G. Gambino

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.

Original article: 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 traveling wavefront. In this case the amplitude of the pattern is modulated in space and the corresponding evolution is governed by the Ginzburg-Landau equation.

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 one-dimensional and two-dimensional 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 supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves: When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and, through a matching procedure, we construct the outer amplitude equation and the inner core solution.

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 compute the complex Ginzburg–Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.

Intermittent and passivity based control strategies for a hyperchaotic system

In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered 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 controller to stabilize the system states to the equilibrium and to achieve the projective synchronization 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 asymptotically to the origin. Using the same passivity-based method, the complete synchronization of the hyperchaotic system is also obtained. Both the intermittent and the passive controllers are feedback, global and easy to implement. Numerical simulations are included to show the effectiveness of the designed controllers in realizing the stabilization and the synchronization of the hyperchaotic system.

A group analysis via weak equivalence transformations for a model of tumour 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 component reactiondiffusion system modeling the Schnakenberg chemical reaction coupled to linear crossdiffusion terms is studied. The linear cross-diffusion terms favors the destabilization of the constant steady state and the mechanism of pattern formation with respect to the standard linear diffusion case, as shown in [1]. Since the subcritical Turing bifurcations of reaction-diffusion systems lead to spontaneous onset of robust, finite-amplitude localized patterns, here a detailed investigation of the Turing pattern forming region is performed to show how the diffusion coefficients for both species (the activator and the inhibitor) influence the occurrence of supercritical or subcritical bifurcations. The weakly nonlinear (WNL) multiple scales analysis is employed to derive the equations for the amplitude of the Turing patterns and to distinguish the supercritical and the subcritical pattern region, both in 1D and 2D domains. Numerical simulations are employed to confirm the WNL theoretical predictions through which a classification of the patterns (squares, rhombi, rectangle and hexagons) is obtained. In particular, due to the hysteretic nature of the subcritical bifurcation, we observe the phenomenon of pattern transition from rolls to hexagons, in agreement with the bifurcation diagram.

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 non-linear 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 settings.We also comment on generalizing the treatment here to parameter regimes where solutions homoclinic to exponentially small periodic orbits are known to exist, as well as another possible extension placing the solutions derived here within the framework of a comprehensive categorization of ALL possible traveling-wave solutions, both smooth and non-smooth, for our governing ODE.

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-diffusion system with nonlinear diffusion terms. 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 how cross-diffusion effects are responsible for the initiation of spatial patterns. Finally, we find a Fisher amplitude equation which describes the weakly nonlinear dynamics of the system near the marginal stability.

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.