A velocity–diffusion method for a Lotka–Volterra system with nonlinear cross and self-diffusion
The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction-diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka-Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293-310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the …
Convergent Analytic Solutions for Homoclinic Orbits in Reversible and Non-reversible Systems
In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many important nonlinear PDEs or PDE systems, for which these analytical solutions would correspond to regular or localized pulses of the PDE. As such, the homoclinic solutions derived here are clearly topical, and they are shown to match closely to earlier results obtained by homoclinic numerical shooting. In addition, the results for the non-reversible case go beyond those that have been typically considered in analyses conducted within bifurcation-theoretic sett…
ON THE BOUSSINESQ HIERARCHY
A new sequence of nonlinear evolution systems satisfying the zero curvature property is constructed, by using the invariant singularity analysis. All these systems are completely integrable and a pseudo-potential (linearization) is explicitly determined for each of them. The second system of the sequence is the Broer-Kaup system, which, as is well known, corresponds to the higher order Boussinesq approximation in describing shallow water waves.
Global linear feedback control for the generalized Lorenz system
Abstract In this paper we show how the chaotic behavior of the Chen system can be controlled via feedback technique. We design both a nonlinear feedback controller and a linear one which globally regulate the closed-loop system states to a given point. We finally show that our approach works also for the whole family of the generalized Lorenz system.
A Subcritical Bifurcation for a Nonlinear Reaction–Diffusion System
In this paper the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms is investigated. Through a linear stability analysis we show that the cross-diffusion term allows the pattern formation. To predict the form and the amplitude of the pattern we perform a weakly nonlinear analysis. In the supercritical case the Stuart-Landau equation is found, which rules the evolution of the amplitude of the most unstable mode. With the increasing distance from the bifurcation value of the cross-diffusion parameter, the weakly nonlinear analysis fails and a Fourier–Galerkin approach is adopted. In the subcritical case the weakly nonlinear analysis must be pushed u…
MR2997965 (Review) 37D45 Viana, R. L.; Lopes, S. R.; Szezech, J.D., Jr.; Caldas, I. L. Synchronization of chaos and the transition to wave turbulence. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), no. 10, 1250234, 9 pp.1793-6551
In this paper the authors analyze the onset of the wave turbulence in the spatially extended threewave interacting model by using the concept of synchronization. In order to work with a set of ordinary differential equations, they make a pseudo-spectral decomposition of the wave field and identify the onset of wave turbulence as the excitation of spatial modes in the presence of underlying temporally chaotic dynamics. Each spatial point is regarded as a nonlinear oscillator and the onset of wave turbulence is the point where the oscillators lose phase synchronization. The authors use an extremely sensitive complex-order parameter to estimate the threshold of weak turbulence and perform a Ly…
Adaptive control of a seven mode truncation of the Kolmogorov flow with drag
Abstract We study a seven dimensional nonlinear dynamical system obtained by a truncation of the Navier–Stokes equations for a two dimensional incompressible fluid with the addition of a linear term modelling the drag friction. We show the bifurcation sequence leading from laminar steady states to chaotic solutions with increasing Reynolds number. Finally, we design an adaptive control which drives the state of the system to the equilibrium point representing the stationary solution.
A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results
Abstract In this paper we study the Degn–Harrison system with a generalized reaction term. Once proved the global existence and boundedness of a unique solution, we address the asymptotic behavior of the system. The conditions for the global asymptotic stability of the steady state solution are derived using the appropriate techniques based on the eigen-analysis, the Poincare–Bendixson theorem and the direct Lyapunov method. Numerical simulations are also shown to corroborate the asymptotic stability predictions. Moreover, we determine the constraints on the size of the reactor and the diffusion coefficient such that the system does not admit non-constant positive steady state solutions.
On a radiating fluid in a general relativistic context
A model for the radiation hydrodynamics in general relativity is analyzed, describing the gravitational collapse and supernovae explosion. As these physical phenomena can be assumed spherically symmetric, the equations of motion for a unique fluid, representing the interaction between matter and radiation, are written in a spherical symmetric space-time with respect to a comoving frame. The system is completed by using the Eddington closure, assuming a local thermodynamical equilibrium for the radiation field. The resulting system is analyzed by the Lie symmetry approach and a reduction to an ODEs system is obtained. Numerical simulations of the solutions are performed, showing a realistic …
Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth
We focus on the morphochemical reaction–diffusion model introduced in Bozzini et al. (2013) and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of Turing instability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis, and derive the normal form equations governing the amplitude of the patterns. These amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations. Hysteretic type phenomena are highlighted also through numerica…
Pattern formation driven by cross–diffusion in a 2D domain
Abstract In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.
Symmetry reduction of a model in spherical symmetry for benign tumor
A PDEs system, describing the expansive growth of a benign tumor and the phe- nomenon of encapsulation, is studied via a group analysis approach. A weak equiv- alence classi¯cation is obtained and the original PDEs system is reduced to an ODEs system. Numerical simulations are performed both for ODEs and PDEs, which turn out to be in perfect agreement between each other, showing a realistic enough description of the biological process.
European Option Pricing with Transaction Costs and Stochastic Volatility: an Asymptotic Analysis
In this paper the valuation problem of a European call option in presence of both stochastic volatility and transaction costs is considered. In the limit of small transaction costs and fast mean reversion, an asymptotic expression for the option price is obtained. While the dominant term in the expansion it is shown to be the classical Black and Scholes solution, the correction terms appear at $O(\varepsilon^{1/2})$ and $O(\varepsilon)$. The optimal hedging strategy is then explicitly obtained for the Scott's model.
Corrigendum to “Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations” [19 (6) (2014) 1746–1769]
Corrigendum Corrigendum to ‘‘Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations’’ [19 (6) (2014) 1746–1769] M. Russo , S. Roy Choudhury , T. Rehman , G. Gambino b University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., Orlando, USA University of Palermo, Department of Mathematics and Computer Science, Via Archirafi 34, 90123 Palermo, Italy
A group analysis via weak equivalence transformations for a model of tumor encapsulation
A symmetry reduction of a PDEs system, describing the expansive growth of a benign tumour, is obtained via a group analysis approach. The presence in the model of three arbitrary functions suggests the use of Lie symmetries by using the weak equivalence transformations. An invariant classification is given which allows us to reduce the initial PDEs system to an ODEs system. Numerical simulations show a realistic enough description of the physical process.
Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion
In this paper the Turing pattern formation mechanism of a two components reaction-diffusion system modeling the Schnakenberg chemical reaction is considered. In Ref. (Madzavamuse et al., J Math Biol 70(4):709–743, 2015) it was shown how the presence of linear cross-diffusion terms favors the destabilization of the constant steady state. We perform the weakly nonlinear multiple scales analysis to derive the equations for the amplitude of the Turing patterns and to show how the cross-diffusion coefficients influence the occurrence of super-critical or sub-critical bifurcations. We present a numerical exploration of far from equilibrium regimes and prove the existence of multistable stationary…
MR3039719 Reviewed Wang, Lidong; Liu, Heng; Gao, Yuelin Chaos for discrete dynamical system. J. Appl. Math. 2013, Art. ID 212036, 4 pp. (Reviewer: Gaetana Gambino) 37D45
In this paper the authors show the relation between the definitions of Li-Yorke chaos and distributional chaos in discrete dynamical systems. In particular, after listing the main definitions and reviewing the known results, the authors prove that: • a discrete dynamical system is chaotic in the sense of Martelli and Wiggins when it exhibits transitive distributional chaos; • a discrete dynamical system is distributively chaotic in a sequence when it is chaotic in the strong sense of Li-Yorke. Finally, the authors prove a sufficient condition for the dynamical system to be chaotic in the strong sense of Li-Yorke.
Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier
Abstract In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane–Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler–Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical result…
Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations
In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa-Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do not support singular traveling waves. The third equation supports four-segmented, non-smooth $M$-wave solutions, while the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. Moreover, sm…
Turing pattern formation in the Brusselator system with nonlinear diffusion.
In this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in 1D and 2D spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supe…
Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation: Phase-plane, multi-infinite series and variational approaches
In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively o…
The chaotic Dadras–Momeni system: control and hyperchaotification
In this paper a novel three-dimensional autonomous chaotic system, the so called Dadras-Momeni system, is considered and two different control techniques are employed to realize chaos control and chaos synchronization. Firstly, the optimal control of the chaotic system is discussed and an open loop feedback controller is proposed to stabilize the system states to one of the system equilibria, minimizing the cost function by virtue of the Pontryagin’s minimum principle. Then, an adaptive control law and an update rule for uncertain parameters, based on Lyapunov stability theory, are designed both to drive the system trajectories to an equilibrium or to realize a complete synchronization of t…
Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
<p style='text-indent:20px;'>We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the exis…
Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion
In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively. When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a travelin…
Pattern selection in the 2D FitzHugh–Nagumo model
We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.
Generalized Camassa-Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions
In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the…
A Seven Mode Truncation of the Kolmogorov Flow with Drag: Analysis and Control
The transition from laminar to chaotic motions in a viscous °uid °ow is in- vestigated by analyzing a seven dimensional dynamical system obtained by a truncation of the Fourier modes for the Kolmogorov °ow with drag friction. An- alytical expressions of the Hopf bifurcation curves are obtained and a sequence of period doubling bifurcations are numerically observed as the Reynolds num- ber is increased for ¯xed values of the drag parameter. An adaptive stabilization of the system trajectories to an equilibrium point or to a periodic orbit is ob- tained through a model reference approach which makes the control global. Finally, the e®ectiveness of this control strategy is numerically illustra…
Cross-diffusion driven instability for a Lotka-Volterra competitive reaction-diffusion system
In this work we investigate the possibility of the pattern formation for a reaction-di®usion system with nonlinear di®usion terms. Through a linear sta- bility analysis we ¯nd the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, we show how cross-di®usion e®ects are responsible for the initiation of spatial patterns. Finally, we ¯nd a Fisher amplitude equation which describes the weakly nonlinear dynamics of the system near the marginal stability.
Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system
In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form …
Intermittent and passivity based control strategies for a hyperchaotic system
In this paper a four-dimensional hyperchaotic system with only one equilibrium is consid- ered and it is shown how the control and the synchronization of this system can be realized via two different control techniques. Firstly, we propose a periodically intermittent con- troller to stabilize the system states to the equilibrium and to achieve the projective syn- chronization of the system both in its periodic and hyperchaotic regime. Then, based on the stability properties of a passive system, we design a linear passive controller, which only requires the knowledge of the system output, to drive the system trajectories asymptoti- cally to the origin. Using the same passivity-based method, …
Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System
In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control.
MR3090050 Reviewed Belabbas, Mohamed Ali On global stability of planar formations. IEEE Trans. Automat. Control 58 (2013), no. 8, 2148–2153. (Reviewer: Gaetana Gambino) 93A14 (93D05 93D21)
The focus of the paper is planar formation control, i.e. the design of control laws to stabilize agents at given distances from each other, under the constraint that the dynamics of each agent only depends on a subset of the other agents. The main contribution of the paper is the following: It is shown that a simple four-agent formation cannot be globally stabilized using twice differentiable control laws (this is not the case for three-agent formations), even up to sets of measure zero of initial conditions. This suggests that for four-agent formations one needs to look for control laws that are either not differentiable (or even not continuous) or of higher order in the dynamics. The appr…
A SUBCRITICAL BIFURCATION FOR A NONLINEAR REACTION–DIFFUSION SYSTEM
In this paper the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms is investigated. Through a linear stability analysis we show that the cross-diffusion term allows the pattern formation. To predict the form and the amplitude of the pattern we perform a weakly nonlinear analysis. In the supercritical case the Stuart-Landau equation is found, which rules the evolution of the amplitude of the most unstable mode. With the increasing distance from the bifurcation value of the cross-diffusion parameter, the weakly nonlinear analysis fails and a Fourier–Galerkin approach is adopted. In the subcritical case the weakly nonlinear analysis must be pushed u…
Turing Instability and Pattern Formation for the Lengyel–Epstein System with Nonlinear Diffusion
In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel---Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we c…
Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system.
In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce …
An equilibrium point regularization for the Chen system
This paper addresses the control of the chaotic Chen system via a feedback technique. We first present a nonlinear feedback controller which drives the trajectories of the Chen system to a given point for any initial conditions. Then, we design a linear feedback controller which still assures the global stability of the Chen system. We moreover achieve the tracking of a reference signal. Numerical simulations are provided to show the effectiveness of the developed controllers.
WAVE PROPAGATION AND PATTERN FORMATION FOR A REACTION-DIFFUSION SYSTEM WITH NONLINEAR DIFFUSION
We investigate the formation of macroscopic spatio-temporal structures (patterns) for a reaction-diffusion system with nonlinear diffusion. We show that cross-diffusion effects are responsible of pattern initiation. Through a weakly nonlinear analysis we are able to predict the shape and the amplitude of the pattern. In the weakly nonlinear regime we derive the Ginzburg-Landau equation which captures the envelope evolution and the progressing of the pattern as a wave. Numerical simulations, performed using both a particle and a spectral method, are in good agreement with the analytical results.
Cross-diffusion driven instability for a nonlinear reaction-diffusion system
In this work we investigate the possibility of the pattern formation for a system of two coupled reaction-diffusion equations. The nonlinear diffusion terms has been introduced to describe the tendency of two competing species to diffuse faster (than predicted by the usual linear diffusion) toward lower densities areas. The reaction terms are chosen of the Lotka-Volterra type in the competitive interaction case. The system is supplemented with the initial conditions and no-flux boundary conditions. Through a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, we show h…