation values. Architecture and operation of the conventional and proposed schemes are described. The BER performance over a wireless channel is theoretically derived and then numerically verified. The bit rate, energy and bandwidth efficiencies are evaluated in comparison with those of MC-DCSK. In particular, the application of the proposed scheme to cognitive radio (CR) in the scenario of using multicarrier modulation and chaotic spread-spectrum is discussed and evaluated. The obtained results prove that QMC-DCSK is a fit technique for CR communications.

During this analysis, as per natural control approach in pest management, a plant-pest dynamics with biological control is proposed, here assuming that the pest and natural enemy are having different levels of gestation delay and harvesting rate of pests by natural enemy follows Holling type-III response function. Boundedness and positivity of the system are studied. Equilibria and stability analysis is carried out for possible equilibrium points. The existence of Hopf bifurcation at interior equilibrium is presented. The sensitivity analysis of the system at interior equilibrium point for model parameters has been explored. Numerical simulations are performed to support our analytic findings.

Detecting chaos in heavy-noise environments is an important issue in many fields of science and engineering. In this paper, first, a new criterion is proposed to recognize chaos from noise based on the distribution of energy. Then, a new method based on stationary wavelet transform (SWT) is presented for chaos detection that is recommended for data that contain more than 60% noise. This method is dependent on the distribution of signal’s energy in different frequency bands based on SWT for chaos detection which is robust to noisy environments. In this method, the effect of white noise and colored noise on the chaotic system is considered. As a case study, the proposed method is applied to detect chaos in two different oscillators based on memristor and memcapacitor. The simulation results are used to display the main points of the paper.

The modified Chua’s circuit, which is first order differentiable, has degree-of-discontinuity $3$. It has $2n-1$ equilibrium points, including two boundary equilibrium points. For them, except boundary equilibrium points, we obtain in theory, conditions under which Hopf bifurcations exist, which implies coexisting periodic solutions. At the same time, we also show that equilibrium points are asymptotically stable when system parameters are within some limits. Furthermore, we theoretically design a linear feedback controller, which will not change the equilibrium points, with appropriate control parameters to control the dynamical behaviors including chaos to these periodic solutions or equilibrium points, and we verify it by numerical simulations.

This paper proposes a new chaotic system with a specific attractor which is bounded in a sphere. The system is offered in the spherical coordinate. Dynamical properties of the system are investigated in this paper. The system shows multistability, and all of its attractors are inside or on the surface of the specific sphere. Bifurcation diagram of the system displays an inverse period-doubling route to chaos. Lyapunov exponents of the system are studied to show its chaotic attractors and predict its bifurcation points.

In this paper, we discuss the double Hopf bifurcation and the existence of quasi-periodic invariant tori in a generalized Gopalsamy neural network model. Regarding the connection weight and the delay as bifurcation parameters of the double Hopf bifurcation, we derive the normal form up to the fifth order near the critical point by using the center manifold theorem and the normal form method, and obtain sufficient conditions on the existence of invariant 2-tori for the truncated normal form. Moreover, we investigate the effect of higher-order terms on these 2-tori by a KAM theorem. It is proved that in a sufficiently small neighborhood of the bifurcation point, the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where its truncated normal form possesses invariant 2-tori. We give a numerical example to verify the conditions on all results in remarks.

Delay coupled smooth maps over a ring network with global coupling have been considered. Proof of the existence of chaos in the sense of Li–Yorke in the globally coupled smooth maps with homogeneous unit delay has been given by defining a suitable matrix norm. Conditions for the existence of Li–Yorke chaos depending on the coupling strength, map parameter and lattice size have been derived analytically. Chaotic regions in the parameter space are plotted numerically for delay coupled logistic maps.

The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system.

In this paper, we investigate the influence of anti-predator behavior in prey due to the fear of predators with a Beddington–DeAngelis prey–predator model analytically and numerically. We give the existence and stability of equilibria of the model, and provide the existence of Hopf bifurcation. In addition, we investigate 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 prevent the occurrence of limit cycle oscillation and increase the stability of the system.

We study the probabilistic behavior of the Hodgkin–Huxley neuron model in the presence of random forcing of the external current parameter. The stochastic excitement in the zone of stable equilibria is illustrated by the statistics of interspike intervals and probabilistic distributions of mixed-mode oscillations. For the parametric analysis of this phenomenon, a constructive method for stochastic sensitivity and confidence ellipsoids is suggested. It is shown how to simplify this analysis using the principal direction approach. A constructive application of this technique is demonstrated by analyzing the stochastic excitement in the Hodgkin–Huxley model.

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden.

Bifurcation diagrams and Lyapunov exponents are the main tools for dynamical systems characterization. However, they are often computationally expensive and complex to calculate. We present two approaches for dynamical characterization of nonlinear systems via the generation of an undirected complex network that is built from their time series. Periodic windows and chaos can be detected by analyzing network statistics like average degree, density and betweenness centrality. Results are assessed in two discrete time nonlinear maps.

This paper takes the reaction–diffusion approach to deal with the quiescent females phase, so as to describe the dynamics of invasion of aedes aegypti mosquitoes, which are divided into three subpopulations: eggs, pupae and female. We mainly investigate whether the time of quiescence (delay) in the females phase can induce Hopf bifurcation. By means of analyzing the eigenvalue spectrum, we show that the persistent positive equilibrium is asymptotically stable in the absence of time delay, but loses its stability via Hopf bifurcation when time delay crosses some critical value. Using normal form and center manifold theory, we investigate the stability of the bifurcating branches of periodic solutions and the direction of the Hopf bifurcation. Numerical simulations are carried out to support our theoretical results.