International Journal of Bifurcation and Chaos

We consider a class of simple two-dimensional discrete models representative of a system incorporating both mutation and predation. A selection of analytic and numerical results are presented, classifying the dynamic behavior of the system by means of Lyapunov exponents over a biologically-reasonable region of parameter space, and illustrating the occurrence of hyperchaos and a Neimark–Sacker bifurcation producing regions of quasiperiodicity.

In this paper, we study the dynamics and bifurcation properties of a three-parameter family of 1D Gompertz's growth functions, which are defined by the population size functions of the Gompertz logistic growth equation. The dynamical behavior is complex leading to a diversified bifurcation structure, leading to the big bang bifurcations of the so-called “box-within-a-box” fractal type. We provide and discuss sufficient conditions for the existence of these bifurcation cascades for 1D Gompertz's growth functions. Moreover, this work concerns the description of some bifurcation properties of a Hénon's map type embedding: a “continuous” embedding of 1D Gompertz's growth functions into a 2D diffeomorphism. More particularly, properties that characterize the big bang bifurcations are considered in relation with this coupling of two population size functions, varying the embedding parameter. The existence of communication areas of crossroad area type or swallowtails are identified for this 2D diffeomorphism.

In this paper, a traditional five-level cascaded H-bridge inverter is studied and regulated by a proportional-resonant (PR) controller. In order to extend the range of the gain of PR controller, for the purpose of achieving a fast response, a time-delayed feedback controller (TDFC) is used. Similar to the pulse width modulation (PWM) current-mode single phase H-bridge inverter that exhibits bifurcation and chaos when parameters vary, we demonstrate for the first time that the cascaded H-bridge inverter also shows similar features. From the perspective of a discontinuous map, the cascaded H-bridge inverter generally displays extraordinary complexity. Moreover, a new virtual ergodic method (VEM) is proposed to establish the mathematical model of the whole system, which helps to understand the observed bifurcation phenomena. Simulation results are given to verify the analysis.

Multiple chaotic attractors, implying several independent chaotic attractors generated simultaneously in a system from different initial values, are a very interesting and important nonlinear phenomenon, but there are few studies that have previously addressed it to our best knowledge. In this paper, we propose a polynomial function method for generating multiple chaotic attractors from the Sprott B system. The polynomial function extends the number of index-2 saddle foci, which determines the emergence of multiple chaotic attractors in the system. The analysis of the equilibria is presented. Two coexisting chaotic attractors, three coexisting chaotic attractors and four coexisting chaotic attractors are investigated for verifying the effectiveness of the method. The chaotic characteristics of the attractors are shown by bifurcation diagrams, Lyapunov exponent spectrum and phase portraits.

The chaotic delta sigma modulator is a way to suppress the limit cycle oscillation caused by the input of a null or constant signal. By changing integrator gains, the output sequence becomes chaotic and the noise characteristic of the output is changed. This noise characteristic is an important factor for evaluating the performance of delta sigma modulation. The aim of this study is to analyze the noise characteristic of chaotic double loop delta sigma modulation when the null signal is input. We use a bifurcation diagram and FFT analysis to obtain the parameter dependence of the output state and noise characteristic, respectively. The output status of the chaotic double loop delta sigma modulation can be guessed from a bifurcation diagram with the brute force method. We also investigate the noise characteristic of the output signal of the chaotic modulator with FFT analysis and classify the various noise characteristics by changing the integrator gains of the double loop delta sigma modulator. We use FFT and the bifurcation diagram to classify these noise characteristics into three categories: suppressed tone (affected by the chaos), divergence, and the appearance of the limit cycle oscillation. We also confirm the existence of an unusual noise-shaping characteristic caused by the intermittent chaos.

In this paper, we consider a general class of discrete multiple-time delayed recurrent neural networks with external inputs. By applying a new transformation, we transform an m-neuron network model into a parameterized map from $l_{\infty }$ to $l_{\infty }$. A chaotic invariant set of the neural networks system is obtained by using a family of projections from $l_{\infty }$ onto $R^p$. Furthermore, we prove that the dynamics of this neural networks system restricted to the chaotic invariant set is topologically conjugate to the dynamics of the full shift map with two symbols, which indicates that chaos occurs. Numerical simulations are presented to illustrate the theoretical outcomes.

In this paper, the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp of order two and a hyperbolic saddle for a planar near-Hamiltonian system is given. Next, we consider the limit cycle bifurcations of a hyper-elliptic Liénard system with this kind of heteroclinic loop and study the least upper bound of limit cycles bifurcated from the period annulus inside the heteroclinic loop, from the heteroclinic loop itself and the center. We find that at most three limit cycles can be bifurcated from the period annulus, also we present different distributions of bifurcated limit cycles.

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.

Lyapunov exponent (LE), fast Lyapunov indicator (FLI), relative finite-time Lyapunov indicator (RLI), smaller alignment index (SALI), and generalized alignment index (GALI) are some of the available methods in most conservative systems. This study focuses on the effects of the above indicators on dissipative chaotic circuit systems such as the Lorenz system and a hyperchaotic model. Numerical experiments show that the performances of the chaos indicators in the hyperchaotic system are almost similar to those in the Lorenz system. These indicators clearly provide transition from chaotic to regular motion. However, FLI, RLI, SALI, and GALI cannot describe transition from chaos to hyperchaos. These indicators are also applied to study a new four-dimensional chaotic circuit system. The basic dynamic behaviors and structures are investigated analytically and numerically. The dynamic qualitative properties of individual orbits are observed using an oscilloscope. Moreover, the entire set of LE about the parameter is found to have three threshold values. Comparisons show that all chaos indicators are able to capture chaotic and periodic motion in chaotic circuit systems, but SALI displays significantly different behavior in several periodic orbits. SALI drops exponentially to zero for “morphologically regular” orbits that are actually unstable and sensitive to perturbation. This conclusion can also be confirmed by GALI.

A nonlinear four oligopolies price game with heterogeneous players, that are boundedly rational and adaptive, is built using two different special demand costs. Based on the theory of complex discrete dynamical system, the stability and the existing equilibrium point are investigated. The complex dynamic behavior is presented via bifurcation diagrams, the Lyapunov exponents to show equilibrium state, bifurcation and chaos with the variation in parameters. As disturbance is ubiquitous in economic systems, this paper focuses on the analysis of delay feedback control method under noise circumstances. Stable dynamics is confirmed to depend mainly on the low price adjustment speed, and if all four players have limited opportunities to stabilize the market, the new adaptive player facing profits of scale are found to be higher than the incumbents of bounded rational.

This paper proposes an efficient but simple method to determine the approximate stationary probability distribution around periodic attractors of nonautonomous nonlinear systems under multiple time-dependent parametric noises and estimate the critical noise intensity for noise-induced explosive bifurcations under a given confidence probability. After adopting a stroboscopic map constructed by a method with higher accuracy and efficiency, nonautonomous dynamical systems around periodic attractors are transformed into mapping ones. Then the mean-square analysis method of discrete systems is used to derive the stochastic sensitivity function. Based on the confidence ellipses of stochastic attractors and the global structure of deterministic nonlinear systems, the critical noise intensity of noise-induced explosive bifurcations under a given confidence probability is estimated. A Mathieu–Duffing oscillator under both multiplicative and additive noises is studied to show the validity of the proposed method.

In this paper, we investigate the bifurcation phenomena of a planar piecewise linear system. This piecewise linear system comprises two linear subsystems. The two linear subsystems have different types of dynamics. One subsystem has node or saddle dynamic and the other has focus dynamic. Some sufficient and necessary conditions for the existence of periodic orbit are given by studying the properties of Poincaré maps. Our results show that two crossing periodic orbits can bifurcate from this piecewise linear system. Moreover, we establish some sufficient and necessary conditions for the existence of sliding periodic orbit, crossing–sliding periodic orbit and sliding homoclinic orbit passing through a pseudo saddle and so on. We find that this piecewise system can appear multiply as two limit cycle bifurcation, buckling bifurcation, critical crossing cycle bifurcation, sliding homoclinic bifurcation, pseudo homoclinic bifurcation and so on. To our knowledge, sliding bifurcation phenomena are usually ignored when people study piecewise linear systems.

Epileptic seizures are generated by abnormal activity of neurons. The prediction of epileptic seizures is an important issue in the field of neurology, since it may improve the quality of life of patients suffering from drug resistant epilepsy. In this study a new similarity index based on symbolic dynamic techniques which can be used for extracting behavior of chaotic time series is presented. Using Freiburg EEG dataset, it is found that the method is able to detect the behavioral changes of the neural activity prior to epileptic seizures, so it can be used for prediction of epileptic seizure. A sensitivity of 63.75% with 0.33 false positive rate (FPR) in all 21 patients and sensitivity of 96.66% with 0.33 FPR in eight patients were achieved using the proposed method. Moreover, the method was evaluated by applying on Logistic and Tent map with different parameters to demonstrate its robustness and ability in determining similarity between two time series with the same chaotic characterization.

In this paper, the dynamical behaviors of a single Hindmarsh–Rose neuron model with multiple time delays are investigated. By linearizing the system at equilibria and analyzing the associated characteristic equation, the conditions for local stability and the existence of local Hopf bifurcation are obtained. To discuss the properties of Hopf bifurcation, we derive explicit formulas to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions occurring through Hopf bifurcation. The qualitative analyses have demonstrated that the values of multiple time delays can affect the stability of equilibrium and play an important role in determining the properties of Hopf bifurcation. Some numerical simulations are given for confirming the qualitative results. Numerical simulations on the effect of delays show that the delays have different scales when the two delay values are not equal. The physiological basis is most likely that Hindmarsh–Rose neuron model has two different time scales. Finally, the bifurcation diagrams of inter-spike intervals of the single Hindmarsh–Rose neuron model are presented. These bifurcation diagrams show the existence of complex bifurcation structures and further indicate that the multiple time delays are very important parameters in determining the dynamical behaviors of the single neuron. Therefore, these results in this paper could be helpful for further understanding the role of multiple time delays in the information transmission and processing of a single neuron.

The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert’s 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family $QT\overline{S}$ of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifurcation diagram yields 27 phase portraits for systems in $QT\overline{S}$ counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincaré disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices.

In literature many chaotic systems, based on third-order jerk equations with different nonlinear functions, are available. A jerk system is taken to be a part of dynamical systems that can exhibit regular and chaotic behavior. By extension, a hyperjerk system can be described as a dynamical system with $n$th-order ordinary differential equations where $n$ is 4 or up to. Hyperjerk systems have been investigated in literature in the last decade. This paper consists of numerical studies and experimental realization on FPAA for fourth-order hyperjerk system with exponential nonlinear function.

This paper is concerned with Devaney chaos in nonautonomous discrete systems. It is shown that in its definition, the two former conditions, i.e. transitivity and density of periodic points, in a set imply the last one, i.e. sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Some chaotic behavior is studied for a class of nonautonomous discrete systems, each of which is governed by a convergent sequence of continuous maps. In addition, the concepts of some pseudo-orbits and shadowing properties are introduced for nonautonomous discrete systems, and it is shown that some shadowing properties of the system and density of periodic points imply that the system is Devaney chaotic under the condition that the sequence of continuous maps is uniformly convergent in a compact metric space.

The existing Block Compressive Sensing (BCS) based image ciphers adopted the same sampling rate for all the blocks, which may lead to the desirable result that after subsampling, significant blocks lose some more-useful information while insignificant blocks still retain some less-useful information. Motivated by this observation, we propose a scalable encryption framework (SEF) based on BCS together with a Sobel Edge Detector and Cascade Chaotic Maps. Our work is firstly dedicated to the design of two new fusion techniques, chaos-based structurally random matrices and chaos-based random convolution and subsampling. The basic idea is to divide an image into some blocks with an equal size and then diagnose their respective significance with the help of the Sobel Edge Detector. For significant block encryption, chaos-based structurally random matrix is applied to significant blocks whereas chaos-based random convolution and subsampling are responsible for the remaining insignificant ones. In comparison with the BCS based image ciphers, the SEF takes lightweight subsampling and severe sensitivity encryption for the significant blocks and severe subsampling and lightweight robustness encryption for the insignificant ones in parallel, thus better protecting significant image regions.

Aiming at resolving the conflict between security and efficiency in the design of chaotic image encryption algorithms, an image encryption algorithm based on information hiding is proposed based on the “one-time pad” idea. A random parameter is introduced to ensure a different keystream for each encryption, which has the characteristics of “one-time pad”, improving the security of the algorithm rapidly without significant increase in algorithm complexity. The random parameter is embedded into the ciphered image with information hiding technology, which avoids negotiation for its transport and makes the application of the algorithm easier. Algorithm analysis and experiments show that the algorithm is secure against chosen plaintext attack, differential attack and divide-and-conquer attack, and has good statistical properties in ciphered images.

Since different regions of an image have different importance, therefore only the important information of the image regions, which the users are really interested in, needs to be encrypted and protected emphatically in some special multimedia applications. However, the regions of interest (ROI) are always some irregular parts, such as the face and the eyes. Assuming the bulk data in transmission without being damaged, we propose a chaotic image encryption algorithm for ROI. ROI with irregular shapes are chosen and detected arbitrarily. Then the chaos-based image encryption algorithm with scrambling, S-box and diffusion parts is used to encrypt the ROI. Further, the whole image is compressed with Huffman coding. At last, a message authentication code (MAC) of the compressed image is generated based on chaotic maps. The simulation results show that the encryption algorithm has a good security level and can resist various attacks. Moreover, the compression method improves the storage and transmission efficiency to some extent, and the MAC ensures the integrity of the transmission data.