International Journal of Bifurcation and Chaos

Complex spatiotemporal patterns in nature significantly challenge reductionism-based modern science. The lack of a paradigm beyond reductionism hinders our understanding of the emergence of complexity. The diversity of countless patterns undermines any notion of universal mechanisms. Here, however, we show that breaking the symmetry of three simple and self-organization rules gives rise to nearly all patterns in nature, such as a wide variety of Turing patterns, fractals, spiral, target and plane waves, as well as chaotic patterns. The symmetry breaking is rooted in the basic physical quantities, such as positive and negative forces, space, time and bounds. Besides reproducing the hallmarks of complexity, we discover some novel phenomena, such as abrupt percolation of Turing patterns, phase transition between fractals and chaos, chaotic edge in traveling waves, etc. Our asymmetric self-organization theory established a simple and unified framework for the origin of complexity in all fields, and unveiled a deep relationship between the first principles of physics and the complex world.

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

In recent years, many image encryption schemes have adopted Hilbert curves for encryption. In this approach, the Hilbert curve is used to encrypt grayscale images by traversal scrambling. However, the correlation between pixels has not been fully considered and those algorithms are not safe enough. To solve this problem, a new image encryption algorithm based on a new chaotic system of 2D-LICM (Two-Dimensional Linear-Infinite-Collapse Chaotic Map) and an improved Hilbert curve is proposed in this paper. First, we propose a new 2D-chaotic system to address the shortcoming that the commonly used chaotic systems are too simple in scope and complexity. Then, a new image encryption algorithm is proposed using the newly designed 2D-LICM and the improved Hilbert curve. The proposed algorithm uses Hilbert curve to reduce the correlation between adjacent pixels of the image at the pixel and bit levels and increase the scrambling and diffusion effects. Simulation and security analysis results show that the proposed scheme has high security and is superior to several advanced image encryption algorithms.

This paper presents a novel discrete memristor model that incorporates exponential and absolute value functions. A discrete coupled memristor neural network model is constructed based on this memristor design. The periodic and chaotic regions of the discrete neural network model are determined using bifurcation and Lyapunov exponent spectrum. Furthermore, by varying the initial values of the discrete memristive neural network, we observe the coexistence of chaos and periodic attractors, as well as periodic attractors. Additionally, an application to color image encryption based on the discrete system model is given. Security analysis is conducted in the aspects of key space, histogram analysis, correlation analysis, sensitivity analysis, Peak Signal-to-Noise Ratio (PSNR), and information entropy analysis. The analysis results show that the algorithm has a key space size of $2^{258}$, and the information entropy of baboon graph is 7.9993, which is very close to the ideal value of 8. It shows that the image encryption algorithm is feasible and effective. Finally, the implementation of the discrete memristive neural network model is realized using Field Programmable Gate Array (FPGA). The experimental implementation is conducted using the Verilog language on the Vivado 2018.3 platform, and the obtained results align with the numerical simulation results obtained through MATLAB.

This contribution investigates a photosensitive circuit based on a mutator for emulating memristor, memcapacitor, and meminductor. The circuit contains a phototube, a mutator, and several basic electronic components. The phototube is employed within the circuit to capture the external illumination for energy injection in order to obtain a continuous signal source. By varying the light intensity on the phototube, chaotic dynamics such as attractors, bifurcation diagrams, and Lyapunov exponents spectrum are analyzed, while the mutator is switched to different mem-elements without changing the structure of the circuit. Especially, the special phenomenon of the coexistence of attractors is discovered. Ultimately, the circuit is realized using DSP. This circuit holds promise in offering potential insights into the dynamics of neural networks, and it could find applications in the development of highly sensitive sensors.

In this work, a new glucose–insulin model incorporating time delay and obesity is developed to gain insights of its dynamical mechanisms. Through the method of multiple scales, we theoretically demonstrate that time delay can drive the system to yield Hopf bifurcation, thereby producing oscillating solutions that are consistent with the simulation results. Moreover, obesity changes the level of glucose, but cannot induce oscillations. In particular, it is found that under the combined effect of obesity and time delay, obesity delays the appearance of Hopf bifurcation which is induced by time delay. Results show that a low calorie diet can achieve therapeutic effects including reducing blood glucose fluctuations and insulin resistance, which can be used as an adjuvant for the treatment of diabetes. In addition, our results indicate that the delay, together with an optimal rate of model parameters can cause a variety of dynamics and induce glucose oscillations. The result obtained in this paper may help to better understand the obesity, diabetes, and the interaction between glucose and insulin, so that control strategies can be designed to better regulate blood glucose levels and fluctuations and mitigate the occurrence of type-2 diabetes.

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree $\frac{6}{5}$. In contrast to the previously studied one of degree $\frac{4}{3}$, the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

In this paper, we proposed a Holling-III predator–prey model considering the perturbation of slow-varying, carrying capacity parameters. The study aims to address how the slow changes in carrying capacity influence the dynamics of the model. Based on the bifurcation theory and the slow–fast analysis method, the existence and the equilibrium of the autonomous system are explored, and then, the critical condition of Hopf bifurcation and transcritical bifurcation is established for the autonomous system. The slow–fast coupled nonautonomous system has quasiperiodic oscillations, single Hopf bursting oscillations, and transcritical–Hopf bursting oscillations within a certain range of perturbation amplitude variation if the carrying capacity perturbation amplitude crosses some critical values, such that the predator–prey management is challenging for the extinction of predator populations under the critical value. The motion pattern of the nonautonomous system is closely related to the transcritical bifurcation, Hopf bifurcation and attractor type of the autonomous system. Finally, the effects of changes in parameters related to predator aggressiveness on system behavior are investigated. These results show how crucial the predator–prey control is for varying carrying capacities.

Phantom attractors in nonlinear systems under additive stochastic excitation have been recently discovered. This paper uncovers the existence of phantom attractors in a single-degree-of-freedom smooth nonlinear equation, which characterizes the vibration of an inextensible beam subjected to lateral stochastic excitation. It also elucidates that the stochastic averaging method, in this context, may lead to qualitatively erroneous probability density functions, identified as one of the reasons why these attractors were previously overlooked. The study then proceeds to analyze the formation process of the phantom attractor and the critical noise intensity associated with it. Subsequently, the key nonlinear term related to the emergence of phantom attractors is identified by observing whether the system still exhibits phantom attractors after the corresponding nonlinear terms are removed. It is revealed that in this system, the presence of phantom attractors is closely linked to the inertia nonlinearity of the hardening type. The system investigated in this paper is simpler compared to previously identified systems capable of generating phantom attractors. This simplicity aids in facilitating research focused on unraveling the general principles behind the formation of phantom attractors.

In this paper, we propose a host–parasitoid model with a Holling type-II functional response and incorporate harvest effort. The Holling type-II response leads to saturation in parasitized hosts, creating a potential economic harvesting opportunity. To address overexploitation risks, we integrate a harvest effort function, determining an optimal threshold to prevent depletion. We explore model dynamics and bifurcations, including co-dimension one behaviors such as flip and Neimark–Sacker bifurcations, we provide numerical examples for validation. Our suggested difference-algebraic model, compared to continuous-time models, exhibits rich dynamics within the Nicholson–Bailey host–parasitoid framework.

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results.

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model.

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.

In a food chain, the role of intake patterns of predators is very influential on the survival and extinction of the interacting species as well as the entire dynamics of the ecological system. In this study, we investigate the affluent and intricate dynamics of a simple three-species food chain model in a discrete-time framework by analyzing the parameter plane of the system with simultaneous changes of two crucial parameters, the predation rates of middle and top predators. From the theoretical viewpoint, we study the model by determining the fixed points’ biological feasibility and local asymptotic stability criteria, and performing some analyses of local bifurcations, namely, transcritical, flip, and Neimark–Sacker bifurcations. Here, we initiate the numerical simulation by plotting the changes of the prey population density in terms of a vital parameter of the system, and we observe the switching among different dynamical behaviors of the system. We also draw some phase portraits and plot the time series solutions to show the diverse characteristics of the system dynamics. Further, we move one step ahead to explore the intricate dynamical scenarios appearing in the parameter plane by forming Lyapunov exponent and isoperiodic diagrams. In the parameter plane of the system, we see the emergence of innumerable Arnold tongues. All these Arnold tongues are organized along a particular direction, and the beautiful arrangement of these tongues forms several kinds of period-adding sequences. The study sheds more light on various types of multistability occurring in the model system. We see the coexistence of three periodic attractors in the parameter plane. In this study, the most striking observation is the coexistence of four periodic attractors, which occurs infrequently in ecological systems. We draw the basins of attraction for the tristable and tetrastable attractors, which are complex Wada basins. A system with Wada basin is very sensitive to initial conditions and more erratic in nature than a system with fractal basin. Also, we plot the density of all interacting species in terms of the predation rates of middle and top predators and observe the variation in the population densities of all species with the variability of these two key parameters. In the parameter plane created by the simultaneous changes of two parameters, the system exhibits a variety of intricate and subtle dynamics, which cannot be found by changing only a single parameter.

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.