Search

The concept of multifunctionality has enabled reservoir computers (RCs), a type of dynamical system that is typically realized as an artificial neural network, to reconstruct multiple attractors simultaneously using the same set of trained weights. However, there are many additional phenomena that arise when training a RC to reconstruct more than one attractor. Previous studies have found that in certain cases, if the RC fails to reconstruct a coexistence of attractors, then it exhibits a form of metastability, whereby, without any external input, the state of the RC switches between different modes of behavior that resemble the properties of the attractors it failed to reconstruct. In this paper, we explore the origins of these switching dynamics in a paradigmatic setting via the "seeing double" problem. [ABSTRACT FROM AUTHOR]

In this article we study the long-time dynamics of the dynamical system generated by the Ericksen–Leslie model. More precisely, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemmas we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the time-step approaches zero. [ABSTRACT FROM AUTHOR]

Chameleon chaotic systems have a special property to display various types of chaotic attractors by tuning system parameters, thereby allowing for the generation of diverse chaotic signals that are suitable for various applications. In this paper, we propose and study a class of three-dimensional quadratic chameleon systems capable of transitioning between self-excited and hidden chaotic regimes. Through systematic analysis of the systems, we can identify hidden chaotic attractors in parameter regions where no equilibria exist, or where there is a line equilibrium, or where a single stable equilibrium exists. In order to study the basic properties of the system, we carried out both local stability analysis and Hopf bifurcation analysis. Further bifurcation analysis and Lyapunov exponent calculation uncovered intricate transitions among periodic, chaotic, and hidden chaotic regimes as the system parameters varied. Through the research, we find that antimonotonicity holds significant implications for creating various types of chaotic dynamics in the chameleon systems. Furthermore, we find that by adjusting the values of parameters, the system can display a self-excited chaotic attractor, a hidden chaotic attractor with no equilibrium, a hidden chaotic attractor with a line equilibrium, or a hidden chaotic attractor with a single stable hyperbolic/nonhyperbolic equilibrium point. We are interested in a hidden chaotic system with a stable nonhyperbolic equilibrium point, for which the practical feasibility is verified through circuit simulations. The chameleon chaotic systems studied in this paper expand our understanding of the chaotic mechanisms with various equilibrium configurations. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 풪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball's idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time. [ABSTRACT FROM AUTHOR]

Stochastic bifurcation of dynamical system induced by Lévy noise is investigated in this paper from the global viewpoint, which is demonstrated based on the global attractor obtained with the compatible cell mapping (CCM) method. Compared with the traditional cell mapping method, the efficiency of CCM method is greatly improved by concentrating the computational domain on the covering set of the global attractor rather than the whole state space. The computation time can be further reduced by a simple parallel procedure with the block matrix strategy. With the proposed method, the evolutionary process of global properties has been presented. It shows that the shape and size of stochastic attractors and stochastic saddles vary along the direction of unstable manifold with the parameters of Lévy noise. Stochastic bifurcation occurs when stochastic attractor collides with stochastic saddle. It can be also found that the global bifurcation under Lévy noise is discontinuous because of the jump property. The results are significantly different from those of Gaussian noise, which reveals a distinct global evolutionary mechanism of stochastic bifurcation. Furthermore, it also indicates that the CCM method with block matrix is an effective tool for global analysis. [ABSTRACT FROM AUTHOR]

The theory of compact global attractors for dynamical systems relies on the existence of a bounded absorbing set. In this paper, suppressing this condition, we present sufficient conditions to ensure the existence and uniqueness of maximal attractors for dynamical systems (both autonomous and nonautonomous). Such attractors are, in general, unbounded. For semigroups satisfying these conditions, when a Lyapunov function exists, we also present the characterization of the unique maximal attractor as the unstable set of the critical elements (not necessarily fixed points). As an example we present a semilinear parabolic equation to illustrate the theory in the autonomous case. For the nonautonomous setting, applying the general theory, we present an abstract evolution equation and a concrete parabolic equation as examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of viscoelastic plate equations with past history. We first transform the original initial-boundary value problem into an equivalent one by means of the history space framework. Then we use the perturbed energy method to establish a stabilizability estimate. By employing the gradient property and quasi-stability of the dynamical system, we obtain the existence of a global attractor with finite fractal dimension [ABSTRACT FROM AUTHOR]

We present a comprehensive investigation of the long-term dynamics generated by a semilinear wave equation with time-dependent coefficients and quintic nonlinearity on a bounded domain subject to Dirichlet boundary conditions. By employing rescaling techniques for time and utilizing the Strichartz estimates applicable to bounded domains, we initially study the global wellposedness of the Shatah–Struwe (S–S) solutions. Subsequently, we establish the existence of a uniform weak global attractor consisting of points on complete bounded trajectories through an approach based on evolutionary systems. Finally, we prove that this uniformly weak attractor is indeed strong by means of a backward asymptotic a priori estimate and the so-called energy method. Moreover, the smoothness of the obtained attractor is also shown with the help of a decomposition technique. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the long-time behavior of the Euler-Bernoulli equations with a localized nonlinear damping. Since the effective damping is located in a small neighborhood of the boundary, we can not obtain the existence of an absorbing set and some kind of compactness of the semigroup associated with the Euler-Bernoulli equations, such that the classical theory of infinite dimensional dynamical systems does not work. To overcome this difficulty, we will first prove the existence of a strict Lyapunov functional by a unique continuation result for beam equation. Furthermore, we will also established the asymptotical quasi-stability property by using the multiplier methods and choosing appropriate cut-off functions, which implies the existence of a global attractor with finite fractal dimension. [ABSTRACT FROM AUTHOR]

The hypothesis of chaotic population dynamics was proposed in ecology by Robert May in the mid-1970s. At that time the idea was controversial, and it remains a fascinating and unsettled issue today. We report the results of a 20-year laboratory research programme that continued in the tradition of the pioneering ecologist Thomas Park using the Tribolium flour beetle model. We present biological evidence of complex population dynamics – including bifurcations, chaos, saddle nodes, phase switching, resonance effects, and multiple attractors – by using a low-dimensional difference equation model for Tribolium together with carefully designed, conducted, and statistically analysed experiments. The model, parameterized with data, also explains the results of historical Tribolium experiments, such as the classical competition studies of Thomas Park and his colleagues. Our research programme has inspired other studies using the Tribolium mathematical and laboratory model. This work was conducted by a multidisciplinary team, which included Jim Cushing. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long term dynamics of the supercritical stochastic wave equation driven by multiplicative noise on unbounded domains. By introducing an appropriate variable, we convert the stochastic equation into a pathwise deterministic wave equation and prove the existence and uniqueness of solutions by the uniform Strichartz estimates. Then we prove the existence and uniqueness of pullback random attractors of the non-autonomous random dynamical system associated with the stochastic equation. The pullback asymptotic compactness of solutions is established by the idea of uniform tail-ends estimates as well as the decomposition method of the solution operator in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamical behavior of nonautonomous MGT‐Fourier system uttt+αutt−βΔut−γΔu+ηΔθ=f(x,t),θt−κΔθ−ηΔutt−ηαΔut=g(x,t),$$ \left\{\begin{array}{l}{u}_{tt t}+\alpha {u}_{tt}-\beta \Delta {u}_t-\gamma \Delta u+\eta \Delta \theta =f\left(x,t\right),\\ {}{\theta}_t-\kappa \Delta \theta -\eta \Delta {u}_{tt}-\eta \alpha \Delta {u}_t=g\left(x,t\right),\end{array}\right. $$We first establish the global existence and asymptotic behavior by semigroup method and multiplicative method. Next, we get the existence of the uniform attractors by the method of uniform contractive functions, which only need to verify compactness condition with the same type of energy estimates for establishing absorbing set. Finally, we also consider the semilinear case by semigroup method. [ABSTRACT FROM AUTHOR]

This article studies past attractors and future repellers of topological cocycles conducted by a group G satisfying Ore's condition. The limit behaviour depends on the Green $ \mathcal {L} $ L-pre-order on a right reversible semigroup S of G. An existence and uniqueness theorem for past attractors is proved. A topological method of extending functions is used in order to describe the past attractors by means of skew-product prolongations in the extended space. Illustrative examples of past attractor and future repeller are provided. [ABSTRACT FROM AUTHOR]

The Leray-alpha model to regularize the forced three-dimensional periodic Boussinesq system is shown to have a global in time unique solution and a universal attractor which is connected. Energy method, compactness argument and absorbing balls are used. [ABSTRACT FROM AUTHOR]

In this article, we investigate the dynamics of a nonautonomous stochastic strongly damped wave equation defined on ℝN$$ {\mathrm{\mathbb{R}}}^N $$. We first use the energy equation and tail‐estimates to prove the asymptotic compactness of the solutions and obtain the existence of a unique pullback random attractor for the equation with critical nonlinearity. Then, we give an upper bound of fractal dimension of the random attractor when the nonlinearity is of subcritical growth. The unboundedness of the physical space will impose difficulties on the estimation for the upper bound of fractal dimension of the random attractor. [ABSTRACT FROM AUTHOR]

Multi-stability and control of hyperchaotic system are researched in this paper. Firstly, a new 4D hyperchaotic system containing hidden attractors is modeled. Secondly, the coexistence of different attractors is confirmed and transient hyperchaotic phenomena is found as the parameters are changed. Then, the Hamiltonian energy function of the system is calculated and the energy feedback controller is designed to control the system. Finally, numerical simulations are performed to verify the validity of the results. [ABSTRACT FROM AUTHOR]

From both microscopic and macroscopic aspects, we study both discrete and random approximations of the global attractor for a nonlocal lattice dynamical system. First, we establish an abstract Taylor expansion and obtain a global attractor for the continuous-time system. Second, for the discrete-time nonlocal lattice equation via the implicit Euler scheme, we prove the existence of numerical attractors for small time-sizes, and establish the upper semicontinuity of the numerical attractors towards the global attractor as the time-size tends to zero, while the abstract Taylor expansion and error estimation play key roles in the proof. Third, for the random nonlocal lattice equation driven by nonlinear colored noise, we show the existence and upper semicontinuity of random attractors towards the global attractor when the noise-size goes to infinity. Fourth, we establish optimal bounds and several continuities of numerical attractors as well as random attractors under the Hausdorff metric. [ABSTRACT FROM AUTHOR]

In this paper, a novel four-wing chaotic system is constructed based on the Sprott-A system. The novel system contains three nonlinearly quadratic terms, which have rich dynamics such as four-wing hidden attractors, hidden extreme multistability and transient transitions. It is found that this system has high complexity by spectral entropy analysis. In addition, a corresponding hardware analog circuit is designed based on this system by operational amplifiers and multipliers. The experimental results agree with those of the theoretical analysis and confirm that this system is practically feasible. This system will have a good application value in secure communication and cryptography. [ABSTRACT FROM AUTHOR]

In this paper, we detect Unstable Periodic Orbits (UPOs) of both self-excited and hidden attractors using the Damped Newton–Raphson–Mees algorithm. This study answers one of the long-standing questions about the size of a hidden attractor. We observe that the UPOs of the self-excited attractor are spread in a thick region within the phase space at varying distances from the fixed point, whereas the UPOs of hidden attractors are confined to a thin layer. This reveals details concerning the size of a hidden attractor. We demonstrate this for a hidden attractor without a fixed point and for a hidden attractor with a single stable fixed point. To compare the results with self-excited attractors, we have taken Lorenz and Rössler attractor as examples. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter ε. In other words, we prove that the family of attractors behaves continuously as ε → 0. We will also provide the convergence rates in terms of the parameter. [ABSTRACT FROM AUTHOR]

This paper is concerned with the pathwise dynamics of stochastic p -Laplacian lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. When the stochastic system is driven by a linear multiplicative noise, we prove the convergence of solutions and the upper semicontinuity of random attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero. [ABSTRACT FROM AUTHOR]

In this paper, we present a concise four-dimensional (4D) conservative chaotic system with a wide parameter range. Since there are no terms higher than first order, the circuit does not contain multipliers, resulting in a simple circuit implementation. The nonlinear dynamic characteristics, such as phase diagrams, equilibrium points, divergence, Poincaré cross-sections, Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension, are analyzed in detail, which illustrates the conservativity. Besides, the system exhibits different offset boosting behaviors. Through offset boosting, the system can propagate along a line, convert signal polarity, control variable amplitude, generate coexisting attractors, and even induce changes in its state. Specially, we realize the transition from a single-vortex attractor to a multivortex one by some changes in the initial values. Furthermore, the Pearson correlation coefficient is used to demonstrate the higher initial value sensitivity of the system. Finally, the system is implemented through Multisim simulation and analog circuit separately, and their consistency validates the system effectively. [ABSTRACT FROM AUTHOR]

The behavioral and neural effects of the endogenous release of acetylcholine following stimulation of the nucleus basalis (NB) of Meynert have been recently examined in two male monkeys (Qi et al., 2021). Counterintuitively, NB stimulation enhanced behavioral performance while broadening neural tuning in the prefrontal cortex (PFC). The mechanism by which a weaker mnemonic neural code could lead to better performance remains unclear. Here, we show that increased neural excitability in a simple continuous bump attractor model can induce broader neural tuning and decrease bump diffusion, provided neural rates are saturated. Increased memory precision in the model overrides memory accuracy, improving overall task performance. Moreover, we show that bump attractor dynamics can account for the nonuniform impact of neuromodulation on distractibility, depending on distractor distance from the target. Finally, we delve into the conditions under which bump attractor tuning and diffusion balance in biologically plausible heterogeneous network models. In these discrete bump attractor networks, we show that reducing spatial correlations or enhancing excitatory transmission can improve memory precision. Altogether, we provide a mechanistic understanding of how cholinergic neuromodulation controls spatial working memory through perturbed attractor dynamics in the PFC. [ABSTRACT FROM AUTHOR]

Traditional neural network models of associative memories were used to store and retrieve static patterns. We develop reservoir-computing based memories for complex dynamical attractors, under two common recalling scenarios in neuropsychology: location-addressable with an index channel and content-addressable without such a channel. We demonstrate that, for location-addressable retrieval, a single reservoir computing machine can memorize a large number of periodic and chaotic attractors, each retrievable with a specific index value. We articulate control strategies to achieve successful switching among the attractors, unveil the mechanism behind failed switching, and uncover various scaling behaviors between the number of stored attractors and the reservoir network size. For content-addressable retrieval, we exploit multistability with cue signals, where the stored attractors coexist in the high-dimensional phase space of the reservoir network. As the length of the cue signal increases through a critical value, a high success rate can be achieved. The work provides foundational insights into developing long-term memories and itinerancy for complex dynamical patterns. Artificial associative memories in neural network models have shown ability to store and retrieve static patterns of complex systems, however analysis of dynamic patterns remains challenging. The authors develop a reservoir computing based memory approach for complex multistable dynamical systems. [ABSTRACT FROM AUTHOR]

For a nonautonomous stochastic discrete plate equation driven by multiplicative noise, we prove the unique existence of a pullback random attractor, which is a family of pullback‐attracting random compact sets parameterized by time and samples. We then establish the residual dense continuity of the pullback random attractor on the time‐sample plane with respect to the Hausdorff metric. Even without the Cantor continuum hypothesis, we show that the set of all points of continuity of the pullback random attractor is an uncountable set with the continuity cardinality. These results explain both nonexplosive and nonimplosive phenomena for the random plate lattice system. [ABSTRACT FROM AUTHOR]

We develop the theory of kernel sections of non-autonomous dynamical systems. Under some sufficient conditions, we establish some abstract results on the asymptotically autonomous stability of kernel sections, which suggests that the time-section of a kernel section is asymptotically stable to a global attractor of the autonomous dynamical system. A lattice plate equation with nonlinear damping and non-autonomous forcing is considered as an application. It is shown that the time-section of kernel sections of the non-autonomous lattice plate equations are nonempty, uniformly compact, pullback attracting, and asymptotically stable to a global attractor of the autonomous lattice plate equations. [ABSTRACT FROM AUTHOR]

In the phenomenon of mixed-mode oscillations, transitions between large-amplitude and small-amplitude oscillations may lead to anomalous jitter in the probe of a tapping mode atomic force microscope (TM-AFM) during the scanning process, thereby affecting the accuracy and clarity of the topographical images of the tested sample's surface. This work delves deeply into various mixed-mode oscillations and the corresponding formation mechanisms in TM-AFM under low-frequency resonant excitation. Through a detailed analysis of bifurcation sets of the fast subsystem, we found that the system's mixed-mode oscillations encompass the typical two coexisting branches and the novel three coexisting branches of equilibrium point attractors. In the stable case, a certain transition pattern in phase trajectory can be observed involving two jumps and four jumps, switching between quiescent and spiking states. In the bi-stable case, the trajectory undergoes distinct transitions decided by whether to pass through or crossover the middle branch of attractors when bifurcation occurs. By applying basin of attraction and fast–slow analysis methods, we unfold the dynamic mechanism of mixed-mode oscillations with distinct switching patterns. Our research contributes to a better understanding of complex oscillations of TM-AFM and provides valuable insights for improving image quality and measurement precision while mitigating detrimental oscillations. [ABSTRACT FROM AUTHOR]

It is well-known that exponential attractor is an important concept to study the exponentially attracting rate of a compact set for dynamical systems. But recently, Zhao (Appl Math Lett, 2022. https://doi.org/10.1016/j.aml.2021.107791) studied the polynomially attracting sets for a class of wave equations: u tt - Δ u + k ‖ u t ‖ p u t + f (u) = ∫ Ω K (x , y) u t (t , y) d y + h (x). Clearly, exponential attracting can imply the polynomial attracting. While the converse is unknown. In this note, we give an example which illustrates the existence of a polynomially attracting set but nonexistence of any exponential attractor. [ABSTRACT FROM AUTHOR]

For a nonautonomous random parabolic equation with nonlocal diffusion and nonlinear colored noise, we show the existence of a pullback random attractor, which can be regarded as a set-valued mapping defined on the time-sample plane. We mainly prove continuity in probability and mean continuity of the pullback random attractor on the time-sample plane with respect to Hausdorff metric. To achieve the above objectives, we establish a unified abstract result for the existence, continuity in probability and mean continuity of a pullback random attractor and verify the five conditions that depend only on the random dynamical system for the model. These conditions ensure local controllability of the attractor in path and in expectation respectively. [ABSTRACT FROM AUTHOR]

This article studies the pullback dynamics and statistical solutions for the dissipative non-autonomous Zakharov equations in a bounded interval. The main obstacle in our investigating comes from the geometric constrain of global regularity of solutions for the Zakharov equations. This geometric constrain is caused essentially by its particular structure which is a special coupling of a hyperbolic equation with a dispersive nonlinear Schrödinger equation. The nonlinear term Δ | E | 2 in the hyperbolic equation is the core ingredient leading to this geometric constrain. Firstly we show the global existence and uniqueness of solutions, and prove the continuous dependence of solutions on the initial data by exploiting a certain coercivity of the operator corresponding to the linear principle part of the Zakharov equations. Then we establish that the generated process of solution mappings possesses a bounded pullback absorbing set and pullback asymptotic compactness by using a delicate decomposition of the original problem, and obtain the existence of a pullback attractor. Next we formulate an appropriate definition of τ -continuity for the process and employ the calculus of variations to prove this τ -continuity. Afterwards, we construct a family of invariant Borel probability measures for the process via the pullback attractor and the notion of generalized Banach limit. We also propose an appropriate definition of statistical solutions for the addressed Zakharov equations, and establish that the constructed family of invariant Borel probability measures is indeed a statistical solution which satisfies the Liouville theorem of Statistical Mechanics. Our definitions and approach present here can overcome effectively the difficulty caused by the geometric constrain of the Zakharov equations. Finally, we propose some open problems related to the existence and singular limiting behavior of the pullback attractors and statistical solutions for the Zakharov equations with varying coefficient. [ABSTRACT FROM AUTHOR]

A new four-dimensional chaotic system with high complexity is proposed, and the analysis of its dynamics reveals that the system shows a rare topological attractor structure symmetry phenomenon. The phenomenon of inverse multiplicative bifurcation and multistability, which changes from a quadruple period to a double period and then to a single period, are found. A complex oscillatory transformation between chaos and period can be observed, highlighting the complexity of the system. The transient hyperchaos and burst oscillations are found in the time series and Lyapunov exponential spectra of the system, and the rich dynamical behavior of the system is verified by the 0–1 test and complexity spectral entropy test. The analog circuitry of the system is simulated based on Multisim, and the actual digital circuitry of the system is implemented through a Field Programmable Gate Array. The experimental results and the numerical simulation results agree and prove the feasibility of the constructed system. An image encryption algorithm is designed by combining the disruption-diffusion algorithm with the DNA encoding algorithm, and the system is analyzed and verified to be highly sensitive to the chaotic sequence key and initial value. Moreover, the key space of the algorithm is large and the information entropy is close to the ideal value, so the new system proposed can contribute to play a role in the field of secure communication. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonlinear pseudo-parabolic equation with fading memory. First, we prove the existence, uniqueness and continuous dependence of weak solutions when ρ and f have polynomial growth of critical order. Then, we establish the existence and optimal regularity of the global attractor. The result extends and improves some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the backward asymptotic autonomy of pullback attractors for asymptotically autonomous processes. Namely, time-components of the pullback attractors semi-converge to the global attractors of the corresponding limiting semigroups as the time-parameter goes to negative infinity. The present article is divided into two parts: theories and applications. In the theoretical part, we establish a sufficient and necessary criterion with respect to the backward asymptotic autonomy via backward compactness of pullback attractors. Moreover, this backward asymptotic autonomy is considered by the periodicity of pullback attractors. As for the applications part, we apply the abstract results to non-autonomous retarded sine-Gordon lattice systems. By backward uniform tail-estimates of solutions, we prove the existence of a pullback and global attractor for such lattice systems such that the backward asymptotic autonomy is satisfied. Furthermore, it is also fulfilled under the assumptions of the periodicity for the non-delay forcing and the convergence for processes. [ABSTRACT FROM AUTHOR]

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree 6 5 . In contrast to the previously studied one of degree 4 3 , the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert's 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed. [ABSTRACT FROM AUTHOR]

A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rule with a decay term and contribute to changes in the states after a short delay. The lattice system is first reformulated as a coupled functional-ordinary differential equation system on an appropriate product space. Then it is shown that the solution of the system exists and is unique. Furthermore, it is demonstrated that the system of equations generates a continuous random dynamical system. Finally, the existence of random attractors for the random dynamical system generated by the Dong–Hopfield model is established. [ABSTRACT FROM AUTHOR]

We outline key points of the concept of ideal turbulence offering novel scenarios for distributed chaos based not on the geometric-dynamical complexity of the attractor but on the extremely complex spatial structure of elements of the attractor. Ideal turbulence is observed in idealized (without internal resistance) models of various processes related to electromagnetic or acoustic oscillations. This idealization significantly simplifies the analysis and, at the same time, in many cases, provides a quite adequate description of real processes. [ABSTRACT FROM AUTHOR]

In this article, we study non-autonomous dynamical behavior of weakly damped wave equation in unbounded domain. First of all, we introduce the time-dependent locally uniform space. After that, the pullback asymptotical compactness is proved by applying the contractive function method. Eventually, we obtain the existence of (H lu 1 (R 3) × L lu 2 (R 3) , H ρ 1 (R 3) × L ρ 1 (R 3)) -time-dependent attractor of wave equation. [ABSTRACT FROM AUTHOR]

On the basis of the chaotic system proposed by Wang et al. in 2023, this paper constructs a 5D fractional-order memristive hyperchaotic system (FOMHS) with multiple coexisting attractors through coupling of magnetic control memristors and dimension expansion. Firstly, the divergence, Kaplan–Yorke dimension, and equilibrium stability of the chaotic model are studied. Subsequently, we explore the construction of the 5D FOMHS, introducing the definitions of the Caputo differential operator and the Riemann–Liouville integral operator and employing the Adomian resolving approach to decompose the linears, the nonlinears, and the constants of the system. The complex dynamic characteristics of the system are analyzed by phase diagrams, Lyapunov exponent spectra, time-domain diagrams, etc. Finally, the hardware circuit of the proposed 5D FOMHS is performed by FPGA, and its randomness is verified using the NIST tool. [ABSTRACT FROM AUTHOR]

In this work, we formalize the effect of mechanical shaking by using various forms of an externally exerted force, which may be constant or may be position-dependent, and we examine the changes in the potential energy surfaces that quantify the chemical reaction. We use a simple toy model to model the potential energy surfaces of a chemical reaction, and we study the effect of a constant or position-dependent externally exerted force for various forms of the force. As we demonstrate, the effect of the force can be quite dramatic on the potential energy surfaces, which acquire new stationary points and new Newton trajectories that are distinct from the original ones that were obtained in the absence of mechanochemical effects. We also introduce a new approach to mechanochemical interactions, using a dynamical systems approach for the Newton trajectories. As we show, the dynamical system attractor properties of the trajectories in the phase space are identical to the stationary points of the potential energy surfaces, but the phase space contains much more information regarding the possible evolution of the chemical reaction—information that is quantified by the existence of unstable or saddle fixed points in the phase space. We also discuss how an experimental method for a suitable symmetric liquid solution substance might formalize the effect of shaking via various forms of external force, even in the form of an extended coordinate-dependent force matrix. This approach may experimentally quantify the Epstein effect of shaking in chemical solutions via mechanochemistry methods. [ABSTRACT FROM AUTHOR]

Neural mass models are a powerful tool for modeling of neural populations. Such models are often used as building blocks for the simulation of large-scale neural networks and the whole brain. Here, we carry out systematic bifurcation analysis of a neural mass model for the basic motif of various neural circuits, a system of two populations, an excitatory, and an inhibitory ones. We describe the scenarios for the emergence of complex collective behavior, including chaotic oscillations and multistability. We also compare the dynamics of the neural mass model and the exact microscopic system and show that their agreement may be far from perfect. The discrepancy can be interpreted as the action of the so-called shot noise originating from finite-size effects. This shot noise can lead to the blurring of the neural mass dynamics or even turn its attractors into metastable states between which the system switches recurrently. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the dynamic bifurcation of the general Brusselator model when the order of reaction is $ p \in (1,\infty) $. We verify that the Turing instability occurs above the critical control number and obtain a rigorous formula for the bifurcated stable patterns. We define a constant $ s_N $ that gives a criterion for the continuous transition. We obtain continuous transitions for $ s_N>0 $, but jump transitions for $ s_N<0 $. By using this criterion, we prove mathematically that higher-molecular reactions are rarely observed. We also provide some numerical results that illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the pullback dynamics of a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model for porous media in the whole space. The existence and uniqueness of weak solutions is achieved firstly, leading to the generation of a continuous non-autonomous cocycle for the BFed model. Based on the global well-posedness, the existence of pullback attractors is proved by using uniform a priori estimates on the tails of solutions outside bounded domains. [ABSTRACT FROM AUTHOR]

This manuscript extends the nonautonomous notions of attractor-repeller pair and Morse decomposition from the setting of nonautonomous dynamical systems to the setting of topological cocycles. It takes into consideration a general notion of cocycle governed by a group action $ G\times P\rightarrow P $. The limit behavior is determined by the directional attractors depending on a filter basis on the subsets of a semigroup $ S\subset G $. The Conley methodology of constructing Morse decompositions is extended by means of directional attractor-repeller pairs. The main result shows the existence of attractor-repeller pairs for topological cocycles on compact spaces. The properties of attractor-repeller pairs and Morse decompositions are presented. [ABSTRACT FROM AUTHOR]

Of concern is a problem of strain gradient porous elastic theory with nonlinear damping terms, whose constitutive equations contain higher-order derivatives of the displacement in the basic postulates. The paper is based on the theory of 'consistency' due to Aouadi et al. [J. Therm. Stress. 43(2)(2020), 191–209] and Ieşan [American Institute of Physics, Conference Proceedings, 1329 (2011), 130–149], and contains four results. We firstly show that the system is global well posed by using maximal monotone operator. The second main result is the existence of global attractors which is proved by the method developed by Chueshov and Lasiecka [Long-time behavior of second order evolution equations with nonlinear damping. Mem. Amer. Math. Soc. vol. 195, no. 912, Providence, 2008; Von Karman evolution equations: well-posedness and long-time dynamics. Springer Monographs in Mathematics, Springer, New York, 2010]. By showing the system is gradient and asymptotically smooth, we establish the existence of global attractors with finite fractal dimension via a stabilizability inequality. Then we study the continuity of global attractors regarding the parameter in a residual dense set. The above results allow the damping terms with polynomial growth. Finally we discuss the exponential decay and global boundedness to the linear case of damping terms of the system. The assumption of equal-speed wave propagations is not needed for all of results obtained. [ABSTRACT FROM AUTHOR]

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in R 3 . We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of R 3 that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of S 3 which arise naturally when considering toroidal sets. [ABSTRACT FROM AUTHOR]

This article is concerned with the asymptotically autonomous robustness (almost surely and in probability) of random attractors for stochastic version of 3D convective Brinkman-Forchheimer (CBF) equations defined on |$\mathbb {R}^{3}$|⁠ : $$ \begin{align*} &\frac{\partial\boldsymbol{v}}{\partial\mathrm{t}}-\mu\Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\alpha\boldsymbol{v}+\beta|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla{p}=\boldsymbol{f}+''\mbox{stochastic terms}",\quad\nabla\cdot\boldsymbol{v}=0,\end{align*}$$ where |$\mu ,\alpha ,\beta > 0$|⁠ , |$r\geq 1$| and |$\boldsymbol {f}(\cdot)$| is a given time-dependent external force field. Our goal is to study the asymptotically autonomous robustness for 3D stochastic CBF equations perturbed by a linear multiplicative or additive noise when time-dependent forcing converges towards a time-independent function. The main procedure to achieve our goal is how to justify that the usual pullback asymptotic compactness of the solution operators is uniform on some uniformly tempered universes over an infinite time-interval |$(-\infty ,\tau ]$|⁠. This can be done by showing the backward uniform "tail-smallness" and "flattening-property" of the solutions over |$(-\infty ,\tau ]$|⁠. [ABSTRACT FROM AUTHOR]