MOMENT LYAPUNOV EXPONENT OF DELAY DIFFERENTIAL EQUATIONS

The aim of this paper is to establish a connecting thread through the probabilistic concepts of pth-moment Lyapunov exponents, the integral averaging method, and Hale’s reduction approach for delay dynamical systems. We demonstrate this connection by studying the stability of perturbed deterministic and stochastic differential equations with fixed time delays in the displacement and derivative functions. Conditions guaranteeing stable and unstable solution response are derived. It is felt that the connecting thread provides a unified framework for the stability study of delay differential equations in the deterministic and stochastic sense.


Introduction. Concepts of pth-moment Lyapunov exponent have been employed
in the study of stability behaviour of structural systems with stochastically perturbed excitations, where governing equations for single-degree-of-freedom systems are typically of nonlinear stochastic ordinary differential equations (ODEs) of the form ẍ + 2δ 0 ω 0 + ε 1/2 σ 0 ξ(t) ẋ + ω 2 0 1 + ε 1/2 σ 1 γ(t) x + εσ 3 x 3 = 0, ( where δ 0 and ω 0 represent the damping ratio and the undamped natural frequency of the excitations, respectively.The processes ξ(t) and γ(t) are independent processes which are typically stationary with zero mean values; σ 0 , σ 1 are the noise intensities of the processes, ε is a small parameter that takes values between 0 and 1, while σ 3 is a real constant denoting the nonlinear perturbation.The name moment Lyapunov exponent comes from the connection of pth-moment stability, which we denote here as (p), and the sample stability or Lyapunov exponent of stochastic solutions which is denoted by π exp .The concepts have become the most attractive aspects for the study of stability behaviour of stochastic dynamical systems.The parameter p ∈ of moment exponent is a unique number and it stands for stability index of the solutions.
To define the concepts of pth-moment Lyapunov exponent, we consider the special linearized case of (1.1), written by means of the transformation x = x 1 , ẋ = x 2 , and along with the assumed equilibrium conditions x 1 (0) = x 0 , x 2 (0) = v 0 , as form of a pair of Itô ODEs, namely x 2 dt − ε 12 σ 0 x 2 dW (t), (1.3) where W (t) is a unit Wiener process.By the multiplicative ergodic theorem of Oseledec [7], the Lyapunov exponent and the pth-moment exponent of solutions to (1.2) and (1.3) are defined, respectively, as (p) = lim t→∞ log E x 1 t, x 0 ,υ 0 2 + x 2 t, x 0 ,υ 0 2 1/2 p , (1.5) in which E [•] stands for the expected value of the quantity within the square bracket.
In (1.5), if the exponent (p) < 0, then by definition E[{•} p ] → 0 as t → ∞, and thus we can say that the solution response to (1.2) and (1.3) is pth-moment stability in the almost-sure sense.On the contrary, that is, for (p) > 0, the expectation E[{•} p ] ≠ 0 as t → ∞, and thus pth-moment instability will occur in the almost-sure sense.The values of π exp and (p) are real and deterministic in nature as long as the random system is ergodic.By the pth-moment Lyapunov exponent of a stochastic dynamical system, we mean that there is a pth-moment stability of the corresponding random linear solutions of the system in the almost-sure sense.This means that, among an n number of exponents, if the maximal Lyapunov exponent is negative (i.e., π exp < 0), the random system is almost-surely stable for small values of the stability index p.However, in this situation pth-moment grows exponentially for large values of p, and thereby indicating that pth-moment response of the system is unstable.A remarkable observation one can infer from this, is that although at an exponential rate we may have π exp < 0, thereby resulting to stability of the solution response in the almost-sure sense, yet for large values of p there is small probability of chance that the response would be large.Corresponding expected values for this rare event are indeed also large, and it is conclusive to say that pth-moment exponent of the system is unstable.Opposite of this situation is when the Lyapunov exponent is positive (i.e., π exp > 0) and the system is almost-surely unstable.
The values for the stability index p are usually determined by the solutions of (p) = 0, and these values are dependent upon the dimension of the random dynamical system.It has been shown by Baxendale [3] that the corresponding values of p when (p) = 0 equal the negative of the dimension of the system.For example a system with a dimension n, if for a solution of (p) = 0, we have p = p 1 ≠ 0, then we can write p 1 = −n.At such a point, p = p 1 , there is an expectation that the sign of the maximal value of the Lyapunov exponent π exp will change from negative to positive, and thus a change in character of the corresponding probability density function will occur as well.Traditionally, there is an obvious computational difficulty if one wishes to determine (p) for many arbitrary values of p. Efforts by Arnold et al. [1,2] and many others in the stochastic community, led to the fact that the pthmoment Lyapunov exponent (p) : → is a convex and analytic function in p in such a way that the expression (p)/p increases and (p) = {0} p=p 1 .Thus by taking the asymptotic expansion of (p) for p = p 1 near zero, and bearing in mind that (d /dp)(0) = π exp , (0) = 0, we have This asymptotic connection indeed brought about the concepts of large deviations of linear random dynamical systems in the stability study of solution responses.The asymptotic expansion has been employed in the literature to determine rare situations where negative maximal Lyapunov exponent, pth-moment solution response grow exponentially for large values of p.
Another attractive aspect of pth-moment Lyapunov exponent is its further connection with the integral stochastic averaging method.It was Has'minskiȋ [5] and Stratonovich [9] who developed formulas for such a connection, and the formulas were based upon the ideas of Bogoliubov and Mitropolski [6] in the context of deterministic averaging method.When the integral stochastic averaging method is applied to equations of the form (1.2) and (1.3), it produces Markovian solutions to the nonlinear systems in terms of amplitude and phase relations, which can be written either in the sense of Stratonovich or Itô using stochastic differential rules.The rules treat differentiation of functions mapped originally from a logarithmic polar transformation of the vector state variables, which depend on the corresponding Itô equations.The connection between these probabilistic concepts and the stability study of delay differential equations (DDEs), by means of Andronov-Hopf bifurcation and centre manifold, is the focus of this investigation.In pursuing this, we expect to provide a unified framework for the study of stability of DDEs with deterministic and stochastic perturbations.First, we will use the Andronov-Hopf bifurcation and centre manifold according to Hale [4] to reduce the infinite-dimensional character of the DDEs into family of ODEs in the space C := C([−τ, 0], 2 ).The integral averaging of the ODEs produces averaged equations in terms of amplitude and phase relations.From the averaged equations, explicit expressions for the occurrence of stable and unstable solution responses will be derived using pth-moment Lyapunov exponent.[4]).Let L = L(x t (θ), µ) : Cx → n and ∆f = ∆f (x t (θ), µ, ε) : × C → n denote accordingly the linear and nonlinear functional mappings depending upon a parameter µ and the state variable x t (θ).The variable x t (θ) is contained in C := C([−τ, 0], n ), the Banach space of all continuous functions equipped with the usual supremum norm • in C and vector norm |•| in n ; x t (θ) ∈ C represents the past history solution of a delay differential equation of the form
Hence, for the estimation of the nonlinear variation of constant integral solutions ) and (2.7) we have from the second inequality the following: where the left-hand side of this equation exponentially converges to zero as t → ∞, namely (2.12) In a similar way, the first inequality produces x P t (φ(θ), µ) → 0 as t → −∞.As firstorder approximation, one can accept these convergencies, and then describe the longterm behaviour of the original delay differential equations (2.1) with the corresponding set of ODEs for Therefore, on the centre manifold M µ ∈ C, we have the corresponding solution of (2.1) as (2.13) Since we know that the exponential estimate for x Q t (φ(θ), µ) in the complementary subspace Q is zero, then the change of variables x P t (φ(θ), µ) = Φ(θ)z(t), −τ ≤ θ ≤ 0, and their differentiation with respect to time t produces where the substitution of X P 0 (θ) := Φ(θ) Ψ (0) into this equation gives the k-dimensional ODEs and B is a k×k matrix.We now illustrate these ideas by two examples with fixed time delays.

An illustrative example I.
The specific single-degree-of-freedom dynamical system considered is represented by the second-order DDEs of the form All the parameters contained in these equations are real and µ is the selected bifurcation parameter, which is set to vary by ε μ in the neighborhood of some critical value µ c , namely µ = µ c + ε μ. τ 1 , τ 2 are the respective time delays in the restoring and damping forces.Equations of the form (3.1) have been encountered in the active controlling of structural systems with earthquake excitations [8], where the parameter µ often stand for the gain of the delayed forces.

Conclusion.
An attempt to establish a unified framework for the study of stability of second-order differential equations with multiple and distinct time delays in the displacement and derivative functions, plus a derivative process of the damping coefficient, has been made.Andronov-Hopf bifurcation, centre manifold theorem, the integral stochastic averaging method, and pth-moment Lyapunov exponents have been employed in the development of the framework.Sufficient conditions for stability in the deterministic and stochastic sense have been presented.It is felt that this framework will uncover a wealth of phenomena of stochastic dynamical systems with delays since the investigations are conducted in the appropriate infinite-dimensional space C([−τ, 0], 2 ) without the assumption of small delay.