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Global Exponential Stability for Complex-Valued Recurrent Neural Networks With Asynchronous Time Delays

Xiwei Liu, Tianping Chen

arXiv:1505.07903v1math.DS

TL;DR

Stability analysis for complex-valued recurrent neural networks with asynchronous time delays remains relatively little studied, despite delays complicating network dynamics. This paper decomposes the networks into real and imaginary parts and applies three generalized norms to derive stability criteria. The results establish exponential convergence to a unique equilibrium under sufficient conditions.

  • Problem

    Few studies had reported stability results for complex-valued neural networks with asynchronous delays, where different interconnections can have different delays.

    Source: §I · p. 2

  • Method

    The paper decomposes the complex-valued network into real and imaginary parts and derives criteria using generalized infinity, 1-, and 2-norms.

    Source: §I · p. 2, §V · p. 9

  • Results

    Under the stated conditions, solutions converge exponentially to a unique equilibrium.

    Source: A. Criteria with {ξ, ∞}-norm · pp. 3–4

  • Takeaways & Limitations

    Considering the signs of coupling-matrix entries yields sufficient conditions described as more general and less restrictive than previous M-matrix results.

    Source: §V · p. 10

    The decomposition assumes activation functions have partial derivatives that exist and are bounded.

    Source: §V · p. 10

Abstract

from arXiv · show

In this paper, we investigate the global exponential stability for complex-valued recurrent neural networks with asynchronous time delays by decomposing complex-valued networks to real and imaginary parts and construct an equivalent real-valued system. The network model is described by a continuous-time equation. There are two main differences of this paper with previous works: (1), time delays can be asynchronous, i.e., delays between different nodes are different, which makes our model more general; (2), we prove the exponential convergence directly, while the existence and uniqueness of the equilibrium point is just a direct consequence of the exponential convergence. By using three generalized norms, we present some sufficient conditions for the uniqueness and global exponential stability of the equilibrium point for delayed complex-valued neural networks. These conditions in our results are less restrictive because of our consideration of the excitatory and inhibitory effects between neurons, so previous works of other researchers can be extended. Finally, some numerical simulations are given to demonstrate the correctness of our obtained results.

I. INTRODUCTION

Complex-valued recurrent neural networks have attracted interest for applications involving complex states, while time delays complicate their dynamics and stability. This paper addresses asynchronous delays and adopts a direct exponential-convergence approach using three generalized norms.

I. INTRODUCTION

  • Complex-valued neural networks extend real-valued recurrent networks and can represent states composed of amplitude and phase.The introduction describes potential applications including nonlinear quantum systems, reaction-advection-diffusion systems, heat equations, and petri nets.Source: §I · p. 1
  • Time delays can contribute to oscillation and instability, and prior work studied delayed complex-valued networks under several delay settings.The introduction notes prior work on time-varying delays, impulsive systems, and global exponential or asymptotical stability.Source: §I · p. 1
  • Asynchronous delays allow each node to receive information from different nodes at different times, but few studies had addressed stability of complex-valued networks with such delays.Earlier asynchronous-delay studies used a two-step approach: establish equilibrium existence, then prove stability.Source: §I · pp. 1–2
  • The paper adopts a direct approach to exponential convergence and presents criteria based on generalized L∞, L1, and L2 norms.The introduction notes that the L∞-norm criteria can address time-varying delays.Source: §I · p. 2
  • The paper organizes its work around real-and-imaginary decomposition, stability criteria, and numerical simulations with constant and time-varying delays.The equivalent real-valued system has twice the dimension of the original complex-valued system.Source: §I · p. 2

II. PRELIMINARIES

The preliminaries define the asynchronous-delay network and its activation-function assumptions, then introduce generalized norms and matrix conditions used in the stability analysis.

II. PRELIMINARIES

  • Asynchronous delays mean a node may receive information from different nodes at different times, unlike synchronous delays shared across incoming connections.The model uses complex-valued neuron states, activation functions, connection weights, external inputs, and asynchronous constant delays.Source: §II · p. 2
  • The network model specifies intrinsic reset rates, complex-valued connection weights with and without delays, activation functions, and external inputs.The asynchronous delays are indexed by both receiving and sending nodes.Source: §II · p. 2
  • The activation-function assumptions trade off restrictiveness against criteria that account for the signs of connection weights.Definition 2 is more restrictive than the usual bounded-derivative assumption associated with Definition 3.Source: §II · p. 3
  • The complex-valued network is decomposed into real and imaginary parts, producing equivalent real-valued systems with twice the original dimension.The authors note that this expansion may increase calculations, while potentially doubling the number or dimension of equilibria.Source: §II · p. 3
  • The analysis uses weighted generalized infinity, 1-, and 2-norms, along with an M-matrix equivalence for matrices with nonpositive off-diagonal entries.The stated M-matrix equivalences include positive real parts of all eigenvalues and the existence of a positive vector satisfying a matrix inequality.Source: §II · p. 3

III. MAIN RESULTS

The section presents criteria for uniqueness and global exponential stability of the equilibrium.

III. MAIN RESULTS

  • The main results establish criteria for uniqueness and global exponential stability of the equilibrium.Source: §III · p. 3

A. Criteria with {ξ, ∞}-norm

The section gives sufficient criteria for exponential convergence to a unique equilibrium using the {ξ, ∞}-norm. It also extends the stability result to systems with time-varying delays.

A. Criteria with {ξ, ∞}-norm

  • The criteria cover H1 and H2 activation-function classes and use positive weights, with one condition requiring a small ε.The H2 statements specify a positive vector ξ and φ; the corollary requires a sufficiently small ε.Source: A. Criteria with {ξ, ∞}-norm · pp. 3–4
  • Under the stated conditions, solutions converge exponentially to a unique equilibrium.The result is stated for systems (4) and (5).Source: A. Criteria with {ξ, ∞}-norm · p. 4
  • Theorem 1 extends to time-varying delays that may be bounded or unbounded, yielding a unique equilibrium that is globally µ stable under conditions (6) and (7).The paper identifies this extension as a result for system (15).Source: A. Criteria with {ξ, ∞}-norm · p. 4

B. Criteria with {ξ, 1}-norm

The section presents criteria using the {ξ, 1}-norm. Under the stated conditions, solutions converge exponentially to a unique equilibrium.

B. Criteria with {ξ, 1}-norm

  • The criteria address H1 and H2 activation-function classes and require a positive vector, with the H2 condition including ε.The theorem and corollary state their respective activation-function classes and positive-vector conditions.Source: B. Criteria with {ξ, 1}-norm · pp. 4–5
  • The stability argument uses the {ξ, 1}-norm for the solution vector Z(t) defined by equation (8).The passage identifies equations (9) and (10) as holding for that vector.Source: B. Criteria with {ξ, 1}-norm · p. 5
  • Under the stated conditions, solutions converge exponentially to a unique equilibrium in both the theorem and corollary results.The results apply to systems (4) and (5).Source: B. Criteria with {ξ, 1}-norm · p. 5

C. Some comparisons

The comparisons show that criteria accounting for connection-matrix signs can be more powerful than criteria that ignore them. Matrix conditions also connect the sufficient criteria to exponential convergence and indicate how connection weights can support stability.

C. Some comparisons

  • When G is a nonsingular M-matrix, solutions of systems (4) and (5) converge exponentially to a unique equilibrium.Source: C. Some comparisons · p. 5
  • A nonsingular M-matrix G implies that matrix (20) is also a nonsingular M-matrix, but the converse need not hold.Source: C. Some comparisons · p. 6
  • Accounting for connection-matrix signs yields a better Corollary 1 criterion than prior work, which ignores those signs.Source: C. Some comparisons · pp. 5–6
  • Larger P1, P2, and P3 support stable equilibria; one suggested choice is making all aR_jj negative.Source: C. Some comparisons · p. 6

D. Criteria with {ξ, 2}-norm

The section gives sufficient criteria using the {ξ, 2}-norm for uniqueness and exponential convergence to an equilibrium. It also notes that the weighted {ξ, 1} and {ξ, 2} norms can be used to analyze time-varying delayed networks.

D. Criteria with {ξ, 2}-norm

IV. NUMERICAL EXAMPLE

The simulations show convergence to an equilibrium across different initial histories, external controls, and delay settings. The example satisfies Corollary 1’s exponential-convergence condition even though Theorem 3’s M-matrix condition fails.

IV. NUMERICAL EXAMPLE

  • The example uses a two-neuron complex-valued recurrent network with distinct delays of 1, 2, 3, and 4 across its connections.The activation functions are reported as belonging to classes H1(1, 0.5, 0.25, 0.5) and H2(0.25, 0.5, 1, 0.5).Source: §IV · pp. 7–8
  • Although G is not an M-matrix, with eigenvalues −0.7655, 18.6670, 20.9701, and 19.8784, Corollary 1 still establishes exponential convergence.The reported Δ values are 0.8488, 20.0717, 22.7947, and 21.5348.Source: §IV · p. 8
  • Across five initial histories, the trajectories converge to the same equilibrium, (−0.0351, 0.1423, 0.0912, 0.2239)T.Figures 1–4 show the real and imaginary components for the two neurons.Source: §IV · p. 8, Figure 1 · p. 8, Figure 2 · p. 8
  • Changing the external control produces different equilibria in the simulated network.The two controller settings are (−3 + i, 2 + 4i)T and (3 + 2i, 4 − i)T.Source: §IV · pp. 8–9, Figure 5 · p. 9
  • With time-varying delays, the system reaches the same equilibrium as the system with different fixed delays.The time-varying delay terms include 1 + sin(t), 2 + cos(t), 3 − sin(t), and 4 − cos(t).Source: §IV · p. 9, Figure 6 · p. 9, Figure 7 · p. 9

V. CONCLUSION AND DISCUSSIONS

The paper proposes complex-valued recurrent neural networks with asynchronous delays and proves exponential convergence directly using three generalized norms. It presents sufficient conditions for uniqueness and global exponential stability, illustrates the results numerically, and identifies assumptions and extensions for future work.

V. CONCLUSION AND DISCUSSIONS

  • The paper proposes a complex-valued recurrent neural network model with asynchronous time delays, then proves exponential convergence directly using the infinity-, 1-, and 2-norms.Existence and uniqueness of the equilibrium follow as consequences of exponential convergence.Source: §V · p. 9
  • Considering coupling-matrix signs, the paper presents sufficient conditions for uniqueness and global exponential stability that are less restrictive than prior criteria.The M-matrix property of G is described as a special case of the exponential-stability criteria.Source: §V · p. 10
  • Three numerical examples demonstrate the correctness of the results, and Figs. 8 and 9 show trajectories for different initial values as evidence of global exponential stability.Fig. 8 plots zR 2(t), while Fig. 9 plots zI 2(t).Source: §V · p. 10, Figure 8 · p. 10, Figure 9 · p. 10
  • The analysis decomposes complex-valued networks into real and imaginary parts and assumes activation-function partial derivatives exist and are bounded.The paper identifies analysis using the system’s complex nature as future work.Source: §V · p. 10
  • The model treats asynchronous delays as discrete; stability with continuously distributed delays and other dynamical behaviors remain future research directions.The paper also lists multistability, robustness of uncertain networks, periodic or almost-periodic solutions, and chaotic behavior.Source: §V · p. 10

APPENDIX A: THE PROOF OF THEOREM 1

The proof constructs an equilibrium and establishes exponential convergence to it, which also gives uniqueness. Its argument uses a monotonically decreasing bound on the solution norm.

APPENDIX A: THE PROOF OF THEOREM 1

  • The proof represents the system state as a 2n-dimensional real vector and uses the maximum delay τ in bounding its norm.The argument distinguishes cases for the index attaining the weighted infinity norm.Source: Appendix A · pp. 10–11
  • The proof shows that M(t) decreases monotonically and that the solution components decay as O(e^-εt).The resulting convergence establishes that the equilibrium is unique.Source: Appendix A · pp. 11–12
  • The proof establishes that Z is an equilibrium point and then considers any equilibrium Z to show uniqueness.Any solution Z(t) converges to Z exponentially.Source: Appendix A · p. 12

APPENDIX B: PROOF OF THEOREM 2

The appendix proofs use Lyapunov functions for the real-valued component dynamics and conclude that the equilibrium is unique in both theorem settings.

APPENDIX B: PROOF OF THEOREM 2

  • A Lyapunov function L1(t) is differentiated along equations (25) and (26) in the proof of Theorem 2.The displayed derivative bound includes a nonpositive term.Source: Appendix B · p. 12
  • Theorem 2’s proof concludes that the equilibrium point is unique by arguments similar to those used for Theorem 1.Source: Appendix B · p. 12
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