Showing papers in "International Journal of Bifurcation and Chaos in 2022"
TL;DR: An optical image encryption scheme based on fractional Fourier transform and five-dimensional host-induced nonlinearity fractional-order laser hyperchaotic system is studied and a novel imageryption scheme is proposed combining BP neural network, GF(17) domain diffusion andHyperchaotic, random point scrambling algorithm.
Abstract: In recent years, image encryption schemes based on optical methods have been extensively studied. However, possible optical encryption methods in combination with fractional-order laser hyperchaotic systems have not been reported. Therefore, an optical image encryption scheme based on fractional Fourier transform and five-dimensional host-induced nonlinearity fractional-order laser hyperchaotic system is studied in this paper. Firstly, the dynamical characteristics of the proposed fractional-order laser hyperchaotic system are analyzed, and the DSP platform is used to realize the system. Then, a novel image encryption scheme is proposed combining BP neural network, GF(17) domain diffusion and hyperchaotic, random point scrambling algorithm. Finally, the performance of the encryption scheme is analyzed in detail. This work provides an experimental basis and theoretical guidance for image secure communication combining fractional-order laser hyperchaotic systems and optical methods and offers a new research perspective for optical image encryption.
51 citations
TL;DR: A new memristive neural network composed of three nodes connected by the simplest circular loop, whose synaptic weights are replaced by hyperbolic memristors is constructed, which gives a positive answer to the interesting question whether chaos can occur in neural network with the simplest cyclic connections.
Abstract: Previous studies have shown that cyclic neural networks which have no autoexcitation and are unidirectional cannot generate chaos. Inspired by this finding, the present paper constructs a new memristive neural network composed of three nodes connected by the simplest circular loop, whose synaptic weights are replaced by hyperbolic memristors. The memristive neural network can generate chaos via period-doubling bifurcation, and generate different stable and periodic states with the variation of parameters. Another remarkable feature of the new memristive neural network is that it coexists with point and periodic attractors, periodic and chaotic attractors from different initial conditions. Detailed dynamic analysis and circuit implementation are given to illustrate the existence of chaos and coexisting attractors, which gives a positive answer to the interesting question whether chaos can occur in neural network with the simplest cyclic connections.
26 citations
TL;DR: Based on a variable-boostable chaotic system, a conservative chaotic system with controllable amplitude and offset is proposed in this paper , which exhibits rich symmetrical dynamics under different parameters and initial conditions.
Abstract: Based on a variable-boostable chaotic system, a conservative chaotic system with controllable amplitude and offset is proposed. The system exhibits rich symmetrical dynamics under different parameters and initial conditions. More interestingly, a parameter of memristor poses a partial amplitude control to a system variable. Furthermore, the derived memristive system has the property of offset boosting, where an independent constant can be introduced for free rescaling of the average value of a system variable. Experimental circuit with a memristor rheostat is designed for amplitude control. Circuit simulation based on Multisim software agrees well with the systematic analysis and numerical exploration. To the best of our knowledge, in the literature there is no 3D conservative memristive system reported with such properties as amplitude control and offset boosting.
25 citations
TL;DR: Experimental simulations and extensive cryptanalysis fully vindicate that superior security effects in addition to satisfactory low time complexity can be simultaneously obtained by the proposed confusion-substitution scheme.
Abstract: In this paper, an image cryptosystem based on genetic central dogma (GCD), Knuth–Morria–Pratt (KMP) algorithm and a chaotic system is developed. The KMP algorithm is firstly used to bind DNA strings to obtain the next array, which participates in the design of the chaotic initial condition, and then the secure chaotic sequences are produced by employing the sliding idea in pattern string matching. In the present procedure, a DNA-level two-way pixel’s shuffle is achieved by a shared stack push operation and it is adopted in the permutation module for the purpose of accelerating the overall pixel’s shuffle. Subsequently, the pixel values are substituted by simulating the process of protein synthesis in GCD, in which the DNA replication and RNA replication form the basis of the DNA-triploid mutation and new RNA mutation rules, respectively. Experimental simulations and extensive cryptanalysis fully vindicate that superior security effects in addition to satisfactory low time complexity can be simultaneously obtained by the proposed confusion-substitution scheme.
19 citations
TL;DR: In this article , the one-and two-parameter bifurcations of a discrete-time prey-predator model with a mixed functional response are investigated by computing their critical normal form coefficients.
Abstract: In this paper, the one- and two-parameter bifurcations of a discrete-time prey–predator model with a mixed functional response are investigated by computing their critical normal form coefficients. The flip, Neimark–Sacker and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complex dynamical behavior of the model up to the 16th iteration is investigated.
18 citations
TL;DR: In this paper , a Leslie-Gower predator-prey model with Allee effect on the prey and a linear functional response was considered and the existence and stability of equilibria were analyzed.
Abstract: In this paper, we consider a Leslie–Gower predator–prey model with Allee effect on the prey and a linear functional response. Here the Allee effect impacts the birth rate of the prey, which is different from the common multiplicative and additive Allee effects. The model is well-posed, that is, all solutions are bounded. Applying the blow-up method indirectly, we prove that the origin which is not an equilibrium of the system is an attractor. Then we study the existence and stability of equilibria, which indicate that the system undergoes bifurcations. With the help of Sotomayor’s theorem, we show the occurrence of saddle-node bifurcation. Moreover, there is degenerate Hopf bifurcation of codimension at least three. By choosing two (three) parameters of the system as bifurcation parameters and calculating a versal unfolding near the cusp, we demonstrate that the system undergoes Bogdanov–Takens bifurcation of codimension two (three). These theoretical results are supported with numerical simulations.
17 citations
TL;DR: The model of the consecutive memcapacitor has been widely used in chaotic circuits and its application in chaotic systems has not been further studied, so a theoretical basis for the application of discrete memapacitor in the design of discrete chaotic systems is supplied.
Abstract: The model of the consecutive memcapacitor has been widely used in chaotic circuits. However, the model of discrete memcapacitor and its application in chaotic systems have not been further studied. In this paper, a model of discrete memcapacitor is proposed. And the dynamical characteristics of the discrete memcapacitor model are analyzed. The memristive Chebyshev map is obtained by coupling the discrete memcapacitor with the Chebyshev map. Since memristive Chebyshev map has linear fixed points, the memristive Chebyshev map is unstable or critically stable, depending on the internal parameters and the initial condition of the chaotic map. The dynamical behavior of control parameter dependence of memristive Chebyshev map is studied by using several analysis methods, and its hyperchaotic attractor is found. The special phenomenon of coexistence of attractors is also found. Finally, the memristive Chebyshev map is realized by DSP. And the results of simulation are further verified. The results of this study supply a theoretical basis for the application of discrete memcapacitor in the design of discrete chaotic systems.
16 citations
TL;DR: In this paper , a new chaotic system in spherical coordinates is presented, where the system's attractors are limited to be located on one side of the sphere and cannot touch it.
Abstract: The investigation of new chaotic systems in spherical coordinates has been one of the present exciting research directions in exploring new chaotic systems. In this paper, a new system in spherical coordinates is presented. The appealing feature of the proposed system is that the dynamics of the system cannot pass through a sphere of a specific radius and stop as soon as the solution crosses the sphere in Cartesian coordinates. So, the system’s attractors are limited to be located on one side of the sphere and cannot touch it. Moreover, the reason for this phenomenon is that the velocity of a system’s variable becomes zero for a specific value of that variable. The proposed system has three unstable equilibrium points and four hidden attractors, including a limit cycle and a strange attractor inside and a limit cycle and a strange attractor outside the sphere. The system’s dynamical properties are investigated with the help of bifurcation diagrams and the calculation of Lyapunov exponents. The basin of attraction for the system’s attractors is also studied. Finally, the system is controlled or stabilized using the impulsive control theory.
13 citations
TL;DR: In this paper , a Leslie-Gower predator-prey model with strong Allee effect on prey and fear effect on predator was proposed and the existence and local stability of equilibria by making full use of qualitative analytical theory.
Abstract: In this paper, we propose a Leslie–Gower predator–prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter [Formula: see text], the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when [Formula: see text], the system will undergo degenerate Bogdanov–Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [González-Olivares et al., 2011; Pal & Mandal, 2014].
11 citations
TL;DR: In this article , the authors consider the dynamical effects of electromagnetic flux on the discrete Chialvo neuron model and show that the model can exhibit rich dynamical behaviors such as multistability, firing patterns, antimonotonicity, closed invariant curves, various routes to chaos, and fingered chaotic attractors.
Abstract: We consider the dynamical effects of electromagnetic flux on the discrete Chialvo neuron model. It is shown that the model can exhibit rich dynamical behaviors such as multistability, firing patterns, antimonotonicity, closed invariant curves, various routes to chaos, and fingered chaotic attractors. The system enters a chaos regime via period-doubling cascades, reverse period-doubling route, antimonotonicity, and via a closed invariant curve to chaos. The results were confirmed using the techniques of bifurcation diagrams, Lyapunov exponent diagram, phase portraits, basins of attraction, and numerical continuation of bifurcations. Different global bifurcations are also shown to exist via numerical continuation. After understanding a single neuron model, a network of Chialvo neurons is explored. A ring-star network of Chialvo neurons is considered and different dynamical regimes such as synchronous, asynchronous, and chimera states are revealed. Different continuous and piecewise continuous wavy patterns were also found during the simulations for negative coupling strengths.
10 citations
TL;DR: In this article , a fractional-order susceptible-infected-recovered-susceptible (SIRS) model is studied, focusing on delay effects and high-order interactions.
Abstract: In this paper, a fractional-order susceptible-infected-recovered-susceptible (SIRS) model is studied, focusing on delay effects and high-order interactions. Two types of time delays are considered to describe latent period and healing cycle, respectively. From the ecological point of view, we found that the increasing delays caused by either the latent period or the healing cycle lead to the periodic outbreak of disease. The finding provided us with an important implication to preventing periodic outbreaks of disease by reducing the time delay, like accelerating the healing process with effective medication and medical intervention. Specifically, taking the time delays as bifurcation parameters, the stability of endemic equilibria and the existence of Hopf bifurcation are studied by analyzing the characteristic equation of the SIRS model. From a general point of view, based on the establishment of a fractional-order SIRS model, we found that the order of the fractional order is critical for describing the dynamic behavior of the model. Typically, the decrease of the order appears to bring about the disappearance of the periodic phenomenon (i.e. the periodic oscillation) of the originally stable system.
TL;DR: In this paper , the authors proposed a new method for estimating the differences between two return maps based on a two-dimensional histogram, and investigated the resistance of chaotic shift keying, parameter modulation, and symmetry coefficient modulation (SCM) against three types of attacks.
Abstract: Many studies show the possibility of transmitting messages in a protected and covert manner using a noise-like chaotic waveform as a carrier. Among popular chaotic communication system (CCS) types, one may distinguish chaotic shift keying (CSK) and parameter modulation (PM) which are based on the manipulation of the transmitting chaotic oscillators. With the development of direct digital synthesis (DDS), it became possible to modulate chaotic signals by varying the properties of the numerical method used in digital chaos generators. The symmetry coefficient modulation (SCM) is such an approach potentially able to provide higher secrecy. However, the actual security of chaos-based communications is still a questionable and controversial feature. To quantitatively evaluate the CCS security level, a certain numerical metric reflecting the difficulty of breaking a communication channel is needed. Return maps are commonly used to attack chaotic communication systems, but the standard algorithm does not involve any kind of quantification. In this study, we propose a new method for estimating the differences between two return maps based on a two-dimensional (2D) histogram. Then, we investigate the resistance of chaotic shift keying, parameter modulation, and SCM communication schemes against three types of attacks: the proposed quantified return map analysis (QRMA), recurrence quantification analysis (RQA), which had not been previously reported for attacking chaos-based communications, and the classical approach based on spectral analysis. In our experiments we managed to recover the plain binary message from the waveform in the channel when transmitted using all three chaos-based messaging techniques; among them, SCM appeared to be the most secure communication scheme. The proposed QRMA turned out to be the most efficient technique for message recovery: the sensitivity of the QRMA appeared to be 2–5 times higher than that in the case of spectral analysis. The proposed QRMA method can be efficiently used for evaluating the difficulty of hacking chaos-based communication systems. Moreover, it is suitable for the evaluation of any other secure data transmission channel.
TL;DR: A judicious image encryption algorithm based on the hyperchaotic Lorenz system is proposed with detailed analysis, and the effectiveness of the proposed approach is confirmed via several security analyses, which yields a secure image encryption application.
Abstract: 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.
TL;DR: Wang et al. as discussed by the authors proposed an irreversible parallel key expansion algorithm based on a chaotic map, which has high sensitivity to initial key with independence between round keys, and performance analysis demonstrate that the algorithm can resist the side channel power attack.
Abstract: The key expansion algorithm is an important part of a general iterated block cipher, because its security determines the security level of the encryption scheme. Thus, generating secure round keys with statistical independence and sensitivity is desirable. An irreversible parallel key expansion algorithm is proposed based on a chaotic map in this paper. First, an enhanced nondegenerate 2D exponential chaotic map (2D-ECM) with ergodicity is constructed, with analysis results demonstrating that it has better dynamical characteristics. Then, an irreversible key expansion algorithm is designed based on the 2D-ECM, which has high sensitivity to initial key with independence between round keys. Experimental results and performance analysis demonstrate that the algorithm is feasible and can resist the side channel power attack.
TL;DR: In this article , a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics, and the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively.
Abstract: Networks of coupled oscillators have been used to model various real-world self-organizing systems. However, the dynamics, especially chaos and bifurcation, of complex-valued networks are rarely investigated. In this paper, a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics. Although there are only stable limit cycles in a complex-valued van der Pol oscillator, chaos, hyperchaos, and coexisting chaotic attractors are observed from the ring network, which are analyzed by using the Lyapunov exponent spectrum, bifurcation diagram and 0–1 test. In addition, complexity analysis on nonlinear coefficients and coupling strengths illustrates that the range of parameters within the chaotic (hyperchaotic) region has positive correlation with the number of oscillators. It is shown that the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, it is demonstrated that complete synchronization and phase synchronization of oscillations are stable in a large-scale ring network, while chaotic phase synchronization is unstable in a small-scale network.
TL;DR: In this article , a non-autonomous memcapacitive oscillator was proposed to discover a new type of extreme multistability due to the infinitely many discrete equilibrium points therein.
Abstract: Extreme multistability usually emerges in a mem-element’s circuit or system that possesses a line or plane equilibrium set closely associated with the internal initial state of the mem-element. To extend the investigation of extreme multistability, this paper proposes a nonautonomous memcapacitive oscillator, discovering a new type of extreme multistability due to the infinitely many discrete equilibrium points therein. This memcapacitive oscillator is constructed by connecting a simple memcapacitor-resistor circuit with a sinusoidal voltage. With its normalized model, the infinitely many discrete equilibrium points are computed and the infinitely many necklace-shaped coexisting attractors that were not yet reported are disclosed by numerical methods. Since the number and stability of the equilibrium points vary with time, the attraction basins with complex ripple structures are formed in the memcapacitive oscillator, resulting in the appearance of a special type of extreme multistability. Furthermore, PSIM circuit simulations and microcontroller-based hardware experiments are performed to verify the numerical results.
TL;DR: A novel multilayer network-based convolutional neural network model is proposed for emotion recognition, from multi-channel nonlinear EEG signals, suggesting an effective approach for analyzing multivariate nonlinear time series, especially multi- channel EEG signals.
Abstract: 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.
TL;DR: To counteract dynamical degradation and make it suitable for a PRNG, the periodic point detection and random impulsive perturbation are applied to lengthen the aperiodic time sequence, and statistical results demonstrate that a full-period sequence can be obtained.
Abstract: Some weaknesses of 1D chaotic maps, such as lacking of ergodicity, multiple bifurcations, dense periodic windows, and short iteration period, limit their practical applications in cryptography. A higher-dimensional chaotic map with ergodicity can solve these problems. Based on 1D quadratic map, a 3D exponential hyperchaotic map (3D-EHCM) is constructed, and its dynamic behaviors, such as phase diagram, Lyapunov exponent spectrum, Kolmogorov entropy (KE), correlation dimension, approximate entropy and randomness, are analyzed and tested. The results demonstrate that the 3D-EHCM has ergodicity in a larger range of control parameter, and its state points have a longer period. To counteract dynamical degradation and make it suitable for a PRNG, the periodic point detection and random impulsive perturbation are applied to lengthen the aperiodic time sequence, and statistical results demonstrate that a full-period sequence can be obtained.
TL;DR: In this paper , the authors considered a Leslie-Gower predator-prey system with Allee effect and prey refuge and analyzed the stability of the equilibria in the system, and found that there are abundant dynamic behaviors.
Abstract: This paper considers a Leslie-Gower predator–prey system with Allee effect and prey refuge. By considering the prey refuge constant as a parameter, we analyze the stability of the equilibria in the system, and find that there are abundant dynamic behaviors. It is shown that the model can undergo a sequence of bifurcations including saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension two or three as the parameters vary. Moreover, the model underdoes a degenerate Hopf bifurcation of codimension two and has two limit cycles, where the inner one is stable and the outer one is unstable. Through some numerical simulations, the occurrence of Bogdanov–Takens bifurcation and Hopf bifurcation of codimension two are confirmed.
TL;DR: In this paper , the existence of intermittent bursts of temporary divergence in the fractional difference logistic map of matrices is studied in an explicit form and the relationship among these bursts is derived.
Abstract: 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.
TL;DR: A fast period detection algorithm is designed to locate all local “periodicities” contained in chaotic binary sequences quickly and accurately and enriches the randomness test methods for binary sequences.
Abstract: When chaotic systems are applied to stream ciphers, chaotic real-valued sequences generally need to be converted into binary sequences with the purpose of encrypting data. However, the performance of binary sequences will be degraded under the joint influence of round-off and quantization errors. In this case, the randomness of some chaotic binary sequences may be weakened in a local range. Taking advantage of parallel computing, a fast period detection algorithm is designed to locate all local “periodicities” contained in chaotic binary sequences quickly and accurately. This algorithm evaluates the randomness of a chaotic binary sequence from a new perspective of periodicity which enriches the randomness test methods for binary sequences. Different logistic binary sequences are analyzed to demonstrate the effectiveness and practicability of the proposed algorithm.
TL;DR: In this article , an initially controlled double-scroll hyperchaotic map is constructed based on two sine functions, and four different modes for attractor growing are demonstrated, and hardware experiments based on STM32 are carried out to verify the theoretical analysis and numerical simulation.
Abstract: Initial condition-dominated offset boosting provides a special channel for coexisting orbits. Due to the nonlinearity and inherent periodicity, sinusoidal function is often introduced into a dynamical system for multistability design. Typically, the distance between two attractors or two petals of an attractor is fixed. Moreover, any chaotic signal and sequence need to be modified with amplitude and offset for a real application. In this paper, an initially-controlled double-scroll hyperchaotic map is constructed based on two sine functions. Four patterns of the double-scroll hyperchaotic orbits are found as 0-degree, 90-degree, 45-degree and 135-degree. Consequently, different modes for attractor growing are demonstrated. In this case, all the coexisting attractors are arranged in phase space in a direction defined by the initial value and the distance between two petals of any double-scroll orbit is adjusted. Finally, hardware experiments based on STM32 are carried out to verify the theoretical analysis and numerical simulation.
TL;DR: In this article , a detailed investigation of stochastic stability and complex dynamics of a centrifugal governor system with approximately uniform color noise is presented, and the results manifest that the effects of noise intensity and correlation time on stationary probability density are opposite.
Abstract: This paper presents a detailed investigation of stochastic stability and complex dynamics of a centrifugal governor system with approximately uniform color noise. The centrifugal governor system excited by noise is transformed into Itö equation using polar coordinate transformation and stochastic average method. According to the boundary conditions of attraction and repulsion, the stochastic stability is ensured. In addition, analyses concerning the influence of parameter variation and validity are carried out by employing numerical method. The results manifest that the effects of noise intensity and correlation time on stationary probability density are opposite. The amplitudes of probability density finally tend to a limit value, and the only limit cycle appears, which shows that when the bifurcation occurs, the trivial solution of the system converges to a limit cycle with a higher probability. Finally, the two-dimensional parameter bifurcation analysis of the centrifugal governor system subject to color noise excitation is studied. An interesting distribution characteristic is found that the periodic region is organized according to the sequence of Stern–Brocot trees, and this typical characteristic is a universal characteristic of the system on the two parameter planes. Furthermore, it is concluded that based on the largest Lyapunov exponent diagram and bifurcation diagram in two-dimensional parameter plane, the effects of noise intensity and correlation time on the periodic oscillation state are opposite, but both of them can transform the quasi-periodic oscillation into periodic oscillation. It should be emphasized that with the increase of noise intensity, the coexisting oscillation behavior of the centrifugal governor system will change, which is manifested by the destruction of coexisting attractors and the generation of chaotic attractors.
TL;DR: In this article , the authors modified the Hastings-Powell (HP) model by considering the cost of fear in middle predator and the Allee effect in top predator, and derived the stability conditions for the biologically feasible equilibria using linear stability analysis.
Abstract: In ecology, predator–prey interactions are very complex in nature. Apart from direct killing, predator induces fear among their prey which affects the life-history, behavioral changes, and reproduction potential of the prey population. On the other hand, the Allee effect has a great impact on regulating the population size, community structure and population dynamics. In the present investigation, we modify the Hastings–Powell (HP) [1991] model by considering the cost of fear in middle predator and the Allee effect in top predator. The stability conditions for the biologically feasible equilibria are derived using linear stability analysis. Considering the cost of fear and the Allee effect as key parameters, the Hopf bifurcation analysis is carried out around the interior equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Our numerical results suggest that the fear effect can stabilize the system. It is observed that high levels of fear among middle predator decrease the population density of top predator. We also observe that if the Allee parameter is increased, then the system becomes stable from chaotic oscillations. However, further increase in the Allee parameter leads to population extinction. We have also drawn several one- and two-parameter bifurcation diagrams which explore rich dynamical behaviors.
TL;DR: In this article , the dynamics of a new 4D chaotic hyper-jerk system with a unique equilibrium point is studied. And the authors demonstrate that it is possible to generate different varieties of two, three, four, or five coexisting hidden and selfexcited attractors in the introduced model.
Abstract: Nonlinear dynamical systems with hidden attractors belong to a recent and hot area of research. Such systems can exist in different forms, such as without equilibrium or with a stable equilibrium point. This paper focuses on the dynamics of a new 4D chaotic hyper-jerk system with a unique equilibrium point. It is shown that the new hyper-jerk system effectively exhibits different hidden behaviors, which are hidden point attractor, hidden periodic attractor, and hidden chaotic state. Collective behaviors of the system are studied in terms of the equilibrium point, bifurcation diagrams, phase portraits, frequency spectra, and two-parameter Lyapunov exponents. Some remarkable and exciting properties are found in the new snap system, such as period-doubling transition, asymmetric bubbles, and coexisting bifurcations. Also, we demonstrate that it is possible to generate different varieties of two, three, four, or five coexisting hidden and self-excited attractors in the introduced model. In addition, the amplitude and offset of the hidden chaotic attractors are perfectly controlled for possible application in engineering. Furthermore, a circuit design has been implemented to support the physical feasibility of the proposed model.
TL;DR: In this article , a complete group classification for the Gilson-Pickering equation is presented, where exact traveling wave solutions and bifurcations of the GPE are studied by the method of dynamical systems.
Abstract: The main purpose of this paper is to form a complete group classification for the Gilson–Pickering equation. Exact traveling wave solutions and bifurcations of the Gilson–Pickering equation are studied by the method of dynamical systems. The study on the wave solutions of the model derives a planar Hamiltonian system. Based on phase portraits, many exact explicit parametric representations of wave solutions are obtained under different parametric conditions.
TL;DR: In this article , a nitrogen fixation game with two time delays under nitrogen limitation is investigated and the existence and local stability of the equilibrium points for the nondelay system is discussed.
Abstract: In this paper, a nitrogen fixation game system with two time delays under nitrogen limitation is investigated. Firstly, we discuss the existence and local stability of the equilibrium points for the nondelay system. Then the nitrogen fixation delay and strategy-dependent delay are used as bifurcation parameters to analyze the local stability and the Hopf bifurcation. In addition, we obtain the direction of Hopf bifurcation, the change of the period for periodic solution and the stability of periodic solution via center manifold theory and normal form method. Finally, numerical simulation is employed to visualize the theoretical analysis results and we find that the nitrogen fixation strategy is the dominant strategy when the values of the two time delays are large enough. This study promotes the investigation of the effects of two time delays on the dynamics of evolutionary games with environmental feedback, especially on stability and bifurcation.
TL;DR: This work constructed a nondegenerate 2D exponential hyper chaotic map (2D-EQCM), derived a recursion formula to calculate the number of S-Boxes, and designed an irreversible parallel key expansion algorithm which could transform the initial key to any number of relatively independent round keys.
Abstract: Cryptanalysis of key expansion algorithms in AES and SM4 revealed that (1) there exists weaknesses in their S-Boxes, and (2) the round key expansion algorithm is reversible, i.e. the initial key can be recovered from any round key, which may be explored by the attacker. To solve these problems, first we constructed a nondegenerate 2D exponential hyper chaotic map (2D-EQCM), derived a recursion formula to calculate the number of S-Boxes, and designed a strong S-Box construction algorithm without such weakness. Then based on 2D-EQCM and S-Box, we designed an irreversible parallel key expansion algorithm, which could transform the initial key to any number of relatively independent round keys. Security and statistical analysis demonstrated the flexibility and effectiveness of the proposed irreversible parallel key expansion algorithm.