Limit Cycle and Conserved Dynamics

Limit Cycle and Conserved Dynamics 1)

X.-M. Zhu1) , L. Yin2) , and P. Ao3)

GenMath, 5525 27th Ave. N.E., Seattle, WA 98105, USA School of Physics, Peking University, 100871 Beijing, PR China 3) Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA (Dated: Dec. 7, 2004)

arXiv:nlin/0412059v1 [nlin.CD] 21 Dec 2004

2)

We demonstrate that a potential coexists with limit cycle. Here the potential determines the final distribution of population. Our demonstration consists of three steps: We first show the existence of limit from a typical physical sciences setting: the potential is a type of Mexican hat type, with the strength of a magnetic field scale with the strength the potential gradient near the limit cycle, and the friction goes to zero faster than the potential near the limit cycle. Hence the dynamics at the limit cycle is conserved. The diffusion matrix is nevertheless finite at the limit cycle. Secondly, we construct the potential in the dynamics with limit cycle in a typical dynamical systems setting. Thirdly, we argue that such a construction can be carried out in a more general situation based on a method discovered by one of us. This method of dealing with stochastic differential equation is in general different from both Ito and Stratonovich calculus. Our result may be useful in many related applications, such as in the discussion of metastability of limit cycle and in the construction of Hopfield potential in the neural network computation.

I.

INTRODUCTION

Noise perturbed dynamics systems with limit cycles are abounded in physical sciences. It is one of the driving forces in the study of nonlinear systems1,2 . Until recently, one of the most useful concepts in physical sciences, the potential energy function, had been found to be not applicable in general2,3 . Hence, it has been concluded that the no Lyapunov function or potential function with periodic attractors, in the neural network computing4 . Similar statements can also be found in many other fields. When the matastability and the decay of metability state are discussed, efforts have been made to avoid the use of potential energy function in the form of Boltzmann-Gibbs distribution. Various methods which can go around this issue, such as the MachlupOnsager functional method, quasipotentials, etc, have been developed5 . They have been successfully applied to situations with limit cycle dynamics, such as in the discussion of topological structure of the decay of metastable limit cycle6 , in the numerical study of trajectory of the escaping path7 . However, even in such settings, there is a need to consider potential energy function, such as indicated in the escape of asteroids in the solar system8 . Hence, it is a natural question to ask what would be the connection between such an approach and the classical approach based on potential energy function9,10 , and whether or not the potential function exists in dynamics with limit cycle. The purpose of the present article is to give an explicit demonstration that at least for the case of limit cycle dynamics, potential function can be explicitly constructed, which provides a critical link between those two ways of dealing with metastability. Our construction is based on a novel way of handling the stochastic differential equation. It can be understood as the zero mass limit of a generalized Klein-Kramers equation2 . In this limit the steady state distribution can be established from the Klein-Kramers equation with-

out the concerns of the Ito or Stratnovich calculus. In this sense it would not be surprising that a different perspective can be obtained other than those in Ref.[6,7,8 ]. We believe that our following demonstration provides an important step for an alternative understanding of the nonequilibrium processes such as limit cycles. We will demonstrate the co-existence of a potential with limit cycle in three steps in three sections, respectively. In section II we explicitly demonstrate how to construct a limit cycle based on our knowledge in physical sciences, with a potential. This is different from the usual approach in dynamical systems. In section III we demonstrate how to construct a potential in the presence of a limit cycle in the dynamical systems setting. In section IV we outline a general construction of potential in a broad class of dynamical systems, including limit cycles. In section V we discuss two mathematical subtleties. In section VI we conclude that the potential can co-exist with limit cycle and with possibly more complicated dynamics.

II.

LIMIT CYCLE: PHYSICAL SCIENCES‘ POINT OF VIEW

In physical sciences, the general dynamical equation for a massless particle in two dimensional state space may be expressed as, with both deterministic and stochastic forces11 : [S(q, t) + T (q, t)]q˙ = −∇ψ(q, t) + ξ(q, t) ,

(1)

and supplemented by the relationship on the stochastic force: hξ(q, t)ξ τ (q, t′ )i = 2S(q, t) ǫ δ(t − t′ ) ,

(2)

and hξ(q, t)i = 0. Here qτ = (q1 , q2 ) with q1 , q2 the two Cartesian coordinates of the state space, which may be

2 perceived as the position space of the massless particle. The transpose is denoted by the superscript τ , and q˙ = dq/dt. The scalar function ψ will is the usual potential energy function. Its graphical representation in the state space is a landscape. The antisymmetric matrix T represents the dynamics which conserves the potential, corresponding to the Lorentz force in physical sciences, determined by the magnetic field. The matrix S represents the dynamics which decreases the potential, the dissipation. This matrix will be called friction matrix. The friction matrix is connected to the stochastic force ξ by Eq.(2), which guarantees that it is nonnegative and symmetric. All T, S, ψ can be nonlinear functions of the state variable q as well as the time t. The numerical parameter ǫ corresponds to an effective temperature, which will be taken to be zero to recover the deterministic dynamics. It has been shown that if a steady state distribution ρ(q) in state space exists,   ψ(q) , (3) ρ(q) ∝ exp − ǫ a Boltzmann-Gibbs type distribution function2 . Eq.(3) implies that for dynamics which repeats itself indefinitely, such as limit cycle, the potential may be the same along such trajectory. It should be pointed that the potential function ψ(q, t) exists from the beginning by construction. This is one of most useful quantitative concepts in physical sciences. If the stochastic force could be set to be zero, ξ(q, t) = 0, that is, the deterministic dynamics, the dynamics of this massless particle always decreases its potential energy: q˙ τ ∇ψ(q, t) = −q˙ τ [S(q, t) + T (q, t)]q˙ = −q˙ τ S(q, t)q˙ ≤ 0.

(4)

The zero occurs only at the limiting set. Hence the potential function has the usual meaning of Lyapunov function. To model a limit cycle, we choose following forms for the friction matrix S, the anti-symmetric matrix T , and the p potential ψ, assuming the limit cycle occurs at q = q12 + q22 = 1: (q 2 − 1)2 1 0 − 1)2 + 1 0 1   q2 0 −1 T = (q − 1) 2 (q − 1)2 + 1 1 0 1 ψ = (q − 1)2 2 S =

(q 2





(5) (6) (7)

The potential ψ given in Eq.(7) is rotational symmetric in the state space. It has a local maximum ψ = 1/2 at q = 0, which is a fixed point, and the minimum ψ = 0 at q = 1, which is a cycle in the state space. Hence the potential takes the shape of Mexican hat type.

Evidently, if the friction matrix S would be zero, the dynamical trajectory of the massless particle would move along the equal potential contour determined by the initial condition, which would be a cycle according to above chosen potential. In the presence of nonzero friction matrix, this is not true. What will be our concern is the behavior near the minimum of the potential function: When q is sufficiently close to 1, does the particle trajectory asymptotically approach the cycle of q = 1 and eventually coincide with it? If the answer is positive, we have a limit cycle dynamics. We will demonstrate below that it is indeed possible. For a deterministic dynamics, we can set ǫ = 0 in Eq.(1) and (2): setting the stochastic force to be zero. The dynamical equation can then be rewritten as q˙ = −[S(q, t) + T (q, t)]−1 ∇ψ(q, t)

(8)

With the choice of Eqs. (5-7), we have    1 (q 2 − 1)2 1 0 −1 [S + T ] = det(S + T ) (q 2 − 1)2 + 1 0 1   q2 0 −1 −(q − 1) 2 (9) ) 1 0 (q − 1)2 + 1 and det(S + T ) = [(q 2 − 1)2 /((q 2 − 1)2 + 1)]2 + [(q − 1)2 q 4 /[(q 2 − 1)2 + 1]2 . Near q = 1, we have    1 0 −1 −(1 − 2(q − 1)) [S + T ]−1 = 1 0 q−1    0 −1 + O((q − 1)2 )(10) + 4(q − 1) 1 0 In terms of radial coordinate q and azimuth angle θ in the polar coordinate representation of the state space, using the small parameter expansion given in Eq.(10) following Eq.(8) we have, to the order of q − 1, q˙ = −4(q − 1) q−1 θ˙ = 1 − 2 q

(11) (12)

The solution is q(t) = 1 + δq0 exp {−4t} 1 1 + δq0 exp {−4t} θ(t) = θ0 + t + ln 4 1 + δq0 1 + ln (1 + δq0 exp {−4t}) 2

(13)

(14)

Here δq0 (|δq0 |