Dimensional Nonlinear Dynamical System Research Articles

Computational analysis of entropy generation in three-dimensional mixed convection flow with thermal dissipation and variable thermal conductivity

Entropy generation in steady three-dimensional thermal flow is investigated under the coupled influence of viscous dissipation and mixed convection. The stretching sheet with an exponential velocity profile induces the flow. In this examination, we presume that the fluid is incompressible and possesses a Newtonian behavior and temperature-dependent thermal conductivity. Using the appropriate classical similarity variables reduces the dimensional nonlinear dynamical system of governing equations to a dimensionless nonlinear dynamical set of ordinary differential equations. It’s important to note that the selection of an exponentially varying surface temperature is deliberate, as it enables precise similarity transformations to be established in situations involving both viscous dissipation and buoyancy forces. The simplified finite difference approach (SFDM) is used to solve the acquired nonlinear dynamical set of ordinary differential equations. The tridiagonal matrix approach is used to solve the resulting system of algebraic equations. In a particular situation, the present numerical simulation is numerically checked with the previously available results and observed a good agreement. The effects of physical parameters on velocity, temperature, surface stresses and heat tranfer rate, entropy generation, and the irreversibility parameter are investigated. The entropy generation rate exhibits a decreasing trend with respect to the mixed convection parameter in the vicinity of the stretching surface. Entropy reduction can be achieved by employing a working fluid with a low Prandtl number and by minimizing the fluid friction.

Evolving Tangent Hyperbolic memristor based 6D chaotic model with fractional order derivative: Analysis and applications

The nonlinear dynamics of fractional order systems and memristor based circuits have attracted an escalating attention in recent years. This paper reports on a novel six dimensional nonlinear dynamical system with a tangent hyperbolic memristor circuit and amalgamated image encryption. The dynamical system is analyzed using standard tools, including phase portraits, equilibrium points, Eigenvalues, Routh–Hurwitz stability criterion, Lyapunov exponents and bifurcation diagrams. The analysis suggests that the developed system is chaotic in nature and has exciting new 2D and 3D trajectories that have not been reported before. The chaotic system is numerically solved for different values of the fractional order q, and the results of equilibrium points and Lyapunov exponents are mentioned in tables. Fractional order circuits are designed for q=1 and q=0.99 utilizing the approximation fractional order technique of the transfer function. The developed circuit is generalized in a way that the circuit can work for both the integer and non-integer fractional order values by just introducing the fractional block to make it work for q=0.99, and removing it with a conventional integrator capacitor to work for q=1. The circuit simulation and numerical results were compared, and signifying that they were both in good agreement with each other. Lastly, the random number generated from the chaotic system is utilized to scramble the image via the scheme of Amalgamated Image Encryption. The scrambled image was tested using different image security test algorithms to support the idea that the chaotic system and image can together form an advantageous key.

Recurrent neural networks enable design of multifunctional synthetic human gut microbiome dynamics.

Predicting the dynamics and functions of microbiomes constructed from the bottom-up is a key challenge in exploiting them to our benefit. Current models based on ecological theory fail to capture complex community behaviors due to higher order interactions, do not scale well with increasing complexity and in considering multiple functions. We develop and apply a long short-term memory (LSTM) framework to advance our understanding of community assembly and health-relevant metabolite production using a synthetic human gut community. A mainstay of recurrent neural networks, the LSTM learns a high dimensional data-driven non-linear dynamical system model. We show that the LSTM model can outperform the widely used generalized Lotka-Volterra model based on ecological theory. We build methods to decipher microbe-microbe and microbe-metabolite interactions from an otherwise black-box model. These methods highlight that Actinobacteria, Firmicutes and Proteobacteria are significant drivers of metabolite production whereas Bacteroides shape community dynamics. We use the LSTM model to navigate a large multidimensional functional landscape to design communities with unique health-relevant metabolite profiles and temporal behaviors. In sum, the accuracy of the LSTM model can be exploited for experimental planning and to guide the design of synthetic microbiomes with target dynamic functions.

An Analysis of a Two Dimensional Continuous Non-Linear Dynamical Systems

An Analysis of a Two Dimensional Continuous Non-Linear Dynamical Systems

A Mathematical Study of a Generalized SEIR Model of COVID-19

In this work (Part I), we reinvestigate the study of the stability of the Covid-19 mathematical model constructed by Shah et al. (2020) [1]. In their paper, the transmission of the virus under different control strategies is modeled thanks to a generalized SEIR model. This model is characterized by a five dimensional nonlinear dynamical system, where the basic reproduction number can be established by using the next generation matrix method. In this work (Part I), it is established that the disease free equilibrium point is locally as well as globally asymptotically stable when . When , the local and global asymptotic stability of the equilibrium are determined employing the second additive compound matrix approach and the Li-Wang’s (1998) stability criterion for real matrices [2]. In the second paper (Part II), some control parameters with uncertainties will be added to stabilize the five-dimensional Covid-19 system studied here, in order to force the trajectories to go to the equilibria. The stability of the Covid-19 system with these new parameters will also be assessed in Intissar (2020) [3] applying the Li-Wang criterion and compound matrices theory. All sophisticated technical calculations including those in part I will be provided in appendices of the part II.

Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation

A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplified model of a reed-like instrument having two quasi-harmonic resonances, represented by a four dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the different solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the first acoustic resonance (first register) or the second acoustic resonance (second register) are successfully identified by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The effect of the inharmonicity between the first two impedance peaks is observed both when the amplitude of the first is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.

Information Reconstruction on an Infinite Tree for a $$4\times 4$$-State Asymmetric Model with Community Effects

The information reconstruction problem on an infinite tree, is to collect and analyze massive data samples at the $n$th level of the tree to identify whether there is non-vanishing information of the root, as $n$ goes to infinity. This problem has wide applications in various fields such as biology, information theory and statistical physics, and its close connections to cluster learning, data mining and deep learning have been well established in recent years. Although it has been studied in numerous contexts, the existing literatures with rigorous reconstruction thresholds established are very limited. In this paper, motivated by a classical deoxyribonucleic acid (DNA) evolution model, the F$81$ model, and taking into consideration of the Chargaff's parity rule by allowing the existence of a guanine-cytosine content bias, we study the noise channel in terms of a $4\times 4$-state asymmetric probability transition matrix with community effects, for four nucleobases of DNA. The corresponding information reconstruction problem in molecular phylogenetics is explored, by means of refined analyses of moment recursion, in-depth concentration estimates, and thorough investigations on an asymptotic $4$-dimensional nonlinear second order dynamical system. We rigorously show that the reconstruction bound is not tight when the sum of the base frequencies of adenine and thymine falls in the interval $\left(0,1/2-\sqrt{3}/6\right)\bigcup \left(1/2+\sqrt{3}/6,1\right)$, which is the first rigorous result on asymmetric noisy channels with community effects.

Emergence of behavior through morphology: a case study on an octopus inspired manipulator

The complex motion abilities of the Octopus vulgaris have been an intriguing research topic for biologists and roboticists alike. Various studies have been conducted on the underlying control architectures employed by these high dimensional biological organisms. Researchers have attempted to replicate these architectures on robotic systems. Contrary to previous approaches, this study focuses on a robotic system, which is only morphologically similar to the Octopus vulgaris, and how it would behave under different control policies. This sheds light on the underlying optimality principles that these biological systems employ. Open loop control policies are obtained through a trajectory optimization method on a learned forward dynamic model. The motion patterns emerging from variations in morphology and environment were then derived to study the role of the body and environment. Results show that for the specific case of dynamic reaching with a soft appendage, the invariance in motion profile is a fundamental constraint imposed by the morphology and environment, independent from the controller. This suggests how morphological design can simplify stable control even for highly dimensional nonlinear dynamical systems and can provide insights into design of new soft robotic mechanisms.

Local and global analysis of a speculative housing market with production lag.

We extend the model of Dieci and Westerhoff [J. Evol. Econ. 22(2), 303-329 (2012)], where the authors analyse a speculative housing market populated by heterogeneous interacting agents described by a two dimensional nonlinear discrete time dynamical system. They show the emergence of complicated dynamics through the occurrence of bifurcations for particular parameter combinations. We enlarge their model in several ways. On one hand, we introduce time lag in the supply side and we consider two new scenarios characterised by agents' expectations formation. First, naive expectations instead of perfect foresight are considered, while in the second scenario, we study a mix between the model of Dieci and Westerhoff [J. Evol. Econ. 22(2), 303-329 (2012)] and the one we propose. As a consequence, we, analytically and numerically, explain the appearance of instability in the housing market providing conditions on the parameters that lead to a bifurcation. On the other hand, thanks to further numerical simulations, we conduct a global analysis providing the structure of the basin of attractions of the map showing coexistence of attractors.

Lie-Algebraic Structure of the Lax-Integrable (2|1 + 1)-Dimensional Supersymmetric Matrix Dynamical Systems

By using a specially constructed Backlund transformation, we obtain the Hamiltonian representation for the hierarchy of Lax-type flows on the space dual to the Lie algebra of matrix superintegrodifferential operators with one anticommutative variable coupled with suitable evolutions of eigenfunctions and adjoint eigenfunctions of the associated spectral problems. We also propose a Hamiltonian description for the corresponding set of the hierarchies of additional homogeneous symmetries (squared eigenfunction symmetries). The relationship between these hierarchies and the Lax-integrable (2|1 + 1)-dimensional supersymmetric matrix nonlinear dynamical systems and their triple Lax-type linearization is analyzed.

Multi-model cross-pollination in time

The predictive skill of complex models is rarely uniform in model-state space; in weather forecasting models, for example, the skill of the model can be greater in the regions of most interest to a particular operational agency than it is in “remote” regions of the globe. Given a collection of models, a multi-model forecast system using the cross-pollination in time approach can be generalized to take advantage of instances where some models produce forecasts with more information regarding specific components of the model-state than other models, systematically. This generalization is stated and then successfully demonstrated in a moderate (∼40) dimensional nonlinear dynamical system, suggested by Lorenz, using four imperfect models with similar global forecast skill. Applications to weather forecasting and in economic forecasting are discussed. Given that the relative importance of different phenomena in shaping the weather changes in latitude, changes in attitude among forecast centers in terms of the resources assigned to each phenomena are to be expected. The demonstration establishes that cross-pollinating elements of forecast trajectories enriches the collection of simulations upon which the forecast is built, and given the same collection of models can yield a new forecast system with significantly more skill than the original forecast system.

An efficient center manifold technique for Hopf bifurcation of n-dimensional multi-parameter systems

• Hopf bifurcation of a n -dimensional nonlinear multi-parametric dynamical system. • Use of the center manifold theory via a proper symbolic form. • Effective computation of the “restricted” normal form throughout the parameter space. • Construction of bifurcation portraits and evaluation of the respective limit cycles. • Application to two specific three-dimensional, three-parametric systems. The center manifold theory with respect to the simple Hopf bifurcation of a n -dimensional nonlinear multi-parametric system is treated via a proper symbolic form. Analytical expressions of the involved quantities are obtained as functions of the parameters of the system via effective algorithms based on the followed procedure and carried out using a symbolic computation software. Moreover the normal form of a codimension 1 Hopf bifurcation, as well as the corresponding Lyapunov coefficient and bifurcation portrait, can be computed for any system under consideration. Here the computational procedure is applied to two nonlinear three-dimensional, three-parametric systems and graphical results are obtained as concerns the stability regions, the bifurcation portraits, as well as emerged limit cycles with respect to both the supercritical and the subcritical case of bifurcation.

On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems

We study the classification schemes for nonoscillatory solutions of a class of nonlinear two dimensional systems of first order delay dynamic equations on time scales. Necessary and sufficient conditions are also given in order to show the existence and nonexistence of such solutions and some of our results are new for the discrete case. Examples will be given to illustrate some of our results.

Generating finite dimensional integrable nonlinear dynamical systems

In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed.

The Sufficient Condition for the Existence of Periodic Solution of Certain Three Dimensional Nonlinear Dynamical Systems

In this paper, the sufficient condition for the existence of periodic solution of a class of three dimensional nonlinear dynamical systems is investigated. The moving Frenet frame is established on the closed orbit and the successor functions near the closed orbit are defined. According to the study of the existence of solution of the equation which obtained from the successor functions with the implicit function theorem, the sufficient condition for the existence of periodic solution of these systems is obtained. The results in this paper have important significance to decide the existence of periodic solution of three dimensional nonlinear dynamical systems.

The Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Nonlinear Dynamical Systems

In this paper, we investigate a class of three dimensional nonlinear dynamical systems whose unperturbed systems have a family of periodic orbits. Firstly, we establish the moving Frenet Frame on these closed orbits. Secondly, the successor functions are defined by the orbits which go through the normal plane. Finally, by judging the existence of solutions of the equations obtained from the Successor functions, we obtain the necessary condition for the existence of periodic solutions of these three dimensional nonlinear dynamical systems. The result has important significance for the basic research of applied mechanics.

A framework for optimal temporal reduced order modeling of nonlinear dynamical systems

An optimal temporal reduced order modeling framework is proposed for nonlinear dynamical systems. The governing equations are modified for an under-resolved simulation with an arbitrary scheme and a coarse temporal grid. Subgrid-scale models are developed to account for the unresolved temporal structure via inclusion of statistical information on a multi-point temporal stencil. These models are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation. In order to validate the proposed framework, we investigate time-periodic solutions for a canonical Duffing oscillator using a coarse harmonic balance scheme. In order to demonstrate application of the proposed framework to a high dimensional nonlinear dynamical system, we also investigate a simply supported, geometrically nonlinear beam under the influence of time-periodic external forcing. For both problems, the subgrid-scale models are shown to significantly improve the accuracy of coarse-grained solutions.

Chaos Behavior and Control of Stay-Cables

Stay-cable is infinite dimensional nonlinear dynamical system with a very complex vibration types and mechanism which are not described reasonably yet. In order to better control its dynamic behavior, it is necessary to study complex dynamic behavior carefully. Fistly, partial differential equation of the cable motion is established based on the parabolic initial configuration and is simplified into n Duffing-equations by using Galerkin method. Secondly, the chaos behaviors of the first order Duffing-equation under periodic excitation are studied by taking advantage of Melnikov method. At last , parameters may lead to chaotic motion of a true cable in laboratory are calculated and the methods of chaos control are discussed briefly. The study shows that: 1. First order vibration of cable under periodic excitation has much more complex behaviors than the freedom vibration; 2. The Melnikov method can be very effective and convenient for the analysis of chaotic motion of cable.

Dynamical origin of complex motor patterns

Behavior emerges as the nervous system generates motor patterns in charge of driving a peripheral biomechanical device. For several cases in the animal kingdom, it has been identified that the motor patterns used in order to accomplish a diversity of tasks are the different solutions of a simple, low dimensional nonlinear dynamical system. Yet, motor patterns emerge from the interaction of an enormous number of individual dynamical units. In this work, we study the dynamics of the average activity of a large set of coupled excitable units which are periodically forced. We show that low dimensional, yet non trivial dynamics emerges. As a case study, we analyze the air sac pressure patterns used by domestic canaries during song, which consists of a succession of repetitions of different syllable types. We show that the pressure patterns used to generate different syllables can be approximated by the solutions of the investigated model. In this way, we are capable of integrating different description scales of our problem.

Dynamics and stability of volume-scavenging drop arrays: Coarsening by capillarity

N drops, pinned by circular contact lines, are arranged in an array and coupled by a network of conduits. Inertialess exchange of volume among drops is driven by capillarity through the minimization of total surface energy. Drops scavenge volume from one another based on pressure differences, proportional to the surface tension, and arising from curvature differences. The system coarsens in the sense that, with time, volume is increasingly localized and ends up in a single ‘winner’ drop. Numerical simulations show that the identity of the winner can depend discontinuously on the initial condition and connectivity network. This motivates a study of the corresponding N -dimensional nonlinear dynamical system. All fixed points and their linear stabilities, obtained analytically, are found to be independent of connectivity. To determine which of the stable fixed points will be the winner, manifolds separating the attracting regions are found using a method which combines local information (eigenvectors at fixed points) with global information (invariant manifolds due to symmetry). This method is demonstrated for three N = 3 systems with various connectivity networks, and is used to explain the numerical observations.