The study of planar polynomial dynamical systems in Hilbert’s 16th problem offers critical insights into nonlinear dynamical systems. However, current research rarely conducts quantitative analysis of the system from the perspective of phase portrait geometric structures, nor does it feature physical implementations. To address this gap, this paper will investigate planar polynomial systems with four, six, and nine equilibrium points from the perspective of geometric structures, revealing intrinsic characteristics such as equilibrium point distributions and homoclinic/heteroclinic orbits. Then, the system is investigated via Hamiltonian energy to explain the fundamental causes behind the geometric structure formation. Most importantly, a method for hardware circuit implementation of geometric structures is successfully developed for the first time. Oscilloscope observations show consistency between experimental results and theoretical predictions, and the hardware implementation lays a foundation for engineering applications. Overall, this study contributes to the interdisciplinary integration of geometric topology, nonlinear dynamics, analog circuit design, and engineering applications, which provides a new methodological framework for exploring complex dynamical systems.

To enable accurate and resource-efficient hardware implementation of fractional-order neural networks for neuromorphic computing, an optimized hardware architecture for field programmable gate arrays (FPGA) is proposed, wherein Grünwald–Letnikov fractional calculus is integrated with Chebyshev optimal approximation techniques. First, a hardware-efficient FPGA-based Grünwald-Letnikov operator is developed by truncating the infinite memory term into a fixed window and quantizing binomial coefficients. Second, Chebyshev-optimized piecewise linear approximation is employed to implement nonlinear activation functions. The proposed approach achieves a 20-30% reduction in maximum error compared to traditional methods, implemented in a fractional-order heterogeneous memristive Hopfield neural network. The systematic integration of temporal optimization into functional modules achieves resource savings. In the experimental outcomes, excellent agreement with numerical simulations is observed. To demonstrate the practicality of the implemented fractional-order network, a pseudorandom number generator is designed, successfully passing all NIST SP 800-22 tests. This research advances computational efficiency in fractional-order neural networks, with substantial implications for their applications in secure communications and related domains.

Chaotic signals in neural networks necessitate sophisticated control strategies in practical engineering applications. The integration of multiple control methods into a unified framework to enhance flexibility remains both attractive and challenging. This article proposes an integrated control framework for small-scale neural networks, in which a chemically coupled Hindmarsh–Rose neural network is employed as an example to verify the framework’s capability for arbitrary signal modulation. First, a global bifurcation analysis is performed using the implicit mapping method, thereby revealing the hidden dynamical behaviors of the coupled model. The types of bifurcation points are determined through eigenvalue analysis. Subsequently, a general control theory is established, enabling direct manipulation of an attractor’s position, scale, and orientation through its parameters. This composite scheme encompasses amplitude, offset, and rotation control, as well as the flexible adjustment of the rotation axis within the rotation control. Furthermore, to substantiate the simulation results, the proposed control strategies are implemented on a microcontroller unit platform. Overall, the controlled chaotic sequences successfully pass the National Institute of Standards and Technology tests, confirming their suitability for multi-angle pseudo-random number generation.

The dynamics of memristive neural networks have incurred significant concern. However, existing research predominantly concentrates on networks with single or multiple neurons, ignoring the large-scale and high-dimensional characteristics inherent to brain networks. To this end, this article first innovatively introduces a family of discrete corsage memristor (DCM) models and scrutinizes their attributes, including nonvolatility, local activity, and edge of chaos. Subsequently, a cross-hemispheric memristive neural network (CMNN) model with dual ring-star structure is proposed, enabling intra-hemispheric information exchange via bidirectional electrical synapses and inter-hemispheric communication through shared memristor synapses. Furthermore, the spatio-temporal dynamics of the CMNN are numerically investigated with the aid of quantitative metrics and visualization techniques, exhibiting phase chimera and complete interlayer synchronization. In particular, the synchronization condition is theoretically estimated utilizing the Lyapunov stability theorem. Moreover, we develop a microcontroller unit (MCU)-based hardware experiment platform to implement the CMNN. Finally, a telemedicine encryption scheme is developed for safeguarding medical images in the Internet of Medical Things (IoMT), with experimental results validating its reliability and security performance.

The Hopfield Neural Network has been widely used to simulate brain electrical activity due to its flexible topology and rich dynamical behaviors. Moreover, the spectral characteristics of neural activity serve as fundamental signal, offering potential to identify novel targets for neurological disorders and advancing next-generation neuromodulation therapies. However, the frequency-domain analyses of HNN have rarely been reported. To further explore the complex periodic motions induced by harmonic terms, this study employs discrete mapping method incorporating finite Fourier series to analyze bifurcation evolutions and coexistence of firing behaviors, providing the quantitative analysis of their amplitude-frequency characteristics for the first time. Additionally, by investigating the relationship between harmonic amplitudes and phases in coexisting attractors, new perspectives on the study of coexisting attractors are provided. Finally, the simulation results are verified through the hardware circuit built using printed circuit board and a high-performance pseudorandom number generator is designed by leveraging chaotic sequences derived from unstable periodic orbits. This research contributes a new approach for data encryption and communication security in Internet of Things.

Chua corsage memristor (CCM) has been instrumental in constructing continuous oscillatory systems. However, locally active memristor with lobes has not yet been found in the discrete-time domain. To address this gap, this article introduces a 2-lobe discrete corsage memristor (DCM) model characterized by the nonvolatility, bistability, and odd-symmetric locally active region, as provided by the local activity principle. The edge-of-chaos regime is further identified through the Jacobi matrix approach. Moreover, a 2-lobe DCM-coupled neuron map (DNM) is developed and its attractor dynamics are numerically revealed. The DNM generates numerous symmetric periodic and chaotic attractors relying on the memristor intrinsic parameter, as well as offset-boosted coexisting attractors with mono-topology and hybrid-topology, controlled by the initial state of the memristor. Finally, an FPGA-based implementation platform is established for numerical simulation verification, and three pseudo-random numbers with compliant performance are acquired.

Nonlinear circuit can exhibit complex dynamical behaviors, especially the coexisting periodic or chaotic oscillations, by employing various electronic elements. However, oscillation circuits with a simple hybrid diode, which can induce richer dynamical behaviors with only a pair of anti-parallel diodes and an RC filter, are rarely reported, thus hindering the deep cognition and potential applications of such nonlinear circuits. This paper focuses on the dynamical behaviors of a Shinriki circuit modified by applying the hybrid diode, in which multiple coexisting oscillations and antimonotonic evolutions are successfully discovered by varying circuit parameters. To deeply study these phenomena, a discrete mapping structure of the proposed circuit is constructed by the semi-analytical method, and the bifurcation trees of diverse coexisting periodic oscillations are thoroughly investigated via bifurcation diagrams and phase portraits. The emergence and disappearance of antimonotonic phenomena with stable and unstable orbits are also clearly revealed. Notably, the stability and bifurcation points of the motions are precisely judged by eigenvalues, and the unstable routes hidden in chaotic regions can be uncovered. Finally, the coexisting stable and unstable periodic orbits are captured from the oscilloscope via a field programmable gate array (FPGA), which verifies the correctness of the analysis. The research may be devoted to the improvement of nonlinear circuit.

When using a synapse as a coupler to connect neurons, parameter-based synchronization transitions have been investigated. However, the dependence on initial conditions has not been comprehensively discussed in the literature. This work presents an electrical-synapse-coupled model consisting of two homogeneous Hopfield neural networks (HNNs), which is the simplest network-to-network coupling model known for HNN. The model possesses several fixed points, which are found to be unstable. Simulation results of peak differences, bifurcation diagrams, and normalized mean synchronization errors indicate that complex synchronization transitions occur, depending on both the electrical coupling strength and initial conditions. Particularly, we focus here on mapping the basins of attraction between periodic and chaotic synchronization for bistable patterns. Finally, a multiplierless electrical neuron circuit is developed to validate initial condition-induced synchronization phenomena, which provides a new perspective for the study of collective dynamics of brain-like networks and the development of lightweight neuromorphic circuits.

Detection of hidden dynamics is of great value in model prediction and control engineering. To explore its effects and control methods in the memristive network model, this paper presents a memristor synapse-driven ReLU-type Hopfield neural network (MRHNN). The generalized Hamilton function is derived from Helmholtz’s theorem and the equilibrium points of the model are analyzed. It is found via numerical computations that because of no existence of equilibrium, the MRHNN model always unfolds hidden dynamics, including hidden bifurcation, hidden mode transition, hidden transient chaos, and hidden multistability. In addition, amplitude and offset boosting control of hidden attractors are executed, illustrating the flexibility of the attractor regulation. Finally, based on digital hardware devices, circuit experiments are deployed and their measurements well agree with the numerical results, certifying the dynamical effects and lossless control of the memristive neural network and physical reliability of the electronic neuron.

The study on the dynamical behaviors of the coupled heterogeneous neural network, including bifurcation orbits, synchronization, especially unstable firing behaviors, may have great significance for diagnosis and guarding against brain diseases. To investigate this matter in depth, the discrete implicit mapping method can be employed for assessing the neural network, which is coupled with the Hindmarsh-Rose and FitzHugh-Nagumo neuron models in this paper. The bifurcation trees of periodic motions, exhibiting intricate dynamic behaviors, are precisely demonstrated by maniputing the coupling strength. The transitions from period-1 to period-8, period-3 to period-12, period-4 to period-16 and period-5 to period-10 will be achieved through saddle bifurcations and period doubling bifurcations. The corresponding stable firing patterns are observed through discrete nodes in phase diagrams, time-histories and deviations of the membrane potential diagrams. Meanwhile, the unstable firing patterns, using the particular method, are also obtained, which cannot be calculated through the numerical method due to its accumulative errors. Moreover, the synchronous and asynchronous behaviors depending on the coupling strength are successively revealed and described. Lastly, the experiment of the heterogeneous neural network is validated by the field-programmable gate array (FPGA) circuit. Such an investigation will also positively contribute to the development of the progress of brain medicine and life science and engineering.

Memristors have been employed in various continuous neural network models through emulating magnetic induction effect, but have not yet been successfully performed in discrete neural network models. To address this issue, this article first proposes a discrete memristive Rulkov neuron model (MRN) and then constructs a large-scale discrete memristive Rulkov ring-star neural network model (MRRSNN). Furthermore, spatiotemporal pattern, snapshot, and recurrence plot of the nodes are adopted to declare the spatio-temporal dynamics of the MRRSNN. Consequently, it can manifest rich network behaviors, including double-well chimera, asynchronized, multiclustered, solitary, synchronized, and continuous traveling wave states. Besides, the influence of the memristive coupling strength and initial conditions of MRNs on the network behaviors is quantified by three metrics, including root mean-square deviation, averaged cross-correlation coefficient, and normalized time-averaged synchronization error, which provides an important basis for state regulation of the MRRSNN. Finally, a spatio-temporal chaos-based pseudorandom number generator is designed, and experiment results from the NIST 800-22 and performance indicators show that the MRRSNN generates better random sequences than the MRN even in the presence of dynamic degeneracy.

Recently, human activity recognition (HAR) that uses wearable sensors has become a research hotspot because its wide applications in real-world scenarios. Essentially, HAR can be treated as multi-channel time series classification problem, where different channels may come from heterogeneous sensor modalities. Deep learning, especially convolutional neural networks (CNNs) have made breakthroughs in ubiquitous HAR scenario. Various normalization methods enable layers of networks to learn more independently by normalizing hybrid sensor features. However, normalization tends to produce a channel collapse phenomenon, where many channels generates tiny values. Most channels are inhibited and contribute very little to output. As a result, the network has to rely on only a few valid channels, which inevitably impair the generality ability. In this paper, we provide an alternative called Channel Equalization to reactivate these inhibited channels by performing whitening or decorrelation operation, which compels all channels to contribute more or less to feature representation. Extensive experiments are conducted on several public HAR benchmarks, which indicate that the proposed method significantly surpasses recent SOTA at negligible computational overhead. To our knowledge, the Channel Equalization is for the first time to be applied in multimodal HAR scenario. Finally, the actual operation is evaluated on an embedded platform.