International Journal of Bifurcation and Chaos

The border-collision normal form is a canonical form for two-dimensional, continuous maps comprised of two affine pieces. In this paper, we provide a guide to the dynamics of this family of maps in the noninvertible case where the two pieces fold onto the same half-plane. Most significantly we identify parameter regimes for the occurrence of key bifurcation structures, such as period-incrementing, period-adding, and robust chaos. We characterize the simplest and most dominant bifurcations and illustrate various dynamical possibilities such as invariant circles, two-dimensional attractors, and several cases of coexisting attractors. We then apply the results to a classic model of a boost converter for adjusting the voltage of direct current. It is known that for one combination of circuit parameters the model exhibits a border-collision bifurcation that mimics supercritical period-doubling and is noninvertible due to the switching mechanism of the converter. We find that over a wide range of parameter values, even though the dynamics created in border-collision bifurcations is in general extremely diverse, the bifurcation in the model can only mimic period-doubling, although it can be subcritical.

Recently, we presented two methods of constructing periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom [Katsanikas & Wiggins, 2021a, 2021b]. These methods were illustrated with an application to a quadratic normal form Hamiltonian system with three degrees of freedom. More precisely, in these papers we constructed a section of the dividing surfaces that intersect with the hypersurface $x=0$. This was motivated by studies in reaction dynamics since in this model reaction occurs when the sign of the $x$ coordinate changes. In this paper, we continue the work of the third paper [Katsanikas & Wiggins, 2023] of this series of papers to construct the full dividing surfaces that are obtained by our algorithms and to prove the no-recrossing property. In the third paper we did this for the dividing surfaces of the first method [Katsanikas & Wiggins, 2021a]. Now we are doing the same for the dividing surfaces of the second method [Katsanikas & Wiggins, 2021b]. In addition, we computed the dividing surfaces of the second method for a coupled case of the quadratic normal form Hamiltonian system and we compared our results with those of the uncoupled case. This paper completes this series of papers about the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom.

This paper proposes a new chaotic system 2D-HLM, which is a combination of Henon map and logistic map. SHA-256 algorithm based on the plaintext image produces the initial value, which enhances the correlation with the plaintext. Therefore, the algorithm avoids the disadvantages of being easily cracked by selected plaintext attacks. The chaotic sequence generated by 2D-HLM is adopted to scramble an image, and the bit plane is extracted and reorganized on the scrambled image. Based on the relationship between two mathematical propositions of the logistic map operations, a novel propositional logic coding algorithm is proposed. The simulation results show that the algorithm has large key space and high key sensitivity, and can resist common attacks such as differential attack.

In this paper, a four-dimensional chaotic system based on a flux-controlled memristor with cosine function is constructed. It has infinitely many equilibria. By changing the initial values $x(0)$, $z(0)$ and $u(0)$ of the system and keeping the parameters constant, we obtained the distribution of infinitely many single-wing and double-wing attractors along the $u$-coordinate, which verifies the initial-offset boosting behavior of the system. Then the complex dynamical behavior of the system is studied in detail through the phase portraits of coexisting attractors, the average value of state variables, Lyapunov exponent spectrum, bifurcation diagram, attraction basin and the complexity of spectral entropy (SE). In addition, the simulation of the Multisim circuit is also carried out, and the results of numerical simulation and analog circuit simulation are consistent. Finally, the chaotic sequence generated by the system is applied to image encryption, and according to the performance analysis, the proposed chaotic system has good security performance.

Electric energy routers (EERs) can effectively deal with the energy management issues caused by the access of multi-sources and multi-loads. Different from the energy transmission form in the existing series architecture EER (SA-EER) for low-voltage distribution networks, a new series-parallel architecture EER (SPA-EER) has the advantages of high-power transmission capability and reactive power flexible operation. However, if controller parameters change beyond their critical stability boundaries, SPA-EER will produce slow- and fast-scale bifurcation behaviors, which are manifested as low- and high-frequency oscillations of port voltages or port currents in varying degrees. To avoid the system instability caused by controller parameter changes, how to discriminate the stability regions of controller parameters for SPA-EER is a key research content in this paper. To this end, the stroboscopic mapping discrete model of SPA-EER system is first established, and system bifurcation behaviors are analyzed based on bifurcation theory. Furthermore, the critical stability boundaries of controller parameters are discriminated by using Jacobian matrix, root locus and bifurcation diagram. After that, a time delay feedback control (TDFC) is employed to suppress system bifurcation behaviors. Finally, the correctness of bifurcation analysis and the effectiveness of TDFC are verified by a hardware-in-the-loop (HIL) experimental platform.

Chaos is a nonlinear phenomenon that reveals itself everywhere in nature and in many fields of science. It has gained increasing attention from researchers and scientists over the last two decades. In this article, the nature of the fixed and periodic states are examined for a discrete two-parameter map; a composition of Euler’s numerical map and the logistic map. Further, the dynamical properties such as fixed states, period-doubling, and stability in fixed and periodic states are also described and the onset of chaos is characterized in detail followed by a few lemmas and remarks. Afterward, some scaling methods such as the bifurcation scale, fork-width scale, and Lyapunov exponent are illustrated to examine the appearance of chaos for the discrete two-parameter map. Experimental and numerical simulations are conducted followed by some bifurcation graphs, tables, and remarks. The scaling property is discussed in two key parameters. In addition, a comparative analysis of fork-width length, bifurcation length, and the maximum Lyapunov exponent is also presented to demonstrate the validity of the results.

We study the influence of the coexisting steady states in high-dimensional systems on the dynamical evolution of the vector field when a slow-varying periodic excitation is introduced. The model under consideration is a coupled system of Bonhöffer–van der Pol (BVP) equations with a slow-varying periodic excitation. We apply the modified slow–fast analysis method to perform a detailed study on all the equilibrium branches and their bifurcations of the generalized autonomous system. According to different dynamical behaviors, we explore the dynamical evolution of existing attractors, which reveals the coexistence of a quasi-periodic attractor with diverse types of bursting attractors. Further investigation shows that the coexisting steady states may cause spiking oscillations to behave in combination of a 2D torus and a limit cycle. We also identify a period-2 cycle bursting attractor as well as a quasi-periodic attractor according to the period-2 limit cycle.

In this paper, we consider two-dimensional cellular automata (CA) with the von Neumann neighborhood. We study the characterization of 2D linear cellular automata defined by the von Neumann neighborhood with new type of boundary conditions over the field ${\mathbb{Z}}_p$. Furthermore, we investigate the rule matrices of 2D von Neumann CA by applying the group of permutations $S_4$. Moreover, the algorithm for computing the rank of rule matrices is given. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for considered two-dimensional cellular automata are obtained.

At present time, sustainable crop production is of prime importance due to the expansion of human population and diminishing cultivable land. Insects attack the plants’ roots, blooms and leaves and lessen the agricultural production across the globe. In this research work, we propose a nonlinear mathematical model to manage the spray of insecticides to control insect population and increase crop production. In the model formulation, we consider that the spraying of insecticides is attributed to both the density of insects and loss in crop production. This study identifies the range of spraying rate of insecticides at which the model system shows bistability behavior and its threshold value after which system stabilizes to the equilibrium with higher crop production. Further, we have also demonstrated that the model undergoes transcritical, saddle-node, Hopf, and Bogdanov–Takens bifurcations. The extensive numerical simulation is performed to validate the analytical findings.

When chaotic systems are used for speech encryption, their chaotic performance largely determines the security of speech encryption. However, traditional chaotic systems have problems such as parameter discontinuity, easy occurrence of chaos degradation, low complexity, and the existence of periodic windows in chaotic intervals. In real applications, chaotic mappings may fall into periodic windows, which is extremely unfavorable for security. In this paper, a new chaotic mapping 2D-LMSM is proposed by improving the chaotic logistic and sine mappings, and applied to speech encryption. Performance evaluation shows that this map can effectively generate robust chaotic signals in a wide parameter range. The 2D-LMSM achieves better robustness and desired chaotic properties than several existing two-dimensional chaotic maps. We propose a novel speech encryption algorithm using this map. First, it performs Fast Fourier Transform (FFT) on the input speech signal to obtain real and imaginary values, which are encrypted by one-time scrambling encryption and XOR diffusion encryption with pseudorandom numbers generated by chaos; then, it performs secondary scrambling encryption by Discrete Wavelet Transform (DWT) and 2D-LMSM; finally, it obtains encrypted speech data by Discrete Wavelet Inverse Transform (IDWT) and Fast Fourier Inverse Transform (IFFT). Experimental results show that this algorithm has good encryption and decryption performances and ensures system security.

In this research article, a three-species food chain model with Allee effect and additional food is proposed and analyzed. The Allee effect and additional food are introduced to the top predator population. The dynamical behavior of the system is studied analytically and numerically. We have performed equilibrium analysis and local stability analysis around the non-negative equilibria. We have also explored different bifurcations in the system. We have drawn several one- and two-parameter bifurcation diagrams to explore complex dynamical behaviors. We observe that top predator goes to extinction when Allee parameter crosses a threshold value, whereas additional food enhances the stability and persistence of the system.

In this paper, we study one type of predator–prey model with simplified Holling type IV functional response by incorporating the fear effect into prey species. The existence and stability of all equilibria of the system are studied. And bifurcation behaviors including saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation of the system are completely explored. Numerical simulation is carried out to illustrate the theoretical analysis. It is shown that the fear effect does affect some dynamic behaviors of the system. Finally, we summarize the findings in a conclusion.

A delay partial difference equation with a delay controller is studied in this paper. We show that it can lead to both Devaney chaos and Li–Yorke chaos by applying the modified Marotto’s theorem, with three criteria established for generating chaos. Computer simulations of chaotic behaviors and spatiotemporal graphs are performed with three examples.

In this paper, the linearizability of a $p:-q$ resonant differential system is studied. First, we describe a method to compute the necessary conditions for linearizability based on blow-up transformation. Using the method, we compute necessary linearizability conditions for a family of $1:-3$ resonant system with quadratic nonlinearities. The sufficiency of the obtained conditions is proven either by the Darboux linearization method or using the recursive procedure after blow-up transformation.

For the generalized Serre–Green–Naghdi system with weak Coriolis effect and surface tension, by using the dynamical system methods and singular traveling wave theory developed by Li and Chen [2007] to its associate traveling wave system, under different parameter conditions, all possible bounded solutions (solitary wave solutions, periodic wave solutions, peakons, periodic peakons as well as compacton solution families) are obtained. Exact explicit parametric representations are given.