Dimensional Nonlinear Systems Research Articles

Null controllability via comparison results for nonlinear age-structured population dynamics

We consider an infinite dimensional nonlinear controlled system describing age-structured population dynamics, where the birth and the mortality rates are nonlinear functions of the population size. The control being active on some age range, we give sharp conditions subject to the age range and the control time horizon to get the null controllability of the nonlinear controlled population dynamics. The main novelty is that we use here as a main ingredient the comparison principle for age-structured population dynamics, and in case of null controllability we provide a feedback control with a very simple structure, while preserving the nonnegativity of the state trajectory. Finally, we establish the lack of the null controllability for the linear Lotka-McKendrick equation with spatial diffusion when the control acts in a subset of the habitat and we want to preserve the positivity of the state trajectory.

Period - control in a coupled system of two genetic oscillators for synthetic biology

Biological complex mechanisms with oscillatory behavior are often modeled by high dimensional nonlinear ODEs systems, which makes the analysis and understanding the dynamics of the system difficult. In this work, we consider two reduced models that mimic the oscillatory dynamics of the cell cycle and the circadian clock, and study their coupling from a synthetic biology perspective. To improve the performance and robustness of the oscillatory dynamics in a living cellular environment, we consider the problem of augmenting the parameter region admitting periodic solutions. Moreover, we study the capacity for mutual period regulation and control of the coupling between the two reduced oscillators.

The Swedish industrial support program of the 1970s revisited

The economy-wide dynamic cost-benefit study of the Swedish industrial subsidy program 1976 through 1984 (Carlsson et al. Res Policy 10(43):336–354 1981; Carlsson J Ind Econ 32(1):9–14, 1983a, b) is revisited in light of later economic development. Since the Swedish Micro to Macro model (Eliasson Am Econ Rev 67(1):277–281 1977a, 2017a) was used for quantification, this article is both (1) a study on the calibration of high dimensional micro-based and nonlinear economic systems models, and (2) a post inquiry into the empirical credibility of the cost-benefit calculations performed. We find that the Micro-based Macro model represents the minimum of detailed resolution necessary for the dynamic cost benefit calculations of the micro interventions in the Swedish economy we study. Even though the increased model complexity meant significant parameter calibration difficulties, a thoroughly researched model specification with exactly defined policy interfaces (with the markets of the economy) should take priority over parameter estimation problems, and always be preferred to estimating the parameters of a wrongly specified model perfectly. The oil price shocks of the 1970s caused radical market disorder in the western economies, bankrupting some 35% of Swedish manufacturing and threatening the Swedish government with massive unemployment. We confirm the earlier results that the government choice of a radical employment rescue policy came at enormous social cost in the form of economic stagnation, and still did not prevent the unemployment of the rest of OECD Europe from hitting Sweden a decade later, and persisting well into the next millennium. According to an alternative simulated policy scenario on the model, had the subsidies been replaced with a general lowering of the payroll tax of the same magnitude and the consequent increase in unemployment taken immediately during 1976–1980, production structures would have been radically and rapidly reorganized, normal employment would have been rapidly restored, and neither the stagnation nor the radical increase in unemployment of the early 1990s would have occurred. In retrospect we see no reason to worry about the empirical credibility of this computed dynamic trade off between Keynesian demand and Schumpeterian supply effects (caused by resource reallocations and endogenous structural change due to the price change), as we did then. We conclude with certainty that this trade-off would not even have been discovered as a possibility had we used a traditional model that did not embody these micro-macro linkages.

Residual symmetry, Bäcklund transformation and CRE solvability of a ( $$\mathbf{2}{\varvec{+}}{} \mathbf{1}$$ 2 + 1 )-dimensional nonlinear system

In this paper, the truncated Painleve expansion is employed to derive a Backlund transformation of a (\(2+1\))-dimensional nonlinear system. This system can be considered as a generalization of the sine-Gordon equation to \(2+1\) dimensions. The residual symmetry is presented, which can be localized to the Lie point symmetry by introducing a prolonged system. The multiple residual symmetries and the nth Backlund transformation in terms of determinant are obtained. Based on the Backlund transformation from the truncated Painleve expansion, lump and lump-type solutions of this system are constructed. Lump wave can be regarded as one kind of rogue wave. It is proved that this system is integrable in the sense of the consistent Riccati expansion (CRE) method. The solitary wave and soliton–cnoidal wave solutions are explicitly given by means of the Backlund transformation derived from the CRE method. The dynamical characteristics of lump solutions, lump-type solutions and soliton–cnoidal wave solutions are discussed through the graphical analysis.

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.

Stable Open Loop Control of Soft Robotic Manipulators

Dynamic control of soft robotic manipulators is a challenging field still in its nascent stages. Modeling is still a major hurdle due to its high dimensional nonlinear dynamic properties. Even if accurate models of these high dimensional nonlinear systems are available, the computational burden of the models or sensory requirements poses problems for robust and stable control. This letter addresses these two problems by using a data-driven model that uses only mechanical feedback for stable control of the soft robotic manipulator. Using a learning-based open loop dynamic controller, we investigate the self-stabilizing behavior that can be obtained from the complex dynamics of a soft manipulator. Experimental findings illustrate the advantage of using open-loop dynamic controllers for accurate self-stabilizing control of soft robotic manipulators without any sensory feedback.

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.

Numerical Solutions of System of Second Order Boundary Value Problems using Galerkin Method

In this paper we derive the formulation of one dimensional linear and nonlinear system of second order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. Here we use Bernstein and Legendre polynomials as basis functions. The proposed method is tested on several examples and reasonable accuracy is found. Finally, the approximate solutions are compared with the exact solutions and also with the solutions of the existing methods.GANIT J. Bangladesh Math. Soc.Vol. 37 (2017) 161-174

Ranking Variables in Nonlinear Nonparametric Additive System Identification

Identification of a high dimensional nonlinear nonparametric system is costly. On the other hand for many real-world problems, they are sparse in the sense that not all variables contribute or contribute significantly. If these variables that do not contribute or contribute little can be identified and removed prior to system identification, the identification problem is of lower dimension. In this paper, depending on the identification purposes, importance measures are defined. Based on these measures, ways to calculate these measures and to rank the importance of variables are proposed. This paper addresses such questions.

Application of Modified Adomian Decomposition Method to (2+1)-dimensional Non-linear Wu-Zhang system

In this paper, the Domain decomposition method (ADM) with modified polynomials is applied for nonlinear (2+1) – dimensional Wu-Zhang system . we compared the solution of the system with MVIM, HPM and RDTM [ 17,13,15]. The numerical results obtained by this polynomial are very effective, convenient and quite accurate to system of partial differential equations. A comparative between the modifications method and the other methods is present from some examples to show the efficiency of each method.

Pemodelan Matematika Perpindahan Panas Konveksi Campuran (Mixed Convection) pada Pelat Horizontal

This study examines and analyzes mathematical model of mixed convection in horizontal plate. The heat transfer uses the model of a two dimensional nonlinear partial differential equations system. Then, this equation is derived first into the dimensionless equation form, and then it is changed into system of nonlinear ordinary differential equations form using similarity transformation. This system of nonlinear ordinary differential equations is solved by using the finite-difference scheme method, also with the mathematics program with software Matlab. The results obtained form this program is to determine skin friction coefficient , wall temperature velocity profiles and temperature profiles

Utilizing null controllable regions to stabilize input-constrained nonlinear systems

In this paper, we present a method for control of input-constrained nonlinear systems that offers guaranteed stabilization from the entire null controllable region (NCR). The controller achieves stabilization by using a constrained control Lyapunov function (CCLF) based on this NCR. Prior to online implementation, the level sets the CCLF are constructed using an iterative algorithm. The algorithm works by using an invariance principle to expand an initial quadratic Lyapunov function-based region of attraction. The level sets of this CCLF are then utilized in the control calculations, and in particular, an MPC is formulated that requires the system to go to lower level sets of the CCLF. The proposed MPC thus achieves stabilization from the entire NCR. The proposed approach is first corroborated against existing results for linear systems using two- and three-dimensional linear systems examples. Subsequently, the implementation is shown for two and three dimensional nonlinear systems.

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.

Generalized Lie symmetry approach for fractional order systems of differential equations. III

The generalized Lie symmetry technique is proposed for the derivation of point symmetries for systems of fractional differential equations with an arbitrary number of independent as well as dependent variables. The efficiency of the method is illustrated by its application to three higher dimensional nonlinear systems of fractional order partial differential equations consisting of the (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov system, (3 + 1)-dimensional Burgers system, and (3 + 1)-dimensional Navier-Stokes equations. With the help of derived Lie point symmetries, the corresponding invariant solutions transform each of the considered systems into a system of lower-dimensional fractional partial differential equations.

Output-Tracking Quantized Explicit Nonlinear Model Predictive Control Using Multiclass Support Vector Machines

In applications involving digital control, the set of admissible control actions is finite/quantized. Coupled with state constraints and fast dynamics, explicit model predictive control (EMPC) provides an attractive control formalism. However, the design of data-driven EMPCs with finite admissible control sets is a challenging and relatively unexplored problem. In this paper, a systematic data-driven method is proposed for the design of quantized EMPCs (Q-EMPCs) for time-varying output tracking in nonlinear systems. The design involves: 1) sampling the admissible state space using low-discrepancy sequences to provide scalability to higher dimensional nonlinear systems; 2) at each sampled data point, solving for optimal quantized model predictive control actions and determining feasibility of the intrinsic mixed-integer nonlinear programming problem; and 3) constructing the Q-EMPC control surface using multiclass support vector machines (MC-SVMs). In particular, four widely used MC-SVM algorithms are employed to construct the proposed data-driven Q-EMPC. Extensive testing and comparison among the different MC-SVM algorithms is performed on 2-D and 5-D benchmark examples to demonstrate the effectiveness and scalability of the proposed methodology.

Modeling the population dynamics and community impacts of Ambystoma tigrinum: A case study of phenotype plasticity

Phenotypic plasticity is the ability of an organism to change its phenotype in response to changes in the environment. General mathematical descriptions of the phenomenon rely on an abstract measure of “viability” that, in this study, is instantiated in the case of the Tiger Salamander, Ambystoma tigrinum. This organism has a point in its development when, upon maturing, it may take two very different forms. One is a terrestrial salamander (metamorph)that visits ponds to reproduce and eat, while the other is an aquatic form (paedomorph) that remains in the pond to breed and which consumes a variety of prey including its own offspring.A seven dimensional nonlinear system of ordinary differential equations is developed, incorporating small (Z) and large (B) invertebrates, Ambystoma young of the year (Y), juveniles (J), terrestrial metamorphs (A) and aquatic paedomorphs (P). One parameter in the model controls the proportion of juveniles maturing into A versus P. Solutions are shown to remain non-negative. Every effort was made to justify parameters biologically through studies reported in the literature.A sensitivity analysis and equilibrium analysis of model parameters demonstrate that morphological choice is critical to the overall composition of the Ambystoma population. Various population viability measures were used to select optimal percentages of juveniles maturing into metamorphs, with optimal choices differing considerably depending on the viability measure. The model suggests that the criteria for viability for this organism vary, both from location to location and also in time. Thus, optimal responses change with spatiotemporal variation, which is consistent with other phenotypically plastic systems.Two competing hypotheses for the conditions under which metamorphosis occurs are examined in light of the model and data from an Ambystoma tigrinum population at Mexican Cut, Colorado. The model clearly supports one of these over the other for this data set. There appears to be a mathematical basis to the general tenet of spatiotemporal variation being important for the maintenance of polyphenisms, and our results suggest that such variation may have cascading effects on population, community, and perhaps ecosystem dynamics because it drives the production of a keystone, cannibalistic predator.

Nonlocal symmetry and consistent Riccati expansion integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system

Under investigation in this paper is nonlocal symmetry, consistent Riccati expansion (CRE) integrability of the (1+1)-dimensional integrable nonlinear dispersive-wave system, which can be used to describes a bidirectional soliton for wave propagation. We construct the Bäcklund transformation and consider the truncated Painlevé expansion of the system. It’s Schwarzian form is derived, whose nonlocal symmetry is localized to provide the corresponding nonlocal group. Furthermore, we verify that the system is solvable via the CRE. Based on the CRE, we further present its soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral.

Mechanisms of nonlinear convective flow of Jeffrey nanofluid due to nonlinear radially stretching sheet with convective conditions and magnetic field

Mathematical modeling for magnetohydrodynamic (MHD) nonlinear convective flow of Jeffrey nanofluid over a nonlinear stretching sheet is introduced. Effect of thermal radiation is considered. Phenomena of heat and mass transfer is based through involvement of convective conditions. The Brownian motion and thermophoresis effects are deliberated in energy and concentration expressions. Dimensional nonlinear systems for momentum, energy and concentration are converted into dimensionless systems by invoking appropriate variables. Series solutions are acquired through convergence domains. Behavior of various physical variables on velocity, temperature and nanoparticle concentration are scrutinized graphically. Skin friction coefficient, local Nusselt and Sherwood numbers are calculated through numerical values. It is concluded that velocity field enhances for Deborah number while reverse situation is noticed regarding power index. Temperature and heat transfer rate are enhanced via thermal radiation. Moreover impact of Brownian motion on the concentration and local Sherwood number are quite reverse.

Nonlocal symmetries and conservation laws of the Sinh-Gordon equation

Nonlocal symmetries of the (1+1)-dimensional Sinh-Gordon (ShG) equation are obtained by requiring it, together with its Bäcklund transformation (BT), to be form invariant under the infinitesimal transformation. Naturally, the spectrum parameter in the BT enters the nonlocal symmetries, and thus through the parameter expansion procedure, infinitely many nonlocal symmetries of the ShG equation can be generated accordingly. Making advantages of the consistent conditions introduced when solving the nonlocal symmetires, some new nonlocal conservation laws of the ShG equation related to the nonlocal symmetries are obtained straightforwardly. Finally, taking the nonlocal symmetries as symmetry constraint conditions imposing on the BT, some new finite and infinite dimensional nonlinear systems are constructed.