Point Attractor Research Articles

Coexisting point attractors, multi-transient behaviors, area-preserving chaotic systems, non-degenerate hyperchaotic systems derived from a simple 3D discrete system

In this paper, we propose a simple 3D discrete system with a variety of interesting dynamic behaviors. When the control parameters of the discrete system are set to different appropriate values, the system is transformed into four distinct systems, namely a discrete system with coexisting point attractors, a discrete system with novel multi-transient behaviors, an area-preserving map, and a non-degenerate hyperchaotic system. This transient transition behavior is manifested as a switch between multiple quasi-periodic flows. This multi-transient behavior is rarely reported in discrete systems. In addition, to meet the requirements of chaotic secure communication, relevant experiments prove that the pixel scrambling effect of the proposed area-preserving map is better than that of the 3D digital Arnold map. Moreover, a PRNG is constructed by quantizing the proposed non-degenerate hyperchaotic system, and the PRNG can pass the NIST SP-800-22 test and show good randomness.

Dynamics and circuit simulation of a fractional-order hyperchaotic system

A hyperchaotic system with fractional terms and fractional-order derivatives is investigated in this paper. Simulations show that different attractors such as equilibrium point, limit cycle and hyperchaotic attractor can be generated by the system. Circuit of fractional-order integrator is designed and it is used to implement the circuit of the studied system. The circuit implementation of the studied system proofs its feasibility. A hyperchaotic system with fractional terms and fractional-order derivatives is investigated in this paper. Simulations show that different attractors such as equilibrium point, limit cycle and hyperchaotic attractor can be generated by the system. Circuit of fractional-order integrator is designed and it is used to implement the circuit of the studied system. The circuit implementation of the studied system proofs its feasibility.

Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity

A scalar–tensor theory of gravity was considered, wherein the gravitational coupling G and the speed of light c were admitted as space–time functions and combined to form the definition of the scalar field ϕ. The varying c participates in the definition of the variation of the matter part of the action; it is related to the effective stress–energy tensor, which is a result of the requirement of symmetry under general coordinate transformations in our gravity model. The effect of the cosmological coupling Λ is accommodated within a possible behavior of ϕ. We analyzed the dynamics of ϕ in the phase space, thereby showing the existence of an attractor point for reasonable hypotheses on the potential V(ϕ) and no particular assumption on the Hubble function. The phase space analysis was performed both with the linear stability theory and via the more general Lyapunov method. Either method led to the conclusion that the condition G˙/G=σc˙/c, where σ=3 must hold for the rest of cosmic evolution after the system arrives at the globally asymptotically stable fixed point and the dynamics of ϕ ceases. This result realized our main motivation: to provide a physical foundation for the phenomenological model admitting G/G0=c/c03, used recently to interpret cosmological and astrophysical data. The thus covarying couplings G and c impact the cosmic evolution after the dynamical system settles to equilibrium. The secondary goal of our work was to investigate how this impact occurs. This was performed by constructing the generalized continuity equation in our scalar–tensor model and considering two possible regimes for the varying speed of light—decreasing c and increasing c—while solving our modified Friedmann equations. The solutions to the latter equations make room for radiation- and matter-dominated eras that progress to a dark-energy-type of accelerated expansion.

The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces

We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.

Torus and fixed point attractors of a new hyperchaotic 4D system

In this article, a new 4D hyperchaotic system with torus attractors is developed and analyzed with integer and non-integer order operators. Various dynamical features of the new system are investigated and discussed. The equilibrium points with stability, Lyapunov spectra, Poincaré section, bifurcations, phase portraits, attractors projection, and inversion properties are studied. The phase inversion property concerning a single parameter, which is very rare for hyperchaotic dynamical systems, is observed in the proposed system. For integer order, the attractor projection shows that there exists a torus attractor, which is attracting all the nearby trajectories towards it. Moreover, the system is theoretically, numerically, and graphically examined from a fractional perspective. The fixed point notions are used to examine the existence of the solution of the proposed hyperchaotic system under the Caputo operator. The system is also studied numerically in fractional sense via a newly introduced numerical technique based on the Newton polynomials. We analyze that with a decrease in fractional order, the system shows a cone-type spiral behavior in which the amplitude of the oscillations decreases and the system tends towards E0. In addition, coexisting fixed point attractors are obtained through numerical simulations at a few fractional orders.

Dynamic analysis and FPGA implementation of a fourth-order chaotic system with coexisting attractor

To further explore the complex dynamical behaviors in coexisting attractors, a fourth-order chaotic system with four types of coexisting attractors and four unstable equilibrium points is constructed in this paper. The dynamic behavior of the new system is analyzed by means of phase trajectory diagram, time domain waveform diagram, Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. The experimental results show that as the parameters change, the system exhibits rich dynamic behaviors such as stable points, period-doubling bifurcations, coexisting bifurcation modes, and chaotic crises. When the system parameters and memristive parameters change, it is found that the system has different types of coexisting attractors, such as the coexistence of two periodic attractors, the coexistence of two single-scroll chaotic attractors, the coexistence of two double-scroll chaotic attractors, the coexistence of two point attractors. In particular, the system also has the rotation phenomenon of coexisting attractors. Finally, a nonlinear feedback controller is designed, which can make the new system achieve chaos synchronization in a short time. The phase diagram captured by the field-programmable gate array (FPGA) hardware platform is consistent with the numerical simulation results, which proves the feasibility of the system.

CLOUDS IN THE BASINS OF FULLY ASYNCHRONOUS CELLULAR AUTOMATA

This paper studies the basins of attraction of fully asynchronous cellular automata (ACAs) that converge to point attractor. Unlike traditional (synchronous) cellular automata (CAs), an ACA may approach to different point attractors from the same initial condition for different runs. In this direction, this paper introduces the notion of cloud behavior in convergence dynamics of ACAs. Depending on the cloud behavior, we classify the convergent ACAs into three classes—eccentric cloud, partially eccentric cloud and deterministic cloud system. In addition, this study observes the computational ability of addressing density classification problem (i.e. distributed consensus problem) of this asynchronous cloud system.

Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: Dynamics analysis and circuit implementation

Electromagnetic induction current is generated between the adjacent neurons in neural network caused by the existence of membrane potential difference. Memristor is the fourth fundamental electric element, which can mimic the behavior of neural synapses and simulate the electromagnetic induction effect. In this paper, in order to simplify practical implementation, the commonly used hyperbolic tangent activation function of Hopfield neural network (HNN) is replaced by a piecewise-linear function, and a simple bi-neuron-based memristor synapse-coupled HNN is proposed. Theoretical analysis and numerical simulation results illustrate that the memsirstive HNN possesses five equilibria including one unstable saddle-focus, two unstable saddle points, and two stable node points (or node-foci). Local attraction basins and phase plane plots show that the memristive HNN model behaves multistability of coexisting chaos, periodic limit cycles, and stable point attractors. Various system dynamical behaviors affected by parameters of the piecewise-linear activation function, memristor coupling strength, and initial conditions of the neurons are investigated by numerical simulations. Furthermore, an analog circuit of the memristive HNN model is simply designed, which is easier for hardware implementation as the piecewise-linear activation function can be implemented by simple op-amp-based module. Finally, the experimental results verify the correctness of the design and analyses.

Application of discrete memristors in logistic map and Hindmarsh–Rose neuron

Continuous memristor (CM) has been widely designed and applied in chaotic and neuromorphic systems. However, discrete memristor (DM) and its application are less studied. This paper reports two kinds of generalized DM model based on sampling discretization of generalized CM model. It is found that the DM models can still retain the characteristics of the CM models. Based on the proposed DM models, the Logistic map (DM-L map) and two-dimensional (2D) discrete Hindmarsh–Rose neuron model (DM-HR model) are constructed. Through multiple numerical measures, it is found that DM-L map has linear fixed points, whose stability are only related to system parameter, and that DM-HR model does not have fixed point thus can generate hidden periodic and chaotic attractors. In particular, the coexistence of point and periodic attractors with different topological structures occur in DM-L map when different initial values are taken, which are rarely reported previously. At the same time, the chaotic region becomes wider when DM is introduced into Logistic map. Therefore, we believe that DM can effectively improve chaotic complexity of Logistic map. However, in addition to hidden complex discharge phenomenon, the coexistence of periodic and chaotic attractors with different positions and shapes are observed in DM-HR model. Finally, the DSP-based hardware platform is constructed to implement these maps, and the experimental results are consistent with the numerical simulation.

Extreme multistability and phase synchronization in a heterogeneous bi-neuron Rulkov network with memristive electromagnetic induction.

Memristive electromagnetic induction effect has been widely explored in bi-neuron network with homogeneous neurons, but rarely in bi-neuron network with heterogeneous ones. This paper builds a bi-neuron network by coupling heterogeneous Rulkov neurons with memristor and investigates the memristive electromagnetic induction effect. Theoretical analysis discloses that the bi-neuron network possesses a line equilibrium state and its stability depends on the memristor coupling strength and initial condition. That is, the stability of the line equilibrium state has a transition between unstable saddle-focus and stable node-focus via Hopf bifurcation. By employing parameters located in the stable node-focus region, dynamical behaviors related to the memristor coupling strength and initial conditions are revealed by Julia- and MATLAB-based multiple numerical tools. Numerical results demonstrate that the proposed heterogeneous bi-neuron Rulkov network can generate point attractor, period, chaos, chaos crisis, and period-doubling bifurcation. Note that extreme multistability are disclosed with respect to initial conditions of memristor and gated ion concentration. Coexisting infinitely multiple firing patterns of periodic firing patterns with different periodicities and chaotic firing patterns for different memristor initial conditions are demonstrated by phase portrait and time-domain waveform. Besides, the phase synchronization related to the memristor coupling strength and its initial condition is explored, which suggests that the two heterogeneous neurons become phase synchronization with large memristor coupling strength and initial condition. This also reflects that the plasticity of memristor synapse enables adaptive regulation in keeping energy balance between the neurons. What's more, MCU-based hardware experiments are executed to further confirm the numerical simulations.

Good point set and double attractors based-QPSO and application in portfolio with transaction fee and financing cost

In order to overcome the premature convergence problem of quantum-behaved particle swarm optimization algorithm (QPSO) in the face of high-dimensional complexity, an improved QPSO based on good point set and double attractor point mechanism (GDQPSO) is presented. In the initialization phase of GDQPSO, the idea of good point set is used to allow particles to explore more uniformly and comprehensively at the beginning of the iteration. To better balance the exploration ability of individuals and the global search capability of the algorithm, the local attraction point is redefined by combining dynamic and fixed methods. Then the global perturbation mechanism is considered to maintain the diversity of the population. And the comparison results with nine intelligent algorithms on 18 benchmark functions indicate that GDQPSO has obvious advantages in convergence accuracy and stability. Finally, the simulation experiments demonstrate very competitive performance of GDQPSO while solving the portfolio model with transaction fee and financing cost when compared with other evolutionary algorithms.

The Emergence of Cultural Attractors: How Dynamic Populations of Learners Achieve Collective Cognitive Alignment.

When a population exhibits collective cognitive alignment, such that group members tend to perceive, remember, and reproduce information in similar ways, the features of socially transmitted variants (i.e., artifacts, behaviors) may converge over time towards culture-specific equilibria points, often called cultural attractors. Because cognition may be plastic, shaped through experience with the cultural products of others, collective cognitive alignment and stable cultural attractors cannot always be taken for granted, but little is known about how these patterns first emerge and stabilize in initially uncoordinated populations. We propose that stable cultural attractors can emerge from general principles of human categorization and communication. We present a model of cultural attractor dynamics, which extends a model of unsupervised category learning in individuals to a multiagent setting wherein learners provide the training input to each other. Agents in our populations spontaneously align their cognitive category structures, producing emergent cultural attractor points. We highlight three interesting behaviors exhibited by our model: (1) noise enhances the stability of cultural category structures; (2) short 'critical' periods of learning early in life enhance stability; and (3) larger populations produce more stable but less complex attractor landscapes, and cliquish network structure can mitigate the latter effect. These results may shed light on how collective cognitive alignment is achieved in the absence of shared, innate cognitive attractors, which we suggest is important to the capacity for cumulative culturalevolution.

Entrainment During Human Locomotion Using a Lightweight Soft Robotic Hip Exosuit (SR-HExo)

A gait entrainment study was conducted using a lightweight soft robotic hip exosuit (SR-HExo) that can apply perturbations at the hip joint during treadmill walking. Periodic mechanical perturbations were applied by flat fabric Pneumatic Artificial Muscle (ff-PAM) actuators starting at a subject’s preferred gait frequency and increasing up to 15% higher in 3% increments. Anterior hip flexion perturbations and posterior hip extension perturbations were tested in two separate experiments. All 11 healthy participants showed successful entrainment in all 12 experimental conditions (i.e., from preferred gait frequency to 15% higher in both flexion and extension perturbation directions). This study confirmed that there exists a single stable point attractor during gait entrainment to unilateral, unidirectional hip perturbations, which is consistent with previous ankle studies. Phase-locking was consistently observed around toe-off phase of the gait cycle (GC). Group averaged results showed gait synchronization with extension perturbations occurred earlier in the gait cycle (around 50% GC where the hip angle reaches maximum extension) than with flexion perturbations (just after 60% GC where the transition from maximum hip extension towards hip flexion occurs). Other gait entrainment characteristics (specifically, success rate of entrainment, basin of entrainment, and transient response) observed in this study posits the potential of the SR-HExo for entrainment-based gait training in rehabilitative contexts.

Synchronization, extreme multistability, and its control for a field coupled neurons with time delays

Neurons are connected through synapses, but it is not reasonable to equate the connection mode of synapses to an edge in the network due to synaptic plasticity whereas the magnetic field coupling can be considered to handle it. Therefore, in this paper, the interaction between HR neurons is realized by magnetic field coupling based on induction coil, and time delay is introduced to represent the lag in information transfer. First, the coexisting firing activities and synchronization behaviors in dual neuronal networks are numerically calculated, respectively, depending on the external stimulation current, coupling strength, time delays, and initial conditions. When the time delays are given, it is interesting to note that the infinite number of firing modes including chaotic firing, periodical firing, and quiescent state is induced by initial conditions. Due to the initial values, the types of synchronization consisting of complete synchronization, delayed synchronization, and asynchronization are then revealed under the framework of extreme multistability. In particular, the state of complete synchronization exhibits only quiescent state and period-1 firing when the time delay is not equal to 0. Furthermore, the linear augmentation method is conceived to control extreme multistability. It can be found that the attractors with different positions and topological structures can be controlled to the point attractors with the same shape but with different positions when the coupling strength of linear system and nonlinear system is increased. That is, the heterogeneous multistability can be successfully controlled to the homogeneous multistability, and the coupled neurons can also be achieved synchronization after control. These conclusions in this paper could be helpful in providing new insights for studying neurodynamics and applying neural circuits.

Storage capacity of networks with discrete synapses and sparsely encoded memories.

Attractor neural networks are one of the leading theoretical frameworks for the formation and retrieval of memories in networks of biological neurons. In this framework, a pattern imposed by external inputs to the network is said to be learned when this pattern becomes a fixed point attractor of the network dynamics. The storage capacity is the maximum number of patterns that can be learned by the network. In this paper, we study the storage capacity of fully connected and sparsely connected networks with a binarized Hebbian rule, for arbitrary coding levels. Our results show that a network with discrete synapses has a similar storage capacity as the model with continuous synapses, and that this capacity tends asymptotically towards the optimal capacity, in the space of all possible binary connectivity matrices, in the sparse coding limit. We also derive finite coding level corrections for the asymptotic solution in the sparse coding limit. The result indicates the capacity of networks with Hebbian learning rules converges to the optimal capacity extremely slowly when the coding level becomes small. Our results also show that in networks with sparse binary connectivity matrices, the information capacity per synapse is larger than in the fully connected case, and thus such networks store information more efficiently.

On the Existence of Scaling Multi-Centered Black Holes

For suitable charges of the constituents, the phase space of multi-centered BPS black holes in N=2 four-dimensional supergravity famously exhibits scaling regions where the distances between the centers can be made arbitrarily small, so that the bound state becomes indistinguishable from a single-centered black hole. In this note we establish necessary conditions on the Dirac product of charges for the existence of such regions for any number of centers, generalizing the standard triangular inequalities in the three-center case. Interestingly, the same conditions are necessary for the existence of multi-centered solutions at the attractor point. We prove that similar conditions are also necessary for the existence of self-stable representations of the corresponding quiver, as suggested by the duality between the Coulomb and Higgs branches of supersymmetric quantum mechanics.

Dynamics and Information Import in Recurrent Neural Networks.

Recurrent neural networks (RNNs) are complex dynamical systems, capable of ongoing activity without any driving input. The long-term behavior of free-running RNNs, described by periodic, chaotic and fixed point attractors, is controlled by the statistics of the neural connection weights, such as the density d of non-zero connections, or the balance b between excitatory and inhibitory connections. However, for information processing purposes, RNNs need to receive external input signals, and it is not clear which of the dynamical regimes is optimal for this information import. We use both the average correlations C and the mutual information I between the momentary input vector and the next system state vector as quantitative measures of information import and analyze their dependence on the balance and density of the network. Remarkably, both resulting phase diagrams C(b, d) and I(b, d) are highly consistent, pointing to a link between the dynamical systems and the information-processing approach to complex systems. Information import is maximal not at the “edge of chaos,” which is optimally suited for computation, but surprisingly in the low-density chaotic regime and at the border between the chaotic and fixed point regime. Moreover, we find a completely new type of resonance phenomenon, which we call “Import Resonance” (IR), where the information import shows a maximum, i.e., a peak-like dependence on the coupling strength between the RNN and its external input. IR complements previously found Recurrence Resonance (RR), where correlation and mutual information of successive system states peak for a certain amplitude of noise added to the system. Both IR and RR can be exploited to optimize information processing in artificial neural networks and might also play a crucial role in biological neural systems.

Fixed Point Attractor Theory Bridges Structure and Function in C. elegans Neuronal Network.

Understanding the structure–function relationship in a neuronal network is one of the major challenges in neuroscience research. Despite increasing researches at circuit connectivity and neural network structure, their structure-based biological interpretability remains unclear. Based on the attractor theory, here we develop an analytical framework that links neural circuit structures and their functions together through fixed point attractor in Caenorhabditis elegans. In this framework, we successfully established the structural condition for the emergence of multiple fixed points in C. elegans connectome. Then we construct a finite state machine to explain how functions related to bistable phenomena at the neural activity and behavioral levels are encoded. By applying the proposed framework to the command circuit in C. elegans, we provide a circuit level interpretation for the forward-reverse switching behaviors. Interestingly, network properties of the command circuit and first layer amphid interneuron circuit can also be inferred from their functions in this framework. Our research indicates the reliability of the fixed point attractor bridging circuit structure and functions, suggesting its potential applicability to more complex neuronal circuits in other species.

Hidden and Self-Excited Collective Dynamics of a New Multistable Hyper-Jerk System with Unique Equilibrium

Nonlinear dynamical systems with hidden attractors belong to a recent and hot area of research. Such systems can exist in different forms, such as without equilibrium or with a stable equilibrium point. This paper focuses on the dynamics of a new 4D chaotic hyper-jerk system with a unique equilibrium point. It is shown that the new hyper-jerk system effectively exhibits different hidden behaviors, which are hidden point attractor, hidden periodic attractor, and hidden chaotic state. Collective behaviors of the system are studied in terms of the equilibrium point, bifurcation diagrams, phase portraits, frequency spectra, and two-parameter Lyapunov exponents. Some remarkable and exciting properties are found in the new snap system, such as period-doubling transition, asymmetric bubbles, and coexisting bifurcations. Also, we demonstrate that it is possible to generate different varieties of two, three, four, or five coexisting hidden and self-excited attractors in the introduced model. In addition, the amplitude and offset of the hidden chaotic attractors are perfectly controlled for possible application in engineering. Furthermore, a circuit design has been implemented to support the physical feasibility of the proposed model.

Dynamics of Hidden Attractors in Four-Dimensional Dynamical Systems with Power Law

Four-dimensional continuous chaotic models with Caputo fractional derivative are presented. Fixed point theory is used to investigate the existence and uniqueness of complex systems. The dynamical properties are studied, including the Lyapunov exponent, phase portrait, and time series analysis. The hyperchaotic nature of each system is revealed by the positive exponents. The numerical method is introduced to describe the influence of the order of the Caputo fractional derivative. The phase portraits are presented to investigate the behavior and effect of some key parameters and fractional orders on model dynamics. The systems approach fixed point attractors for fractional-order and increase the visibility of the attractor by decreasing fractional order. This means that a change in fractional order has a significant impact on the dynamics of the models. When the order of the derivative is equal to one, both systems oscillate frequently. However, as the fractional order is reduced, the system oscillations decrease as compared to the integer-order, and the system moves towards its fixed point, reveals the hidden attractors inherent in the system, and enabling it to develop to a stable state more efficiently.