International Journal of Bifurcation and Chaos

We present the first circuit realization of an animal (quadruped) robot controlled by a Central Pattern Generator (CPG) network of neurons, whose model and design are biologically-inspired [Golubitsky et al., 1998, 1999]. We demonstrate, through hardware simulations of the CPG network and through video snapshots, that our electronic CPG and our animal robot can indeed reproduce, via symmetry-breaking Hopf bifurcations, all primary gaits — walk, jump, trot, bound, pace, and pronk — just as they were originally predicted by theory. The design and development of the circuit realization are discussed in great detail.

Intermittent bursting in the fractional difference logistic map of matrices is studied in this paper. Analytic relationships governing the dynamics of the fractional difference map of matrices are derived in an explicit form. Computational experiments are used to prove the existence of intermittent bursts located far away from the initial conditions. Such isolated waves of temporary divergence in the fractional difference logistic map of matrices are demonstrated for the first time. Intermittent bursts of temporary divergence offer new possibilities for designing advanced information hiding algorithms.

The present work focuses on the existence of chimera states in two-layer networks of locally, nonlocally, and globally coupled Hindmarsh–Rose neurons. We show that chimera states occur in all three coupled neuronal networks by changing synaptic coupling strength and the number of coupled neighbors, and traveling chimera state exists in networks by using local and nonlocal couplings. Particularly, we find that in the nonlocally coupled network, there exist regions of coherent state, which can produce a wave such that the period becomes longer with the increase of the number of coupled neighbors. Interestingly, we observe a new chimera state with the coexistence of interlayer synchronous and asynchronous chimera states, which we named as the interlayer semi-synchronous chimera state. In addition, the results indicate that the interaction between layers in neuronal networks can induce different types of chimera states and firing patterns, which could be useful to control neural firing patterns and deepen the understanding of neuronal evolution where coherent and incoherent dynamics coexist.

Using an effective nonlinear feedback controller, a novel 4D hyperchaotic Lorenz system is built. Dynamical analyses show that it has interesting properties. Using some well-known analysis tools like Lyapunov spectrum, bifurcation analysis, chaos diagram, and phase space trajectories, it is found that several bifurcations enable the hyperchaotic dynamics to occur in the introduced model. Also, many windows of heterogeneous multistability are found in the parameter space (i.e. coexistence of a pair of chaotic attractors, coexistence of a periodic and a chaotic attractor). Besides, DSP implementation is successfully used to support the results of the theoretical prediction. Finally, a judicious image encryption algorithm based on the hyperchaotic Lorenz system is proposed with detailed analysis. The effectiveness of the proposed approach is confirmed via several security analyses, which yields a secure image encryption application.

Most biological systems have long-range temporal memory. Such systems can be modeled using fractional-order differential equations. The combination of fractional-order derivative and time delay provides the system more consistency with the reality of the interactions and higher degree of freedom. A fractional-order delayed prey–predator system with the fear effect has been proposed in this work. The time delay is considered in the cost of fear; therefore, there are no dynamical changes observed in the system due to time delay in the absence of fear. The existence and uniqueness of the solutions of the proposed system are studied along with non-negativity and boundedness. The existence of biologically relevant equilibria is discussed, and the conditions for local asymptotic stability are derived. Hopf bifurcation occurs in the system with respect to delay parameter. Further, a spatially extended system is proposed and analyzed. Hopf bifurcation also occurs in the extended system due to the delay parameter. Numerical examples are provided in support of analytical findings. Fractional-order derivative improves the stability and damps the oscillatory behaviors of the solutions of the system. Bistability behavior of the system admits stable dynamics by decreasing the fractional-order. Also, chaotic behavior is destroyed by decreasing fractional-order.

First, we study the planar continuous piecewise differential systems separated by the straight line $x=0$ formed by a linear isochronous center in $x>0$ and an isochronous quadratic center in $x<0$. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles.

Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line $x=0$ having in $x>0$ the general quadratic isochronous center $ẋ=-y+x^2-y^2$, $ẏ=x(1+2y)$ after an affine transformation, and in $x<0$ an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles.

In short for these families of continuous piecewise differential systems we have solved the extension of the 16th Hilbert problem.

Chaotic maps have been widely used in image encryption due to their complexity, pseudorandomness and high sensitivity to initial values. In this paper, an image encryption algorithm based on a 4D chaotic map and steganography is proposed. The algorithm consists of two rounds of encryption and one operation of embedding hash. In the first round of encryption, the hash value of plaintext image is used to control the generation of chaotic sequences, which makes the encryption algorithm highly relevant to the content of the image. Then, the hash value is embedded into the intermediate image according to the idea of steganography. In the second round of encryption, the image is encrypted under the control of another set of chaotic sequences to hide the hash value and further enhance the security of the algorithm. Different from other algorithms, our algorithm does not need to additionally transmit the hash value to the decryptor through a special channel. It has good availability and adaptability. Experimental results and security analysis demonstrate that the algorithm has high security performance and can resist various attacks.

The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.

Our starting point is a discontinuous piecewise linear dynamical system in ${ℝ}^3$ with two zones that present a single invisible two-fold straight line and invariant cylinders. We broke the dynamics of this vector field to obtain a new four-parameter piecewise linear vector field with a $T$-singularity. For this vector field, we prove that the upper bound of simple crossing limit cycles (simple CLCs) is three, and provide conditions for the existence of 1, 2, or 3 simple CLCs. We also show the occurrence of three distinct bifurcations involving such simple CLCs that are derived from a bifurcation set with two parameters: (i) a Teixeira Singularity bifurcation (TS-bifurcation); (ii) a Fold bifurcation; and (iii) a Cusp bifurcation.

In this paper, we investigate the relations between distributional chaos in a sequence and distributional chaos ($\omega$-chaos, R–T chaos, DC3, respectively). Firstly, we prove a sufficient condition that the distributional chaos is equivalent to the distributional chaos in a sequence. Besides, we prove that distributional chaos in a sequence and $\omega$-chaos (R–T chaos, DC3, respectively) do not imply each other. Finally, we give a new definition of chaos, named DC2${}^{\prime }$, which is similar to DC2, and show that DC2${}^{\prime }$ is an invariant of topological conjugacy and an iteration invariant (that is, for any integer $N>0$, $f$ is DC2${}^{\prime }$ if and only if $f^N$ is DC2${}^{\prime }$).

In this paper, we mainly investigate an infected-age structured HIV model with logistic growth for target cells, which simultaneously takes the latent factor and different transmission modes into consideration. On the basis of integrated semigroup theory and Hopf bifurcation theory for semilinear equations in a nondense domain, we reach a conclusion that the system undergoes a Hopf bifurcation around the positive equilibrium, which implies that there exists a nontrivial periodic oscillation phenomenon when the Hopf bifurcation parameter $\tau$ passes through some critical values. Namely, the HIV infection will periodically erupt when the mature period $\tau$ crosses the critical value ${\tau }_k$ ($k=0,1,2,\dots$), that is, HIV virus will always survive in the patient’s body. In order to verify the fact that the biological maturation period $\tau$ does play an essential role in our system, we also offer the relevant numerical simulation results by selecting several decisive parameters.

To systemically study the dynamic characteristics of the gear system, a single degree-of-freedom coupled thermal-elasto-hydrodynamic-nonlinear (TEHDN) dynamic model is developed with temperature, lubrication and time-varying parameters. Based on the mathematical expressions of thermal deformation, elastohydrodynamic lubrication and other nonlinear factors, the governing differential equation of the coupled system is deduced via the mechanical relationships. The vibration response is obtained by numerical integration and the accuracy and feasibility are verified with the published literature. Parameter analysis on nonlinear dynamic behaviors is conducted to reveal the influences of nonlinear backlash, excitation fluctuation, comprehensive transmission error and surface roughness. The results show that due to the influence of nonlinear backlash, the chaotic motion appears in advance and the chaotic region narrows. The first appearance of chaotic motion lags and shifts to higher excitation amplitude. The transmission error is capable of controlling the chaotic region and vibration intensity, whereas the surface roughness has less effect on the dynamic behavior in lower value. Besides, different meshing impact and bifurcation routes are observed with various parameters. This study could provide a better understand of the stable operation for gear system.

The bifurcations and chaos in a ring of three nonmonotonic piecewise constant neurons with unidirectional inhibitory coupling are examined. Periodic solutions in the system undergo border collision bifurcations, which are characteristic of piecewise smooth systems. Conditions for the bifurcations of periodic solutions are derived analytically by using a piecewise constant function as the output function of neurons. Multiple periodic solutions are generated through border collision bifurcations and saddle-node bifurcations. Chaotic attractors are generated through border collision bifurcations and degenerate period-doubling bifurcations. The existence of multiple periodic solutions and chaotic attractors is proven to be common in rings of unidirectionally coupled nonmonotonic neurons.

Human emotions are an important part in daily life. In this paper, a novel multilayer network-based convolutional neural network (CNN) model is proposed for emotion recognition, from multi-channel nonlinear EEG signals. Firstly, in response to the multi-rhythm properties of brain, a multilayer brain network with five rhythm-based layers are derived, where each layer can pertinently describe one specific frequency band. Subsequently, a novel CNN model is carefully designed, which uses the multilayer brain network as input and allows deep learning of the classifiable nonlinear features from the channel and frequency views. Moreover, one DenseNet model is developed as another branch to study time-domain nonlinear features from the EEG signals. All the learned features are eventually concatenated together for emotion recognition. Publicly available SEED dataset is used to test the proposed method, and it shows good results on all 15 subjects, with average accuracy of 91.31%. Our method builds a bridge between the multilayer network and deep learning, suggesting an effective approach for analyzing multivariate nonlinear time series, especially multi-channel EEG signals.

Different from single-input single-output (SISO) converters, strong nonlinear coupling effect between input ports is one of the most salient features in multi-input converters (MICs), which leads to at least three-piecewise ripple of inductor current. This paper deals with the analysis of bifurcation-based dynamical stabilities and nonlinear coupling effects in the dual-input Buck-Boost DC–DC converter. Firstly, a sampling-based averaged continuous model is proposed to describe the coupling characteristics of the dual-input DC–DC converters with three-piecewise current ripple. Based on the proposed model, the inherent mechanism of low-frequency oscillation is revealed in terms of the movement of the eigenvalue loci. It shows that Hopf bifurcation takes the responsibility for the low-frequency oscillation. Moreover, multiparameters stability boundaries are identified to establish the nonlinear mapping relations between parameter space and state space. Particularly, Gershgorin band analysis is used to give a qualitative analysis in order to evaluate the coupling effects between different state variables of the system. In addition, Gershgorin band sensitivity is proposed to reveal the influence of parameters on the coupling effects of the system. Finally, the correctness of these theoretical analyses is verified by PSpice circuit experiments. These results are beneficial to the practical optimization design as well as the understanding of coupling effect of MICs.

In this paper, we study a dual-channel closed-loop supply chain (CLSC) consisting of one manufacturer, one retailer and one third-party firm or platform (3P). The manufacturer wholesales new products through the traditional retail channel and distributes remanufactured products via 3P. We focus on establishing the dynamic Stackelberg game models for nondelayed and delayed cases, respectively. The existence and local stability of Nash equilibrium are examined as well as the complex dynamical behaviors of each model under various scenarios are investigated by numerical simulations, such as stability region, bifurcations, chaos, strange attractors, and so on. Moreover, the impacts of some key parameters on the performance of chain members are analyzed. In addition, the variable feedback control method is utilized to eliminate the system chaos. The results reveal that the high value of the consumer discount perception for remanufactured products and excessively fast price adjustment speed have a destabilization effect on the Nash equilibrium point. In addition, adopting delay decisions by manufacturer does not always make the system more stable because it can exert either positive or negative effect on the system’s stability, while an intermediate delay weight is conducive to the system have a higher chance to stay stable. Furthermore, the manufacturer’s profits will be declined significantly while the profits of retailer and 3P will be elevated to some extent when the system falls into periodic and chaotic motions, so chaos is not always necessarily detrimental to all the decision makers in the dual-channel CLSC.