Phase space reconstruction of one-dimension time series on Nilaparvata lugens(st a l)occurrence and determination of chaotic attactor dimension
Ma Fei · 2001
The brown planthopper(BPH), Nilaparvata lugens (st a l),is an important insect pest of rice corps both in tropical and temperate areas in the East and South Asia.Rice damage caused by BPH has become an increasingly serious problem since the 1970s in China.Outbreaks have increased in frequency and the area regularly infested has extended into Jiangsu Province (between the Yangtse and Huaihe Rivers)and even north of the Huaihe River.On average,some 13.3 million hm 2 of the crop are likely to be affected,with an annual loss of some half a million tones of grain.Despite the substantial manpower and material resources are invested to study,the long term forecasting power is still weak in these days.For the sake of further studying and providing theoretical proof for prediction,based on time series data,phase space reconstruction in time delay coordinates and correlation dimension[ D 2( m )],the chaotic phenomenon of BPH in Yangtse River Valley are first studied in this paper. Time series data from June to November on BPH occurrence in 1979～1990 observed by Wuxian station and those in 1986～1998 by Taihu District Institute of Agricultral Science are used to extend the 1 D time series of BPH occurrence into a multi dimensional phase space in Yangtse River Valley. Takens(1981)proposed “embedding theorem”,and has proved that strange attractor of D 0 dimensions could be depicted with higher dimensions d of phase space,usually d=2D 0+1 . Supposing one dimension time series x(t 0),x(t 1),…x(t i),…x(t n), is extended to a phase type of m dimensions phase spaceX(t 0)X(t 1)…X(t i)…X(t n-(m-1)τ) X(t 0+τ)X(t 1+τ)…X(t i+τ)…X(t n-(m-2)τ) X(t 0+2τ)X(t 1+2τ)…X(t i+2τ)…X(t n-(m-3)τ) :::::: X(t 0+(m-1)τ)X(t 1+(m-1)τ)…X(t i+(m-1)τ)…X(t n)(1)Where τ=kΔt(k=1,2,…) is delay time,a phase point of phase space is made up of every row in (1) formula.Every phase point X(t i) has m weights X(t i)、X(t i+τ)、X(t i+2τ)、…X(t i+(m-1)τ). Every phase point of m dimension phase space embodies a certainly instantaneous state,and the point's trajectory of phase space is composed of the link line of phase point,whereas it exhibits system state evolution with time.And then,the system dynamics can be studied in multiple dimension phase space. A pair of phase point in m dimensions phase space (usually it is bigger)isX m(t i)=X(t i),X(t i+τ),X(t i+2τ),…,X(t i+(m-1)τ) X m(t j)=X(t j),X(t j+τ),X(t j+2τ),…,X(t j+(m-1)τ)Where the distance is r ij (m)=‖X m(t i)-X m(t j)‖Given a critical distance r ,examining less than r ′ phase point pair( X i,X j ),and less than r ′ phase point pair ( X i,X j )in proportion to whole phase point pair,thus cumulative distribution function is followed asC 2(r,m)=1N(N-1)∑Ni,j=1(i≠j)θ(r-‖X i-X j‖)(2)Here θ is Heaviside function,if z 0, θ(z) =0; ifz 0, θ(z) =1. N is all points. Obviously, C 2( r,m )not only describes the probability of distance between two attractor of phase space r ,but also depicts X i phase point'assemble degree in r .Where it is called incidence function of affractor.Essentially,if r is too small,all ‖X i-X j‖r,θ(z)=0,C 2(r,m) =0,and vice versa C 2(r,m) =1.Too big and small r ,hence,can't reflect system inherence property.Generally, r value is contented with 0≤ C 2(r,m) ≤1. To illustrate how to compute the fractal dimension,let us suppose that we wish to measure the length of a curve.Suppose we have a set of rulers of size { r i }.Determining that C 2(r i,m) of rulers will “cover” the curve to be measured.To each rulers r i ,we get a C 2(r i m) .If the curve is fractal,the following relationship holdsC 2(r,m)∝r -D2 OrD 2= lg (C 2(r,m)) lg (r)= ln (C 2(r,m)) ln (r)(3)Where D 2 is the corresponding fractal dimensions.In logarithm coordinates system,we may get an approximate straight line of whi