International Journal of Bifurcation and Chaos

Chaos is ubiquitous in nonlinear dynamical systems. It is a hidden state of deterministic disorder in certain regions of parameters and initial conditions of a nonlinear system. The path taken by the system to reach chaos is known as its route to chaos. The state of the system can drastically change from a stable fixed point or a periodic solution to chaos due to the effect of local or global bifurcation. The available tools, such as bifurcation diagrams, Lyapunov spectra, Poincaré maps, phase plots, and time series plots, explore the routes for such qualitative changes. These are tiresome procedures. For more efficient verifications, this work offers a guide or catalog for several routes to chaos (RTC) through which a continuous-time dynamical system enters or exits from chaos. Additionally, this review paper highlights various signatures associated with common RTC to make identification easier. The main emphasis is on enabling a reader to assess these RTC by merely observing the explained signatures in the bifurcation diagrams and the Poincaré maps.

Hopfield neural networks were widely studied due to their capability of simulating properties of the brain and its sub-networks, as well as their extensive applications for signal and image processing, pattern recognition, and optimization. While a significant number of investigations were devoted to the stability analysis of the Hopfield neural networks, less attention was paid to the analysis of the network behavior in the instability regions. The purpose of this study is to investigate the activity patterns that are generated by the Hopfield neural network composed of three sub-networks with three time delays and coupled by excitatory or inhibitory connections. It is found that both types of neural networks (excitatory and inhibitory) demonstrate synchronous behavior at the steady-state. An increase of the time delay in one or two inhibitory sub-networks results in multiresonances, discrete oscillation periods, and a period-tripling bifurcation. The novel findings are: (1) while the stability threshold of the purely excitatory network does not depend on the time delays, the threshold for purely inhibitory network nonmonotonically depends on the time delays and sub-network configuration; (2) numerical and analytical studies demonstrate that in the neural network that contains a sub-network with much smaller time delay than those in other sub-networks, the interval until the output saturation dramatically decreases compared to the neural network with larger and identical time delays; (3) numerical and analytical studies reveal the existence of quasi-steady-state sub-levels during neural network output development until saturation, which are connected to the neural network structure; and (4) maximum oscillation amplitudes and oscillation periods in the purely inhibitory neural network with three time delays as 2D functions of the time delays demonstrate qualitatively more complex behavior than the neural network with two time delays.

To explore the implementation of different neurons using one type of Locally Active Memristor (LAM), this paper proposes a novel LAM model featuring two S-type, two N-type, and two C-type Locally Active Domains (LADs). The local activity of the memristor is analyzed via the DC V–I plot. By employing the small-signal analysis method, the small-signal equivalent circuits for the memristor operating within each of the distinct LADs are constructed. By analyzing the inductive and capacitive properties of the memristor in different LADs, two second-order different neurons are constructed. According to the complexity theory, two third-order different neuronal circuits are built to analyze the action potential and possible abnormal neural signals (chaos). Based on the Edge of Chaos (EOC) theory and Hopf bifurcation theorem, the study demonstrates that action potentials and chaos occur near or in the EOC domains, thereby elucidating the underlying physical mechanisms of neuromorphic behavior. Finally, field-programmable gate array (FPGA) simulations are conducted to illustrate spike activities, self-sustained oscillations, and chaos within the second-order and third-order neuronal circuits. The outcomes of the simulated circuits are consistent with the theoretical analysis, thereby validating the proposed model.

Coupled logistic map systems have been used in a wide variety of fields including population dynamics, COVID detection, memristors, chemical physics, detection of natural resources such as oil and digital currency forecasting, to name a few. One phenomenon which is of considerable interest in coupled systems is synchronization and then the question of the critical coupling constant beyond which the system synchronizes. The usual method to estimate this coupling constant is via a linear stability analysis of the synchronization state which is local. It is necessary to develop global methods to study the synchronization transition and one possible way can be through the study of invariant measures. With this general aim, we study dissipatively coupled chaotic logistic map systems. The invariant measure of such a system of two coupled logistic maps is generated numerically in a regime where the transition to chaotic asynchronous state has taken place and is shown to have a multifractal nature. We have considered both types of systems in which the parameters of the logistic maps are identical as well as nonidentical. The invariant measure restricted to the synchronization manifold is then analyzed using the wavelet leaders algorithm preceded by fractional integration. The obtained multifractal spectrum is studied while varying the coupling strength. The results obtained are discussed.

In this paper, we present an improved simplified mathematical model for the acute inflammatory response by incorporating two time delays: one for phagocyte migration and another for immune response activation. The model’s dynamic behavior is analyzed by studying the existence and local stability of equilibrium points, as well as the emergence of Hopf bifurcation at the positive equilibrium point. Results show that the two delays have the potential to destabilize the equilibrium point associated with septic death and induce oscillation of the two state variables through Hopf bifurcation. Additionally, the normal form of Hopf bifurcation at specific time delays is obtained by using the method of multiple scales. It is discovered that the amplitude of bifurcating periodic solution is sensitive to time delays. Ultimately, numerical simulations of various bifurcation diagrams and attraction basins suggest that proper selection of model parameters can steer the disease toward a healthy state. The correctness of the theoretical analysis is confirmed by using numerical solutions. Overall, this study not only enhances our understanding of the underlying biological mechanisms of acute inflammatory response, but also provides guidance for the clinical treatment of inflammatory response.

In this study, we investigate the occurrence of a three-frequency quasi-periodic torus in a three-dimensional Lotka–Volterra map. Our analysis extends to the observation of a doubling bifurcation of a closed invariant curve, leading to a subsequent transition into a state of hyperchaos. The absorption of various saddle periodic orbits into the hyperchaotic attractor is demonstrated through distance computation, and we explore the dimensionality of both stable and unstable manifolds. Various routes to cyclic and disjoint quasi-periodic structures are presented. Specifically, we showcase the transition from a saddle-node connection to a saddle-focus connection, leading to the formation of quasi-periodic closed cyclic disjoint curves, as revealed by the computation of one-dimensional unstable manifold. Additionally, we show an unusual transition from a period-2 orbit to a period-6 orbit and uncover the mechanism related to two subsequent bifurcations: (a) sub-critical Neimark–Sacker bifurcation, and (b) saddle-node bifurcation. Our approach involves the use of computational methods for constructing one-dimensional manifolds, extending saddle periodic orbits through a one-parameter continuation, and employing a multidimensional Newton–Raphson approach for pinpointing the saddle periodic orbits in the three-dimensional map.

In this paper, we consider a five-dimensional dynamic hyperjerk system. This kind of system illustrates the use of chaos induction for lightweight picture encryption. Significant advances in internet and multimedia technologies have facilitated the seamless exchange and transmission of confidential information, particularly in the context of images. Here, the local bifurcation of the system is examined, with explicit conditions provided to determine when the zero and zero-double Hopf equilibrium points undergo changes in this particular type of system. Furthermore, the existence of periodic solutions that bifurcate from these types of equilibrium points for the five-dimensional network hyperjerk system is studied. Consequently, a periodic solution of the system is obtained, and its stability is classified. This periodic solution bifurcates from the zero-double Hopf bifurcation equilibrium point, and the first-order averaging method is employed to determine it. Furthermore, two periodic solutions are obtained, and their stability is classified for the system from the projection approach. Notably, these two periodic solutions could not be identified using the averaging method of the first order.

Plants transmit messages through chemical blends, namely Volatile Organic Compounds (VOCs), regarding the news of herbivory to the neighboring plants and natural carnivorous enemies of herbivores. Qualitatively and quantitatively, the blends of Herbivore-induced Plant Volatile (HIPV) help to improve plant growth by recruiting natural enemies of herbivores. We present HIPV-mediated model-based analysis of plant–herbivore–carnivorous natural enemy interaction and explore the complex dynamical behavior. We emphasize the interplay between system parameters, mainly the predation rate and HIPV-induced recruitment rate of natural enemies of herbivores, and emergent dynamical states. The proposed system experiences various changes like oscillatory state to steady-state through Hopf-bifurcation, alternative steady states, and collision of two coexistence equilibria through saddle-node bifurcation. We simultaneously explore the characteristics of basin of attraction, basin transition, and its correlation with system parameters. We thoroughly explain the importance of alternative steady states on plant fitness and the existence of all populations. Based on the high-dimensional Bendixson criterion, we show the global stability of the coexistence equilibrium point. Moreover, we perform numerical simulation to support all possible analytical outcomes and discuss some concluding remarks on the positive impact of HIPV in enhancing plant growth after herbivory damage by signaling natural enemies of herbivores as a reinforcement.

We investigate pattern formation driven by Turing instability in the Lengyel–Epstein model incorporating the fractional Laplacian operator. This approach simulates anomalous superdiffusion, where particle clouds spread more rapidly than predicted by the classical diffusion models. Such phase phenomena are often observed in viruses or biochemical substances that require unusually rapid diffusion. Our study allows for the estimation of superdiffusion effects in biochemistry or thermodynamics by analyzing temporal changes in fractional dimensions that exhibit abnormal patterns. Specifically, we employ the fractal dimension to quantify the complexity of these patterns using the box-counting method. Since this method is based on examining neighboring patterns, that is self-similarity, we observe that the formation of patterns varies with the bifurcations induced by Turing instability. Interestingly, despite these instabilities and variations, the fractal dimension stabilizes to a certain value once the patterns are formed. We also investigate how fractional diffusion affects the fractal dimension. Numerical experiments were performed with various initial conditions, including random values and circles centered with a fixed distance. The fractal dimension was calculated for Turing instability of the Lengyel–Epstein model, and is analyzed for both classical and fractional diffusion following a Lévy process. These tests revealed the formation of different patterns under the Turing instability by varying parameters, including a time variable t, and mass-conserving the chemical concentrations of the activator iodide u and the inhibitor chlorite v. Then, we analyzed these results in terms of the fractal dimension of the patterns. Furthermore, we analyze the consistency with the mesh refinement numerically and examine parameter sensitivity. Our numerical simulation of the fractional-Lengyel–Epstein model examined the relationship between the fractal dimension and the parameters b, c, and the fractional order s, while holding the scaling parameter a as a constant. In general, in regions of instability, higher values of parameter b are associated with lower fractal dimensions, while higher values of the parameter c lead to an increase in fractal dimension. In addition, as the fractional order s increases, the fractal dimension tends to decrease. Although specific parameter values around the bifurcation point influence the fractal dimension, the overall evolutionary trend follows the aforementioned patterns.

This paper investigates unbiased directed transport, known as the ratchet effect, within a system coupled to a fluid modeled by the bailout embedding technique. The system dynamics under the fluid influence are described by a four-dimensional mapping governed by parameters such as dissipation ($\gamma$), ratchet kicking (K), and the relationship between fluid and particle densities ($\alpha$). We explore the behavior of the Ratchet Current (RC) in the parameter spaces $K\times \gamma$ and $K\times \alpha$, particularly focusing on the aerosol case ($\alpha <1$). Our findings reveal a complex interplay of parameters, with larger RCs observed for parameter pairs that induce periodic dynamics in the ratchet system. Furthermore, we observe RC reversal as a function of $\gamma$ and $\alpha$, and identify coexisting attractors (multistability) at higher $\gamma$ values. These results shed light on the intriguing phenomenon of directed transport in fluid dynamics, offering insights that may contribute to understanding particle transport in nature, especially in the aerosol regime.

Numerous image encryption schemes are hindered by weak correlations with plaintex images, inadequate image reconstruction quality, and inefficient transmission and storage. To address these problems, this paper proposes an image compression–encryption algorithm that combines a three-dimensional discrete chaotic system, Josephus circle and compression perception. First, a new three-dimensional discrete hyperchaotic system is constructed, utilizing a hash function to generate the initial value of the chaotic system, through which the measurement matrix and the sequences required for the encryption process are constructed. An encryption algorithm based on chaotic sequence and Josephus circle is designed. Finally, the proposed compressed encryption algorithm is simulated and tested, and the results of experimental tests indicate that the proposed algorithm provides robust security and high-quality reconstruction performance.

This paper discusses reflection invariant properties exist in some idealized mathematical systems without any external perturbations. This study is devoted to quantifying the symmetry breaking of the attractor for a three-dimensional perturbed reflection invariant system, which permits a further detailed investigation of the effect of the perturbation on the symmetry breaking. First, the attractor of a three-dimensional reflection invariant system with extremely tiny perturbation is reproduced employing the symplectic Runge–Kutta method. In the simulation of the attractor, the simulation results are validated by the tiny relative energy error in the conservative field. Subsequently, a general method for characterizing the degree of the symmetry breaking for the perturbed reflection invariant system is proposed, which is achieved by summing the quantized errors between the discrete points after the reflection transformation. This method may be beneficial for the design of complex encryption algorithms and pseudo-random number generators by controlling the perturbed strength to adjust the symmetry-breaking effect of the system. By means of the proposed quantitative method for the reflection invariant property, two important findings are reported. One is that, the strength and the action duration of the perturbation are the key factors affecting the reflection invariant property of the considered system. Another is that the symmetry breaking of the attractor for the perturbed reflection invariant system is irreversible.

In population ecology, the interactions between predators and prey encompass more than just direct predation and killing. Various indirect effects also significantly influence the system’s dynamics. In this paper, we consider a refuge proportional to the prey population and assume that only unsuccessful refuge prey engage in group defense mechanisms. We introduce an exponential response function to model this behavior, taking into account the negative impact of defense measures on the prey growth rate. Subsequently, we analyze the non-negativity, boundedness, and persistence of the system. Furthermore, we study the existence and stability of different types of equilibrium points, and provide theoretical proofs for bifurcations in the system, such as transcritical, pitchfork, saddle-node, Hopf, and Bogdanov–Takens bifurcations. Finally, we verify our theoretical results through numerical simulations. Our study reveals that variations in refuge size and the intensity of group defense can alter the number and stability of equilibrium points, leading to phenomena such as bubbling and periodic oscillations. Additionally, group defense can induce bistability. Both prey refuge behavior and excessive group defense can lead to a reduction in predator numbers and eventually to their extinction. Moreover, compared to prey refuge, group defense as an anti-predator behavior has a greater effect on predators.

In population biology, prey species try to maintain a delicate balance between reproduction and safety. The prey species subject to the mate-finding Allee effect may face a tough choice: risking life for more certain mating opportunities or decreasing mating activity to evade predators in the hope that a longer lifespan will compensate the reduced mating opportunities. In this paper, we consider a discrete-time predator–prey model where the logistically growing prey species experiences the mate-finding Allee effect. We assume that as predator density increases, prey individuals become more vigilant, leading to a decrease in their mate searching activity. We also consider that a constant proportion of prey individuals seek refuge from their predators due to the fear of predation risk. Our numerical simulations reveal that refuge has a stabilizing effect on the population dynamics, while for a certain amount of prey refuge, a decrease in the searching efficiency of mating partners due to the fear of predation risk has a destabilizing effect. We vary the parameters related to the predation-driven Allee effect and refuge simultaneously to explore system dynamics in the parameter plane and observe various intricate and striking properties of the system dynamics. The most significant discovery of our study is the novel organization of Arnold tongues in quasiperiodic and chaotic regimes. The organization of Arnold tongues looks like a “wing of a flapping bird”. Furthermore, we observe bistability between coexisting periodic attractors and notice intriguing patterns in the basins of attraction of these attractors. Lastly, we analyze the effect of the parameters related to the predation-driven Allee effect and refuge on the equilibrium densities of both species and also explore the dynamics of the analogous continuous-time system in the parameter plane.

Chaos regulation, including polarity control and offset boosting, is a new topic with active research for chaos-related applications. Chaotic systems with multiple offset-boostable variables provide more signal candidates with desired polarities and average values for application of chaos to secure communication. In this paper, we propose a new class of two-dimensional offset-boostable chaotic systems with exponential or logarithmic feedback functions. These systems provide two independent constants for nonbifurcating offset boosting in different regimes. The systems with transcendental functions are realized under the combinations of transistor and operational amplifiers, proving directly two-dimensional offset boosting.