Chaotic semigroups from second order partial differential equations

Conejero, J. Alberto; Lizama, Carlos; Murillo‐Arcila, Marina

doi:10.1016/j.jmaa.2017.07.013

Cited by 10 publications

(5 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In contrast to the recent research studies of J. A. Conejero, C. Lizama et al [6]- [8], where the authors have studied the hypercyclic and chaotic solutions of certain kinds of abstract second and third order differential equations in the spaces of Herzog analytic functions by employing, primarily, the Desch-Schappacher-Webb criterion [10], the operator matrix under our consideration is not bounded and as such does not generate a strongly continuous semigrop on E n a priori. This is the main reason why we use the theory of C-regularized semigroups in this paper.…”

Section: Introductionmentioning

confidence: 73%

“…also [17,Theorem 2.10.45] for a generalization of the above-mentioned theorem to abstract time-fractional differential equations). In our approach, we almost always face the situation Ẽ = E n , which indicates a certain type of subspace frequent hypercyclicity of constructed solutions to (ACP n ) (in [6]- [8], the situation in which Ẽ = E n can really occur). At the end of paper, we provide several examples and applications of our results.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Frequently hypercyclic abstract higher-order differential equations

Chaouchi¹,

Kostić²

2018

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Frequently hypercyclic abstract higher-order differential equations

Chaouchi¹,

Kostić²

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Since PDEs have numerous types, it is difficult to give a particular definition of chaos that is consistently significant for the bulk of PDE. A lot of research has been done in the field of chaos analysis for PDEs [1][2][3][4][5][6][7][8][9]. Knoblock et al [1] showed a transition from periodic oscillations to chaos using the two-dimensional thermosolutal convection numerical studies.…”

Section: Introductionmentioning

confidence: 99%

“…Stability analysis for a class of PDE chaotic models is introduced by Xiang et al [3]. Conejero et al [4] provided broad criteria on parameters to guarantee Devaney and distributional chaos for the semigroup solution corresponding to a class of second order PDE. Chaotic models for neural PDE were established [5].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

Abed-Elhameed,

Otefy,

Mahmoud

2024

Phys. Scr.

View full text Add to dashboard Cite

We present in this paper four versions of chaotic and hyperchaotic modified nonlinear Schr"{o}dinger equations (MNSEs). These versions are hyperchaotic integer order, hyperchaotic commensurate fractional order, chaotic non-commensurate fractional order, and chaotic distributed order MNSEs. These models are regarded as extensions of previous models found in literature. We also studied their dynamics which include symmetry, stability, chaotic and hyperchaotic solutions. The sufficient condition is stated as a theorem to study the existence and uniqueness of the solutions of hyperchaotic integer order MNSE. We state and prove another theorem to test the dependence of the solution of hyperchaotic integer order MNSE on initial conditions. By similar way, we can introduce the previous two theorems for the other versions of MNSEs. The Runge–Kutta of the order 4, the Predictor-Corrector and the modified spectral numerical methods are used to evaluate the numerical solutions for integer, fractional and distributed orders MNSEs, respectively. We calculate numerically using the Lyapunov exponents the intervals of parameters of the purposed models at which hyperchaotic, chaotic and stable solutions are exist. The MNSEs have an important role in many fields of science and technology, such as nonlinear optics, electromagnetic theory, superconductivity, chemical and biological dynamics, lasers and plasmas. The compound synchronization for these chaotic and hyperchaotic models is investigated. We state its scheme using the tracking control technique among three integer commensurate and non-commensurate orders as the derive models and one distributed order as a slave model. We presented and proved a theorem that provides us with the analytical formula for the control functions which are required to achieve compound synchronization. The analytical results are supported by numerical calculations and agreement is found.

show abstract

“…Particular attention deserves the case of C 0 -semigroups of operators, since many of them are originated in the analysis of the asymptotic behaviour of solutions to certain linear partial differential equations and to infinite systems of linear differential equations. Especially, different notions of chaos for C 0 -semigroups have experienced a great development in recent years (see, e.g., [1,2,5,16,22,23,24,25,27,29,38,46,48]).…”

Section: Introductionmentioning

confidence: 99%

The Specification Property for $C_0$-Semigroups

Bartoll

Martínez-Giménez

Peris

et al. 2016

Preprint

View full text Add to dashboard Cite

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, C 0 -semigroups. In addition, we study the relationships of the specification property for C 0 -semigroups (SgSP) with other dynamical properties: mixing, Devaney's chaos, distributional chaos and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black-Scholes equation.

show abstract

Chaotic semigroups from second order partial differential equations

Cited by 10 publications

References 39 publications

Frequently hypercyclic abstract higher-order differential equations

Frequently hypercyclic abstract higher-order differential equations

Dynamics of chaotic and hyperchaotic modified nonlinear Schrödinger equations and their compound synchronization

The Specification Property for $C_0$-Semigroups

Contact Info

Product

Resources

About