Showing papers on "Bifurcation diagram published in 2022"
TL;DR: In this paper , the authors proposed a locally active discrete memristor model for the first time and proved the three fingerprints characteristics of this model according to the definition of generalized memristors.
Abstract: The continuous memristor is a popular topic of research in recent years, however, there is rare discussion about the discrete memristor model, especially the locally active discrete memristor model. This paper proposes a locally active discrete memristor model for the first time and proves the three fingerprints characteristics of this model according to the definition of generalized memristor. A novel hyperchaotic map is constructed by coupling the discrete memristor with a two-dimensional generalized square map. The dynamical behaviors are analyzed with attractor phase diagram, bifurcation diagram, Lyapunov exponent spectrum, and dynamic behavior distribution diagram. Numerical simulation analysis shows that there is significant improvement in the hyperchaotic area, the quasi periodic area and the chaotic complexity of the two-dimensional map when applying the locally active discrete memristor. In addition, antimonotonicity and transient chaos behaviors of system are reported. In particular, the coexisting attractors can be observed in this discrete memristive system, resulting from the different initial values of the memristor. Results of theoretical analysis are well verified with hardware experimental measurements. This paper lays a great foundation for future analysis and engineering application of the discrete memristor and relevant the study of other hyperchaotic maps.
39 citations
TL;DR: In this paper , a new fractional-order 4D neural networks incorporating two different time delays is proposed and the stability and the emergence of Hopf bifurcation are explored.
38 citations
24 citations
TL;DR: The obtained experimental results indicate that the proposed model can capture the main nonlinear behaviors of looseness fault and nonlinear supporting stiffness, verifying the accuracy of theoretical and numerical results.
14 citations
TL;DR: In this paper , the authors analyzed the dynamics of the improved discretised version of the well known Izhikevich neuron model under the action of external electromagnetic field and found that the improved three dimensional IZH map shows rich dynamics.
Abstract: We analyse the dynamics of the improved discretised version of the well known Izhikevich neuron model under the action of external electromagnetic field. It is found that the improved three dimensional IZH map shows rich dynamics. With the variation of the electromagnetic field, period-doubling route to chaos in a repeating fashion is observed from the bifurcation diagram. Even the forward and backward continuation bifurcation diagram which do not completely overlap suggests that there is multistability in the system. The phenomenon of bistability (coexistence of periodic and chaotic attractors) is observed. The presence of periodic and chaotic attractor is aided by the maximal Lyapunov exponent diagram. The Lyapunov phase diagram of electromagnetic field and synapses current shows a large parameter region of chaotic and periodic behaviours with the presence of unbounded regions as well. The IZH map shows a plethora of spiking and bursting patterns such as mixed-mode patterns, tonic spiking, phasic spiking, steady spikes, regular spikes, spike bursting, periodic bursting, phasic bursting, chaotic firing etc with the variation of electromagnetic coupling strength and the synapses current. We also investigate the presence of chimera states in a ring-star, ring, star networks of IZH map neurons. Chimera states are found in the case of ring-star and ring network while synchronised clusters were found in the case of star network and are aided by the spatiotemporal plots, space-time plot, recurrence plots. The rich dynamics shown by the discretised IZH map makes it a promising research model to study about neurodynamics.
14 citations
12 citations
TL;DR: In this article , a star-shaped structure of elastic elements and variable configuration was used for the energy harvesting system with a selected range of the dimensionless excitation frequency and the results of computer simulations were limited to zero initial conditions as it is the natural position of the static equilibrium.
Abstract: The subject of the model research contained in this paper is a new design solution of the energy harvesting system with a star-shaped structure of elastic elements and variable configuration. Numerical experiments focused mainly on the assessment of the configuration of elastic elements in the context of energy harvesting efficiency. The results of computer simulations were limited to zero initial conditions as it is the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, four cases of elastic element configurations were compared. The results are visualized based on the diagram of RMS voltage induced on piezoelectric electrodes, bifurcation diagrams, Lyapunov exponents, and Poincaré maps, showing the impact of individual solutions on the efficiency of energy harvesting. The results of the simulations show that the harvester’s efficiency ranges from 4 V to 20 V depending on the configuration and the frequency range of the excitation, but the design allows for a smooth adjustment to the given conditions.
12 citations
01 Jan 2022
TL;DR: Wang et al. as mentioned in this paper used a gradient adjustment mechanism to establish a dynamic green competition model based on bounded rationality, and the stability of four equilibrium points is researched through different methods, such as 1-D bifurcation diagrams, the largest Lyapunov exponent, as well as 2-dimensional bifurancation diagrams.
Abstract: Green and low carbon is an important way to coordinate economic and social development, which is also true for enterprises. Hence, the essay mainly builds up a green supply chain consisting of one supplier and two manufacturers. Gradient adjustment mechanism is used to establish a dynamic green competition model based on bounded rationality in this paper. Subsequently, the stability of four equilibrium points is researched through different methods. Dynamic behaviors are analyzed by 1-D bifurcation diagrams, the largest Lyapunov exponent, as well as 2-D bifurcation diagrams. We have found that the system goes from a steady state to chaos mainly via flip bifurcation. The chaotic state of system can be alleviated as the increase of adjustment speed in a certain range. Furthermore, invariant sets and natural transverse Lyapunov exponent are used to study synchronization behavior. Basins of attraction under the coexistence of attractors are simulated numerically. The results indicate that the system will have two and four groups of coexisting attractors, where the structure and size of the attractor and its basin will change as the speed of adjustment increases. When the attractor contacts with the boundary of its basin of attraction, the global bifurcation occurs.
12 citations
TL;DR: In this article , a local/global dynamic instability study of post-buckled nanotubes transporting pulsatile flow is conducted for the first time, where the size-dependence of both the solid and fluid is modeled by the nonlocal strain gradient beam model coupled with surface elasticity.
10 citations
10 citations
TL;DR: In this paper , the variable-coefficient Davey-Stewartson system is reduced to two-dimensional nonlinear ordinary differential equations, and the traveling wave transformation is employed to obtain the exact traveling wave solution.
Abstract: This paper mainly studies the bifurcation and single traveling wave solutions of the variable-coefficient Davey–Stewartson system. By employing the traveling wave transformation, the variable-coefficient Davey–Stewartson system is reduced to two-dimensional nonlinear ordinary differential equations. On the one hand, we use the bifurcation theory of planar dynamical systems to draw the phase diagram of the variable-coefficient Davey–Stewartson system. On the other hand, we use the polynomial complete discriminant method to obtain the exact traveling wave solution of the variable-coefficient Davey–Stewartson system.
TL;DR: In this article , a visualization experiment was conducted to study the formation of droplets in the cross-junctions with various bifurcation angles, and a regime diagram was provided to characterize the droplet formation regimes.
TL;DR: In this article , a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method.
Abstract: In this paper, a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained. Then, the mathematical model is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method (ADM) algorithm. So as to study the dynamic behavior in detail, not only the equilibrium stability of the system is analyzed, but also the dynamic characteristics are analyzed by means of a Bifurcation diagram and Lyapunov exponents (Les). By analyzing the dynamic behavior of the system, some special phenomena, such as the coexistence of attractor and state transition, are found in the system. In the end, the circuit implementation of the system is implemented on a Digital Signal Processing (DSP) platform. According to the numerical simulation results of the system, it is found that the system has abundant dynamical characteristics.
TL;DR: In this paper , the authors address the question of which small, bimolecular, mass action chemical reaction networks (CRNs) are capable of Andronov-Hopf bifurcation.
Abstract: We address the question of which small, bimolecular, mass action chemical reaction networks (CRNs) are capable of Andronov–Hopf bifurcation (from here on abbreviated to ‘Hopf bifurcation’). It is easily shown that any such network must have at least three species and at least four irreversible reactions, and one example of such a network with exactly three species and four reactions was previously known due to Wilhelm. In this paper, we develop both theory and computational tools to fully classify three-species, four-reaction, bimolecular CRNs, according to whether they admit or forbid Hopf bifurcation. We show that there are, up to a natural equivalence, 86 minimal networks which admit nondegenerate Hopf bifurcation. Amongst these, we are able to decide which admit supercritical and subcritical bifurcations. Indeed, there are 25 networks which admit both supercritical and subcritical bifurcations, and we can confirm that all 25 admit a nondegenerate Bautin bifurcation. A total of 31 networks can admit more than one nondegenerate periodic orbit. Moreover, 29 of these networks admit the coexistence of a stable equilibrium with a stable periodic orbit. Thus, fairly complex behaviours are not very rare in these small, bimolecular networks. Finally, we can use previously developed theory on the inheritance of dynamical behaviours in CRNs to predict the occurrence of Hopf bifurcation in larger networks which include the networks we find here as subnetworks in a natural sense.
TL;DR: In this article , a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points, is presented.
Abstract: The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.
TL;DR: In this paper , a class of two-dimensional memristive maps with a cosine memristor is presented, which can produce hidden periodic, chaotic, and hyper-chaotic attractors.
Abstract:
In this work, we present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are demonstrated and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map has been carried out to reveal the bifurcation mechanism of the dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors. They can exhibit extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, rarely shown in memristive maps. Potentially, this work can be used for secure communication, such as data and image encryption.
TL;DR: In this paper , the authors consider the dynamics of a system composed of a van der Pol oscillator linearly coupled to a Duffing oscillator and analyze how the explicit symmetry break modifies the phase space location and nature of equilibrium points of the coupled system.
Abstract: Owing to recently published works, the issues of multistability and symmetry breaking can be listed amongst the most followed ongoing research topics in nonlinear science. In this contribution, we consider the dynamics of a system composed of a van der Pol oscillator linearly coupled to a Duffing oscillator (Han, 2000; Kengne et al., 2012). Mention that coupled attractors of different types serve as convenient models for real world systems such as electromechanical, biological, physical, or economic systems. We analyze how the explicit symmetry break modifies the phase space location and nature of equilibrium points of the coupled system, the topology and number of competing attractors, the bifurcation modes, and the shape of the basins of attraction. These investigations are executed by resorting to classical nonlinear tools such as basins of attraction, phase portraits, plots of 1D and 2D largest Lyapunov exponent diagrams, and 1D bifurcation diagrams as well. We report intricate dynamical features such as critical transitions, hysteresis, the coexistence of (symmetric or asymmetric) bubbles of bifurcation and the occurrence of multiple coexisting dynamics (i.e. two, three, four or five coexisting attractors) resulting from the variation of both initial states and parameters of the coupled system.
TL;DR: Inspired by the non-autonomous chaos system and the transient chaotic oscillation, based on the chaos systems of the Lorenz family, the system structure, parameters, and orders are adjusted, and the damping oscillation and sustained chaotic oscillations are found as mentioned in this paper .
Abstract: Inspired by the non-autonomous chaos system and the transient chaotic oscillation, based on the chaos systems of the Lorenz family, the system structure, parameters, and orders are adjusted, and the damping oscillation, sustained chaotic oscillation, transient chaotic oscillation, and chaotic oscillation with multiple chaotic trajectories are found. Firstly, the constrain conditions of the system parameters and fractional-orders are derived. Secondly, one external driving force signal and two control parameters are added to the system, different dynamic behaviors of the system under the excitement of different types, features of the external driving force signals are analyzed, simulated, and discussed. The phase diagram, bifurcation diagram, and Lyapunov exponents, etc are adopted for analysis. Finally, based on the adjusted fractional-order nonlinear systems, a kind of signal detection scheme that can be used under a strong noise environment is designed, some key information of the signal to be detected can be acquired. The simulation results show the effectiveness of the designed signal detection scheme.
TL;DR: In this article , a novel four-dimensional hyper-chaotic system (4D-HCS) based on a tri-valued memristor is found, which has complex hyper-chaos dynamics such as hidden attractors and coexistent attractors.
Abstract: Recently, the nonlinear dynamics of memristor has attracted much attention. In this paper, a novel four-dimensional hyper-chaotic system (4D-HCS) based on a tri-valued memristor is found. Theoretical analysis shows that the 4D-HCS has complex hyper-chaotic dynamics such as hidden attractors and coexistent attractors. We also experimentally analyze the dynamics behaviors of the 4D-HCS in aspects of the phase diagram , bifurcation diagram, Lyapunov exponential spectrum, power spectrum and the correlation coefficient . To rigorously verify the chaotic behavior, we analyze the topological horseshoe of the system and calculate the topological entropy. In addition, the comparison with binary-valued memristor-based chaotic system shows that a chaotic system with a tri-valued memristor can generate hyper-chaos and coexistent attractors, while the one with a binary-valued memristor cannot. This finding suggests that applying three- or multi-value memristors in chaotic circuits can produce more complex dynamic properties than binary-valued memristors. To show the easy implementation of the 4D-HCS, we implement the 4D-HCS in an analogue circuit-based hardware platform, and the implementation results are consistent with the theoretical analysis. Finally, using the 4D-HCS, we design a pseudorandom number generator to explore its potential application in cryptography .
TL;DR: In this paper , two discrete memristors with cosine with amplitude memristance are designed based on the discrete Memristor model to improve the dynamic of a classic chaotic map.
Abstract: The discrete memristor has aroused increasing interest. In this paper, two discrete memristors with cosine with amplitude memristance are designed based on the discrete memristor model. The Simulink models of the two discrete memristors are built to verify that they meet the definition of the memristor. To improve the dynamic of a classic chaotic map, the discrete memristors are introduced into two chaotic maps: a Logistic map and a Hénon Map. Through the trajectory analysis, Lyapunov exponent, bifurcation diagram, and complexity analysis, it is shown that discrete memristors can indeed make the dynamical behaviors of chaotic maps richer and more complex.
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.
01 Jun 2022
TL;DR: In this article , the stability and Hopf bifurcation analysis for a type of fractional-order bidirectional associative memory (BAM) neural network involving two neurons in the X-layer and n neurons in Y-layer, respectively, were analyzed for the case of no leakage delay.
Abstract: This article aims to deal with the stability and Hopf bifurcation analysis for a type of fractional-order bidirectional associative memory (BAM) neural network involving two neurons in the X-layer and n neurons in the Y-layer, respectively. In view of the universal existence and multiplicity of time delay in many real systems, leakage delay and nonuniform communication delays are both taken into account. Coates’s flow-graph formula is efficiently adopted to solve the high-order characteristic equation of the associated linearized system. By making some assumptions on the mixed time delays, the obtained characteristic equation only contains the leakage delay, which is selected as the bifurcation parameter. Utilizing the discriminated criteria of stability for fractional-order dynamical systems and Hopf bifurcation theory, we obtain the critical value of the bifurcation point, greater than which the Hopf bifurcation would occur. Particularly, the stability and Hopf bifurcation is also analyzed for the case of no leakage delay to get an insight into the effect of the leakage delay. Finally, the validity of our theoretical results is substantiated through a simulation example.
TL;DR: In this article , the dynamical behaviors of fractional q-deformation chaotic map are analyzed by employing the Caputo delta difference operator, and two controllers are designed to study the chaos control and synchronization of the chaotic map.
TL;DR: The obtained bifurcation diagram and the plot of the Lyapunov exponent suggest that the so-called robust chaos characterizes this map, and the article proposes a new image encryption algorithm that uses, among others, cyclically shifted S-box or saving encrypted pixels on the first or last free space in the cipher-image.
Abstract: One of the applications of dynamical systems with chaotic behavior is data encryption. Chaos-based cryptography uses chaotic dynamical systems as the basis for creating algorithms. The present article discusses a new dynamical system called M-map with its analysis: fixed points, bifurcation diagram, Lyapunov exponent, and invariant density. The obtained bifurcation diagram and the plot of the Lyapunov exponent (with a minimum value of ln2 and a maximum value of ln4) suggest that the so-called robust chaos characterizes this map. Moreover, the obtained results are compared with other dynamical systems used in cryptography. Additionally, the article proposes a new image encryption algorithm. It uses, among others, cyclically shifted S-box or saving encrypted pixels on the first or last free space in the cipher-image. The conducted analysis shows that the cipher-images are characterized by an entropy value close to 8, a correlation of adjacent pixels value close to 0, or values of Number of Pixel of Change Rate (NPCR) and Unified Average Changing Intensity (UACI) measures close to 100% and 33%, respectively.
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 paper , a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors is reported, which has some interesting characteristics for different parameters and initial conditions.
Abstract: Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We discuss the rich dynamics of the proposed system, which have some interesting characteristics for different parameters and initial conditions, through the use of dynamic analysis tools such as the phase portrait, Lyapunov exponent spectrum, and bifurcation diagram. The topological classification of the periodic orbits of the system is investigated by a recently devised variational method. Symbolic dynamics of four and six letters are successfully established under two sets of system parameters, including hidden and self-excited chaotic attractors. The system is implemented by a corresponding analog electronic circuit to verify its realizability.
TL;DR: In this article , the authors investigate the dynamics of a discrete-time stage-structured population model where juveniles and adults may be subject to threshold harvesting, which allows the harvesting of the target population only if its size exceeds a predetermined threshold.
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.