In this paper, by the example of 3D model of enzyme reaction, we study mechanisms of noise-induced generation of complex multimodal chaotic oscillations in the monostability zone where only simple deterministic cycles are observed. In such a generation, a constructive role of deterministic toroidal transients is revealed. We perform a statistical analysis of these phenomena and localize the intensity range of the noise that causes stochastic P- and D-bifurcations connected with transitions to chaos and qualitative changes in the probability density. Constructive possibilities of the stochastic sensitivity function technique in the analytical study of these phenomena are demonstrated.

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.

This paper studies the limit cycle bifurcations of a class of planar cubic isochronous centers. For different values of two key parameters, we give an estimate of the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems under arbitrarily small piecewise smooth polynomial perturbation. The main method and technique are based on the first order averaging theory for discontinuous systems and the Argument Principle in complex analysis.

This paper proposes a new nonlinear transverse-torsional coupled model for single-stage gear transmission system, by taking transmission error, time-varying meshing stiffness, backlash, bearing clearances and the self-adaptive double-parameter control module into account. The nonlinear differential governing equation of system motion is derived and solved by applying variable step-size Runge–Kutta numerical integration method. The system’s nonlinear dynamic characteristics and stability are investigated systematically by a bifurcation diagram of the Poincaré map and parameter stability region. Firstly, the velocity bifurcation diagrams have shown that, under the same damping ratio and backlash and with the increase of control parameter Ph1, the route to chaos in the subcritical velocity region is first experienced from crisis to periodic doubling, and to crisis again, but the route that reverts to periodic motion in the super-critical velocity region is not affected. Additionally, the backlash is found to be the key parameter to affect the route to chaos as well. With the increase of the backlash, the crisis becomes the unique route to chaos in sub-critical region no matter what the Ph1 is, but the increase of Ph1 could change the route that reverts to periodic motion from 3T-periodic attractor to 2T-periodic attractor. Secondly, with the increase of the control parameter Ph2, the system starts to enter the chaotic motion and exit the chaos state at different critical points and through different routes. Besides, the unstable region could shrink dramatically and the route to crisis is suppressed as well with the increase of damping ratio. Thirdly, the motion stability region analysis established in full range of double-parameter and velocity provides a mathematical reference model and is stored in control module, which could be utilized to make the control module seek a nearest parameter set automatically that could make the motion stable again in the quickest way under unstable working condition. Finally, according to global motion stability diagram, the forbidden zones that cannot make the system motion stable by adjusting single control parameter are revealed, which has remarkable guiding value during the practical operation especially under the manual adjusting working condition.

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, we propose a fast hyperchaotic image encryption scheme based on RSVM and step-by-step scrambling-diffusion. In this scheme, we firstly propose a new algorithm named ring shrinkage with variable modulo (RSVM), which can randomly scramble the elements in a one-dimensional array, which are composed of the row numbers or column numbers of the pixel matrix to be encrypted. Before encryption, we use RSVM algorithm to generate two random one-dimensional arrays of pixel matrix (i.e. row array As and column array Bs), and each element in the As/Bs represents the row/column number in the pixel matrix. Then the rows/columns of the pixel matrix are scrambled-diffused step-by-step according to the row/column numbers in the As/Bs. The initial control parameters of RSVM algorithm are controlled by SHA-256 of plaintext pixels, and RSVM algorithm controls the step-by-step scrambling-diffusion process of pixel matrix, rows and columns, so the small changes of plaintext pixels will lead to great differences in ciphertext images. In addition, the overall time complexity of the image encryption scheme is only Θ(1.5(M+N)), which can greatly reduce the time cost. Finally, the experimental results and extensive security analysis prove the efficiency and feasibility of this image encryption method.

Periodicity analysis of sequences generated by a deterministic system is a long-standing challenge in both theoretical research and engineering applications. To overcome the inevitable degradation of the logistic map on a finite-precision circuit, its numerical domain is commonly converted from a real number field to a ring or a finite field. This paper studies the period of sequences generated by iterating the logistic map over ring ℤ3n from the perspective of its associated functional network, where every number in the ring is considered as a node, and the existing mapping relation between any two nodes is regarded as a directed edge. The complete explicit form of the period of the sequences starting from any initial value is given theoretically and verified experimentally. Moreover, conditions on the control parameter and initial value are derived, ensuring the corresponding sequences to achieve the maximum period over the ring. The results can be used as ground truth for dynamical analysis and cryptographical applications of the logistic map over various domains.

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.

For a class of three-dimensional piecewise linear systems with an admissible saddle-focus, the existence of three kinds of homoclinic loops is shown. Moreover, the birth and number of the periodic orbits induced by homoclinic bifurcation are investigated, and various sufficient conditions are obtained to guarantee the appearance of only one periodic orbit, finitely many periodic orbits or countably infinitely many periodic orbits. Furthermore, the stability of these newborn periodic orbits is analyzed in detail and some conclusions are made about them to be periodic saddle orbits or periodic sinks. Finally, some examples are given.

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter h in an open interval under small perturbation. Furthermore, we prove that the wave speed c(h) of periodic wave is strictly monotonically increasing with respect to h by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems.

Feed-forward effect strongly modulates collective behavior of a multiple-layer neuron network and usually facilitates synchronization as signals are propagated to deep layers. However, a full synchronization of neuron system corresponds to functional disorder. In this work, we focus on a network containing two layers as the simplest model for multiple layers to investigate pattern selection during interaction between two layers. We first confirm that the chimera state emerges in layer 1 and it also induces chimera in layer 2 when the feed-forward effect is strong enough. A cluster is discovered as a transient state which separates full synchronization and chimera state and occupy a narrow region. Second, both feed-forward and back-forward effects are considered and we discover chimera states in both layers 1 and 2 under the same parameter for a large range of parameters selection. Finally, we introduce adaptive dynamics into inter-layer rather than intra-layer couplings. Under this circumstance, chimera state can still be induced and coupling matrix will be self-organized under suitable phase parameter to guarantee chimera formation. Indeed, chimera, cluster and synchronization can propagate from one layer to another in a regular multiple network for a corresponding parameter selection. More importantly, adaptive coupling is proved to control pattern selection of neuron firing in a network and this plays a key role in encoding scheme.

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

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.

In this paper, we consider a pentagonal lattice and we investigate the rule matrix with null boundary condition for two-dimensional cellular automata with the field ℤp (the set of integers modulo p) and analyze their characteristics. Moreover, an algorithm of computing the rank of rule matrix with null boundary condition for von Neumann neighborhood is developed. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for two-dimensional cellular automata are obtained.

The stochastic nature of magnetization dynamics of dipole–dipole interactions described by the Landau–Lifshitz–Gilbert equation without considering the Gilbert damping parameter is investigated. It is shown that the occurrence of the complex dynamic states depends on the spatial anisotropy of interactions on one hand and the lattice geometry on the other. It is observed from the higher-order moments of the magnetization fluctuations that two significant dynamical regimes, regular and chaos, may be obtained depending on the perturbation strength. Relying on the Hurst exponent obtained by the standard deviation principle, the correlation and persistence of the magnetization fluctuations are analyzed. The results also exhibit a transition from an anti-correlated to a positively correlated system as the relevant parameters of the system vary.

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.

In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space x=0. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom.

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.

This paper proposes a technique based on unsupervised machine learning to find phases and phase transitions characterizing the dynamics of global terrorism. A dataset of worldwide terrorist incidents, covering the period from 1970 up to 2019 is analyzed. Multidimensional time-series concerning casualties and events are generated from a public domain database and are interpreted as the state of a complex system. The time-series are sliced, and the segments generated are objects that characterize the dynamical process. The objects are compared with each other by means of several distances and classified by means of the multidimensional scaling (MDS) method. The MDS generates loci of objects, where time is displayed as a parametric variable. The obtained portraits are analyzed in terms of the patterns of objects, characterizing the nature of the system dynamics. Complex dynamics are revealed, with periods resembling chaotic behavior, phases and phase transitions. The results demonstrate that the MDS is an effective tool to analyze global terrorism and can be adopted with other complex systems.

We apply the �1-equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of �1-equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described.