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Random Attractor for Stochastic Hindmarsh-Rose Equations with Multiplicative Noise

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Pith's one-line read The stochastic Hindmarsh-Rose equations with multiplicative white noise are proved to possess a random attractor in $L^2$, pulling every tempered random set of initial data into one compact, noise-dependent attracting set.

desk verdict The theorem—existence of a random attractor in L^2 for the stochastic Hindmarsh-Rose system—is probably true, and the written proof has two repairable but load-bearing gaps: a wrong sign in Lemma 2.2's constant and a missing compactness step in Lemma 3.2. read the letter →

arxiv 1908.01220 v2 pith:QSBBPZPI submitted 2019-08-03 math.AP

Chi Phan This is my paper · ORCID
classification math.AP MSC 35K5535Q8037L3037L5537N2535B4060H1592B20
keywords stochasticHindmarsh-RoseequationsrandomattractormultiplicativenoisepullbackabsorbingsetasymptoticcompactnessdynamicalsystemneuronalburstingStratonovich
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the stochastic Hindmarsh-Rose equations—three coupled reaction-diffusion equations for a neuron's membrane potential, spiking variable, and bursting variable—driven by multiplicative white noise on a bounded three-dimensional domain always possess a random attractor in the energy space $H=L^2(\Omega;\mathbb{R}^3)$. The attractor is a compact, invariant, noise-dependent set that pulls in every tempered random set of initial data in the pullback sense as the initial time recedes to $-\infty$. This matters because the deterministic Hindmarsh-Rose model is a standard source of chaotic bursting; the theorem says that under multiplicative noise the long-term random dynamics are captured by a single attracting set rather than spreading out. The result holds for arbitrary positive diffusion, nonlinearity, and noise parameters, with only the reference potential $c$ allowed to be any real number.

What carries the argument

The load-bearing object is the Hindmarsh-Rose cocycle $\Phi(t,\omega,g_0)$ built from the stochastic semiflow $S(t,\tau,\omega)$ after removing the noise factor $Q(t,\omega)=e^{-\varepsilon W(t)}$. The exponential transformation converts multiplicative Stratonovich noise into random coefficients in a nonautonomous parabolic system; then the proof's two-step estimate scheme—pulling back from $-\infty$ to $-1$, then from $-1$ to $0$—establishes an absorbing ball and an $H^1$ bound, and compactness comes from $H^1\hookrightarrow H$. This two-interval split is what makes the uniform Gronwall argument work.

What would settle it

Run the transformed system (2.5)–(2.9) numerically with the paper's parameter set ($J=3.281$, $r=0.0021$, $c=-1.6$, $\phi(s)=3s^2-s^3$, $\psi(s)=1-5s^2$) and a large $\varepsilon$; if for some positive diffusion coefficients a solution leaves every bounded set or blows up in finite time before the absorbing time, the cocycle $\Phi$ is not globally defined and the attractor conclusion would not follow from this argument.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.3: for all positive $d_1,d_2,d_3,a,b,\alpha,\beta,q,r,J,\varepsilon$ and any $c\in\mathbb{R}$, the Hindmarsh-Rose cocycle $\Phi$ on $H=L^2(\Omega;\mathbb{R}^3)$ has a random attractor $\omega\mapsto A(\omega)$ with respect to the universe $\mathcal{D}_H$ of tempered random sets. The proof works by the exponential substitution $Q(t,\omega)=e^{-\varepsilon W(t)}$, which converts the Stratonovich stochastic PDEs (1.1)–(1.3) into the pathwise random PDEs (2.5)–(2.7). Uniform energy estimates produce a pullback absorbing ball in $H$; a second round of estimates using the uniform Gronwall inequality produces a pullback bound in $E=H^1(\Omega;\mathbb{R}^3)$, and the compact embedding $E\hookrightarrow H$ gives pullback asymptotic compactness. A standard existence criterion for random attractors then yields the attractor, which is additionally bounded in $E$.

Load-bearing premise

The whole construction presupposes that the random PDE system obtained after the exponential change of variables has solutions that exist for all future time and depend continuously on the initial data; if that global existence fails, the absorbing estimates and the cocycle itself are not available.

Editorial extensions

If this is right

  • Every tempered random set of initial data is eventually pulled into the random attractor $A(\omega)$, so the asymptotic state of the neuron field is a compact, noise-dependent set rather than an unbounded spread.
  • The attractor is bounded in $H^1(\Omega;\mathbb{R}^3)$, giving square-integrable gradients for the attracting states.
  • The result extends to a vector white noise with three independent scalar noises, as the paper notes, so the single-noise proof covers component-wise multiplicative noise.
  • The theorem supplies the random analogue of the known deterministic global attractor for this system, so the stochastic case is covered by the same attractor framework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same exponential-transform strategy should apply to other three-component neuron models with cubic or quadratic nonlinearities, provided the energy estimates can be replayed; a testable consequence is that FitzHugh-Nagumo-type stochastic systems with multiplicative noise also get random attractors.
  • Beyond the paper, the compactness of the attractor invites a dimension estimate; if one could bound its fractal dimension, that would quantify how many effective degrees of freedom survive the noise.
  • A rigorous reader should note that the paper's global-existence assertion for the transformed system is compressed into a single sentence; if global well-posedness fails for some parameter regime, the attractor existence would need re-examination.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims to prove the existence of a pullback random attractor in H=L^2(Ω;R^3) for the stochastic Hindmarsh-Rose system with multiplicative Stratonovich noise on a bounded domain of dimension n≤3, for all positive parameter values. The proof follows the standard route: an exponential transformation converts the SPDE into a pathwise random PDE, a priori estimates yield a pullback absorbing ball in H, further H^1 estimates yield pullback asymptotic compactness, and the Crauel-Flandoli/Schenk-Hoppé theorem produces the attractor. The main result is Theorem 3.3.

Significance. If the proof is completed, the result is a meaningful extension of the author's deterministic global attractor result to stochastic multiplicative noise, and it covers the full parameter range with no fitted constants, which is a strength. The argument is standard in structure, but it is self-contained apart from standard references and gives explicit absorbing radii and regularity estimates. The paper does not provide machine-checked proofs or numerics; its value lies in the analytical existence theorem for a widely used neurodynamical model.

major comments (3)
  1. [§2.1, after Eq. (2.13)] The choice c1 = 1/(b(β^2+3)) is algebraically inconsistent with the claimed inequality. With this value, the U^4 coefficient in (2.13) is −b c1 + β^2/2 = −1/(β^2+3) + β^2/2, which is positive for β^2 sufficiently large (e.g. β^2=5 gives 1/4) and therefore cannot yield the claimed bound −3∫ U^4/Q^2 dx. A correct choice is c1 ≥ (β^2+6)/(2b); the larger value c1=(β^2+3)/b also works. Since Lemma 2.2 underpins the pullback absorbing estimates in Theorem 2.6, this algebraic error must be corrected and all subsequent constants rechecked.
  2. [Lemma 3.2 and Theorem 3.3] The proof of pullback asymptotic compactness has a gap at the point where Lemma 3.2 is invoked. Lemma 3.2 estimates Φ(t,θ_{-t}ω,g0) for initial data satisfying ‖g0‖≤ρ(ω), and its proof only treats initial data in the absorbing ball B0(ω). For a universe set B∈D_H, however, pullback initial data satisfy ‖g0‖≤ρ_B(θ_{-t}ω), which is tempered but not bounded by a fixed ρ(ω); the sentence 'it suffices to take ρ=R0' is not justified. The standard repair is a two-step argument: first use the H-absorbing property at an intermediate time to obtain ‖G(-2,ω;-t,Q(-t,ω)g0)‖≤R0(θ_{-2}ω), then apply the regularity estimate on [-2,0] with that fixed radius. This intermediate step is absent, so Theorem 3.3's compactness conclusion does not follow from the lemmas as written.
  3. [Section 2, before Lemma 2.2] Global well-posedness of the transformed system is load-bearing: the cocycle Φ in Lemma 2.5 and all subsequent estimates presuppose that the pathwise weak solutions of (2.10) exist, are unique, and depend continuously on initial data on [τ,∞). The manuscript asserts this in a single sentence referring to a Galerkin compactness argument 'with some adaptations' and to [8,43]. Because the coefficients Q(t,ω) are only continuous and the nonlinearity is locally Lipschitz from E to H, this step is not entirely routine and should be either proved or supported by precise verification of the hypotheses of the cited theorems.
minor comments (4)
  1. [Throughout] There are several typos: 'weal solution' should be 'weak solution', 'Wiender' should be 'Wiener', 'reps.' should be 'resp.', 'inquality' should be 'inequality', and Theorem 3.3 refers to 'Lemma 2.6' where it should refer to 'Theorem 2.6'.
  2. [Equation (3.27) and Lemma 3.2 statement] Equation (3.27) has a missing parenthesis: it should be ‖G(0,ω;−t,Q(−t,ω)g0)‖E. In the statement of Lemma 3.2, 'θ−t,ω' has an extra comma and should read 'θ−tω'.
  3. [Theorem 2.6 proof] In (2.38)–(2.39) the transformed solution is written with sample θ_{-t}ω, while the pullback point Φ(t,θ_{-t}ω,g0) equals G(0,ω;−t,Q(−t,ω)g0). The proof should consistently use ω in these displays, or explain explicitly the change of sample point.
  4. [Equation (3.14)] The uniform Gronwall step states ∫ σ(s)ds≤N1, but σ was earlier defined as a constant; the intended quantity is ξ(s)=‖∇G(s)‖² as in (3.11).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the random-attractor proof is derived from the PDE via independent energy estimates; the sole self-citation is contextual and not load-bearing.

full rationale

The paper's derivation chain is a standard random-attractor argument: exponential transformation to the random PDE (2.5)-(2.10), a priori H-estimates (Lemmas 2.2 and 2.3), a pullback absorbing ball (Theorem 2.6), an E-regularity estimate (Lemma 3.1), and pullback asymptotic compactness via the compact embedding E into H (Lemmas 3.2 and Theorem 3.3). Each estimate is obtained from the PDE by energy inequalities and Gronwall arguments; no parameter is fitted to any target output, and no 'prediction' is a renamed input. The only self-citation, [27] by C. Phan, Y. You and J. Su, appears as context ('In the recent paper [27], we have shown the existence of a global attractor...') and is not used as a load-bearing premise for any stochastic step. The random-attractor existence rests on the external Crauel-Flandoli theorem (Theorem 1.8, cited to [12,31]) plus the paper's own uniform estimates. Lemma 3.2's reduction from arbitrary pullback data to data in B0(omega) relies on the already-proved absorbing property of Theorem 2.6, so it is not definitional circularity; whether the two-time-step argument is fully written out is a correctness or missing-proof concern, not a circularity concern. Global well-posedness of the transformed problem (2.10) is asserted via a Galerkin compactness reference ([8], with adaptations) rather than proved in detail; that is a completeness gap, not a circular reduction. No fitted inputs, no renaming of known results, and no load-bearing self-citation chain were found.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. The only hand-chosen auxiliary constant is c1 in the energy estimates, and its printed value appears to be a typo. All model parameters are inputs from the Hindmarsh-Rose literature. The random attractor is a derived object, not a postulated entity.

free parameters (1)
  • c1 = 1/(b(beta^2+3)) as printed; should be (beta^2+3)/b
    Hand-chosen constant in the L2 energy estimate. As printed, the sign of the U^4 dissipation term is wrong, which breaks Lemma 2.2.
assumptions (4)
  • standard math Sobolev embeddings H1(Omega) into L6(Omega) and L4(Omega) hold for space dimension n<=3.
    Used in Section 1 to justify local Lipschitz continuity of the nonlinearity and in Lemma 3.1 to control the L4 norm of U.
  • standard math The uniform Gronwall lemma applies to the differential inequality (3.9) with the integral bounds (3.14).
    Invoked in Lemma 3.1 with a reference to Sell-You Lemma D.3; the lemma itself is not proved in the paper.
  • domain assumption Pathwise weak solutions of the random PDE system (2.10) exist globally, are unique, and depend continuously on initial data.
    Asserted in Section 2 via a one-sentence Galerkin compactness argument. All later cocycle and absorbing estimates rely on this well-posedness.
  • standard math The Wiener process has asymptotically sublinear growth and local Holder regularity.
    Proposition 2.1 uses the law of the iterated logarithm and the Kolmogorov criterion to justify convergence of the random integrals defining r0(omega) and R0(omega).

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Cite this review

Pith. "Pith review of Random Attractor for Stochastic Hindmarsh-Rose Equations with Multiplicative Noise." pith.science (2026). https://pith.science/paper/QSBBPZPI

@misc{pith2026190801220,
  author       = {Pith},
  title        = {Pith review of: Random Attractor for Stochastic Hindmarsh-Rose Equations with Multiplicative Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QSBBPZPI}},
  note         = {Machine review of arXiv:1908.01220}
}
abstract

The longtime and global pullback dynamics of stochastic Hindmarsh-Rose equations with multiplicative noise on a three-dimensional bounded domain in neurodynamics is investigated in this work. The existence of a random attractor for this random dynamical system is proved through the exponential transformation and uniform estimates showing the pullback absorbing property and the pullback asymptotically compactness of this cocycle in the $L^2$ Hilbert space.

Figures

Figures reproduced from arXiv: 1908.01220 by the authors.

Figure 1
Figure 1. Time responses of the membrane potential for various value of the stimulated current: (a) resting state when J = 0, (b) tonic spiking when J = 1.2, (c) regular bursting when J = 2.2, (d) chaotic bursting when J = 3.1, (e) the x-z phase portrait when J = 2.2, (f) the x-z phase portrait when J = 3.1. Source: [25] [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Random Attractor for Stochastic Hindmarsh-Rose Equations with Additive Noise

    math.AP 2019-09 reject novelty 4.0 of 10

    For a two-dimensional bounded domain, the stochastic Hindmarsh-Rose equations with additive noise are claimed to possess a unique random pullback attractor in L2 space.

Reference graph

Works this paper leans on

49 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    Arnold, Random Dynamical Systems, Springer, Berlin, 1998

    L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998

    work page 1998

  2. [2]

    Bates, K

    P.W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845–869

    work page 2009

  3. [3]

    Bertram, M.J

    R. Bertram, M.J. Butte, T. Kiemel and A. Sherman, Topologica and phenomenological classi- fication of bursting oscillations , Bulletin of Mathematical Biology, 57 (1995), 413–439

    1995

  4. [4]

    Buters, J

    R.J. Buters, J. Rinzel and J.C. Smith, Models respiratory rhythm generation in the pre- B¨ otzinger complex, I. Bursting pacemaker neurons, J. Neurophysiology, 81 (1999), 382–397

    1999

  5. [5]

    Castaing and M

    C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions , Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977

    work page 1977

  6. [6]

    Caraballo, M.J

    T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equations without uniqueness of solutions , Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 439–455

    work page 2010

  7. [7]

    Chay and J

    T.R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell , Biophysiology Journal, 42 (1983), 181–189

    work page 1983

  8. [8]

    V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics , AMS Colloquium Publications, Vol. 49, AMS, Providence, RI, 2002

    2002

Show all 49 references
  1. [9]

    Chueshov, Monotone Random Systems Theory and Applications , Lect

    I. Chueshov, Monotone Random Systems Theory and Applications , Lect. Notes of Math., Vol. 1779, Springer-Verlag, Berlin, 2002

    2002

  2. [10]

    Cornelisse, W.J

    L.N. Cornelisse, W.J. Scheenen, W.J. Koopman, E.W. Roubos and S.C. Gielen, Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus Laevis, Neural Computations, 13 (2000), 113–137

    2000

  3. [11]

    Crauel, A

    H. Crauel, A. Debusche and F. Flandoli, Random attractors, J. Dynamics and Differential Equations, 9 (1997), 307–341

    1997

  4. [12]

    Crauel and F

    H. Crauel and F. Flandoli, Attractors for random dynamical systems , Probability Theory and Related Fields, 100 (1994), 365–393

    1994

  5. [13]

    Dhamala, V.K

    M. Dhamala, V.K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons , Physical Review Letters, 92 (2004), 028101

    2004

  6. [14]

    Efendiev and S

    M. Efendiev and S. Zelik, Upper and lower bounds for the Kolmogorov entropy of the attractor for an RDE in an unbounded domain , J. Dynamics and Differential Equations, 14 (2002), 369–403

    2002

  7. [15]

    Ementrout and D.H

    G.B. Ementrout and D.H. Terman, Mathematical Foundations of Neurosciences, Springer, 2010

    2010

  8. [16]

    FitzHugh, Impulses and physiological states in theoretical models of nerve membrane , Bio- physical Journal, 1 (1961), 445–466

    R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane , Bio- physical Journal, 1 (1961), 445–466

    1961

  9. [17]

    Flandoli and B

    F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise , Stoch. Stoch. Rep., 59 (1996), 21–45

    1996

  10. [18]

    X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235–1266

    2011

  11. [19]

    Hindmarsh and R.M

    J.L. Hindmarsh and R.M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 206 (1982), 162–164

    1982

  12. [20]

    Hindmarsh and R.M

    J.L. Hindmarsh and R.M. Rose, A model of neuronal bursting using three coupled first-order differential equations, Proceedings of the Royal Society London, Ser. B: Biological Sciences, 221 (1984), 87–102

    1984

  13. [21]

    Hodgkin and A

    A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve , J. Physiology, Ser. B, 117 (1952), 500–544. 28 C. PHAN

    1952

  14. [22]

    Innocenti and R

    G. Innocenti and R. Genesio, On the dynamics of chaotic spiking-bursting transition in the Hindmarsh-Rose neuron, Chaos, 19 (2009), 023124

    2009

  15. [23]

    Izhikecich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Burst- ing, MIT Press, Cambridge, Massachusetts, 2007

    E.M. Izhikecich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Burst- ing, MIT Press, Cambridge, Massachusetts, 2007

    2007

  16. [24]

    S.Q. Ma, Z. Feng and Q. Lu, Dynamics and double Hopf bifurcations of the Rose-Hindmarsh model with time delay , International Journal of Bifurcation and Chaos, 19 (2009), 3733–3751

    2009

  17. [25]

    Nguyen and K.-S

    L.H. Nguyen and K.-S. Hong. Lyapunov-based synchronization of two coupled chaotic Hindmarsh-Rose neurons, Journal of Computer Science and Cybernetics, 30.4 (2014): 335

    2014

  18. [26]

    Øksendal, Stochastic Differential Equations, 6th edition, Springer-Verlag, Berlin, 2003

    B. Øksendal, Stochastic Differential Equations, 6th edition, Springer-Verlag, Berlin, 2003

    2003

  19. [27]

    C. Phan, Y. You and J. Su, Global attractors for Hindmarsh-Rose equations in neurodynamics , arXiv:1907.13225, July 2019

    1907 arXiv

  20. [28]

    Rinzel, A formal classification of bursting mechanism in excitable systems , Proceedings of International Congress of Mathematics, 1 (1987), 1578–1593

    J. Rinzel, A formal classification of bursting mechanism in excitable systems , Proceedings of International Congress of Mathematics, 1 (1987), 1578–1593

    1987

  21. [29]

    Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters , Physics Review E, 74 (2006), 021917

    J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters , Physics Review E, 74 (2006), 021917

    2006

  22. [30]

    Rulkov, Regularization of synchronized chaotic bursts, Physical Review Letters, 86 (2001), 183–186

    N.F. Rulkov, Regularization of synchronized chaotic bursts, Physical Review Letters, 86 (2001), 183–186

    2001

  23. [31]

    Schenk-Hopp´ e,Random attractors - general properties, existence and applications to sto- chastic bifurcation theory, Discrete and Continuous Dynamical Systems, 4 (1998), 99-130

    K.R. Schenk-Hopp´ e,Random attractors - general properties, existence and applications to sto- chastic bifurcation theory, Discrete and Continuous Dynamical Systems, 4 (1998), 99-130

    1998

  24. [32]

    B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations , Interna- tional Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, (1992), 185–192

    1992

  25. [33]

    G. R. Sell and Y. You, Dynamics of Evolutionary Equations , Applied Mathematical Sciences, Volume 143, Springer, New York, 2002

    2002

  26. [34]

    Shapiro, R

    A. Shapiro, R. Curtu, J. Rinzel and N. Rubin, Dynamical characteristics common to neuronal competition models, J. Neurophysiology, 97 (2007), 462–473

    2007

  27. [35]

    Sherman and J

    A. Sherman and J. Rinzel, Rhythmogenetic effects of weak electrotonic coupling in neuronal models, Proceedings of National Academy of Sciences, 89 (1992), 2471–2474

    1992

  28. [36]

    L. Shi, R. Wang. K, Lu and B. Wang, Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains , J. Differential Equations, 267 (2019), 4373–4409

    2019

  29. [37]

    J. Su, H. Perez-Gonzalez and M. He, Regular bursting emerging from coupled chaotic neurons , Discrete and Continuous Dynamical Systems, Supplement 2007, 946–955

    2007

  30. [38]

    Terman, Chaotic spikes arising from a model of bursting in excitable membrane , J

    D. Terman, Chaotic spikes arising from a model of bursting in excitable membrane , J. Appl. Math., 51 (1991), 1418–1450

    1991

  31. [39]

    Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J

    B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253, (2012), 1544–1583

    2012

  32. [40]

    Wang, Random attractors for non-autonomous stochastic wave equations , Discrete and Con- tinuous Dynamical Systems, Series A, 34 (2014), 269–300

    B. Wang, Random attractors for non-autonomous stochastic wave equations , Discrete and Con- tinuous Dynamical Systems, Series A, 34 (2014), 269–300

    2014

  33. [41]

    Wang and B

    R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional stochastic p- Laplacian equations, Computers and Mathematics with Applications, http://doi.org/10.1016/ j.camwa.2019.05.024

    2019

  34. [42]

    Wang and X.R

    Z.L. Wang and X.R. Shi, Chaotic bursting lag synchronization of Hindmarsh-Rose system via a single controller , Applied Mathematics and Computation, 215 (2009), 1091-1097

    2009

  35. [43]

    You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems , Nonlinear Analysis, Series A, 75 (2012), 3049–3071

    Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems , Nonlinear Analysis, Series A, 75 (2012), 3049–3071. STOCHASTIC HINDMARSH-ROSE EQUATIONS 29

    2012

  36. [44]

    You, Random attractors and robustness for stochastic reversible reaction-diffusion systems , Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 301–333

    Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems , Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 301–333

    2014

  37. [45]

    You, Random attractors for stochastic reversible Schnackenberg equations, Discrete and Con- tinuous Dynamical Systems, Series S, 7 (2014), 1347–1362

    Y. You, Random attractors for stochastic reversible Schnackenberg equations, Discrete and Con- tinuous Dynamical Systems, Series S, 7 (2014), 1347–1362

    2014

  38. [46]

    You, Random dynamics of stochastic reaction-diffusion systems with additive noise , Journal of Dynamics and Differential Equations, 29 (2017), 83-112

    Y. You, Random dynamics of stochastic reaction-diffusion systems with additive noise , Journal of Dynamics and Differential Equations, 29 (2017), 83-112

    2017

  39. [47]

    Zhang, A

    F. Zhang, A. Lubbe, Q. Lu and J. Su, On bursting solutions near chaotic regimes in a neuron model, Discrete and Continuous Dynamical Systems, Ser. S, 7 (2014), 1363–1383

    2014

  40. [48]

    Zhou, Random exponential attractor for stochastic reaction-diffusion equations with multi- plicative noise in R3, J

    S. Zhou, Random exponential attractor for stochastic reaction-diffusion equations with multi- plicative noise in R3, J. Differential Equations, 263 (2017), 6347–6383

    2017

  41. [49]

    J. Zhou, X. Zhu, J. Liu, Y. Zhai and Z. Wang, Tracking the state of the Hindmarsh-Rose neuron by using the Coullet chaotic system based on a single input ,Journal of Information and Computing Science,11 (2016), 083–092. Department of Mathematics and Statistics, University of S...

    2016

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