IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i7p602-d246268.html
   My bibliography  Save this article

Dependence on the Initial Data for the Continuous Thermostatted Framework

Author

Listed:
  • Bruno Carbonaro

    (Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy)

  • Marco Menale

    (Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy
    Laboratoire Quartz EA 7393, École Supérieure d’Ingénieurs en Génie Électrique, Productique et Management Industriel, 95092 Cergy Pontoise Cedex, France)

Abstract

The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the condition that the second order moment of the activity variable (see Section 1) is a constant. We are able to prove that these solutions are stable with respect to the initial conditions in the Hadamard’s sense. In this connection, two remarks spontaneously arise and must be carefully considered: first, one could complain the lack of information about the “distance” between solutions at any time t ∈ [ 0 , + ∞ ) ; next, one cannot expect any more complete information without taking into account the possible distribution of the transition probabiliy densities and the interaction rates (see Section 1 again). This work must be viewed as a first step of a research which will require many more steps to give a sufficiently complete picture of the relations between solutions (see Section 5).

Suggested Citation

  • Bruno Carbonaro & Marco Menale, 2019. "Dependence on the Initial Data for the Continuous Thermostatted Framework," Mathematics, MDPI, vol. 7(7), pages 1-11, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:602-:d:246268
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/7/602/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/7/602/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Kacperski, Krzysztof & Hoł yst, Janusz A., 1999. "Opinion formation model with strong leader and external impact: a mean field approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(2), pages 511-526.
    2. Carlo Bianca & Caterina Mogno, 2018. "Modelling pedestrian dynamics into a metro station by thermostatted kinetic theory methods," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 24(2), pages 207-235, March.
    3. Masurel, Léon & Bianca, Carlo & Lemarchand, Annie, 2018. "On the learning control effects in the cancer-immune system competition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 462-475.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Carlo Bianca & Marco Menale, 2022. "On the Existence of Self-Similar Solutions in the Thermostatted Kinetic Theory with Unbounded Activity Domain," Mathematics, MDPI, vol. 10(9), pages 1-14, April.
    2. Carlo Bianca, 2022. "On the Modeling of Energy-Multisource Networks by the Thermostatted Kinetic Theory Approach: A Review with Research Perspectives," Energies, MDPI, vol. 15(21), pages 1-22, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mikhail Kolev, 2019. "Mathematical Analysis of an Autoimmune Diseases Model: Kinetic Approach," Mathematics, MDPI, vol. 7(11), pages 1-14, October.
    2. Sana Abdulkream Alharbi & Azmin Sham Rambely, 2020. "A New ODE-Based Model for Tumor Cells and Immune System Competition," Mathematics, MDPI, vol. 8(8), pages 1-14, August.
    3. Piotr Przybyła & Katarzyna Sznajd-Weron & Rafał Weron, 2014. "Diffusion Of Innovation Within An Agent-Based Model: Spinsons, Independence And Advertising," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 17(01), pages 1-22.
    4. Carlo Bianca & Marco Menale, 2022. "On the Existence of Self-Similar Solutions in the Thermostatted Kinetic Theory with Unbounded Activity Domain," Mathematics, MDPI, vol. 10(9), pages 1-14, April.
    5. Carlo Bianca & Marco Menale, 2020. "Mathematical Analysis of a Thermostatted Equation with a Discrete Real Activity Variable," Mathematics, MDPI, vol. 8(1), pages 1-8, January.
    6. Gabriel Morgado & Annie Lemarchand & Carlo Bianca, 2023. "From Cell–Cell Interaction to Stochastic and Deterministic Descriptions of a Cancer–Immune System Competition Model," Mathematics, MDPI, vol. 11(9), pages 1-25, May.
    7. Pawel Sobkowicz, 2011. "Simulations of opinion changes in scientific communities," Scientometrics, Springer;Akadémiai Kiadó, vol. 87(2), pages 233-250, May.
    8. Nowak, Andrzej & Kuś, Marek & Urbaniak, Jakub & Zarycki, Tomasz, 2000. "Simulating the coordination of individual economic decisions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 613-630.
    9. Joanna Kwiatkowska & Joanna Gutowska, 2012. "Memetics On The Facebook," Polish Journal of Management Studies, Czestochowa Technical University, Department of Management, vol. 5(1), pages 305-314, June.
    10. Liu, Shang & Li, Peiyu, 2020. "Nonlinear analysis of pedestrian flow Reynolds number in video scenes," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    11. Grinfeld, M. & Piscitelli, L. & Cross, R., 2000. "A probabilistic framework for hysteresis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 577-586.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:602-:d:246268. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.