THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES AND ITS APPLICATIONS

Journal of Pure and Applied Mathematics: Advances and Applications Volume, Numer,, Pages 5-77 THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES AND ITS APPLICATIONS Graduate School of Business Fordham University New York, NY 9 U. S. A. Department of Physics Yunnan University Kunming, Yunnan 659 P. R. China e-mail: pengkd99@yahoo.com Astract We propose an analytical model on the chaotic ehavior of stock prices. Based on the logistic equation and the Melnikov s method, we have shown circumstances under which stock prices exhiit chaotic ehavior. We illustrate that the prices of Chinese listed stocks are more likely to fall into chaos due to the features of Chinese stock markets. Our theoretical predictions are consistent with the empirical results showing the multifractal properties of Chinese stock prices.. Introduction The linear normal form and the normality assumption have een the foundation for contemporary financial theories and practices for decades. It is essential to the efficient-market hypothesis (EMH). EMH has made Mathematics Suject Classification: 9B55, 7C9, 7J5. Keywords and phrases: chaotic analysis, Melnikov s method, homoclinic orit, logistic equation, Chinese stock prices. Received March, Scientific Advances Pulishers

54 considerale headway and otained a series of valuale results [5-7]. The linear normal forms think that every action produces a reaction and the reaction is direct proportion to the action. However, a group of prior studies have provided evidence that the distriution functions of price changes have fat tails and high peak, and financial markets react in a nonlinear fashion [8,, 9,, ]. For example, Friedman et al. [] point out that, the variation of stock prices does not match the normal distriution nicely, since there are too many extreme movements. Consequently, many studies have turned to nonlinear dynamic systems, such as chaos theory, to descrie market data [9, 8, 6, 7, 9, ]. Inconsistent with the EMH, Peters [8] shows empirical results that the standard and poor 5 index exhiits a long memory, which is an important characteristic of fractal time series. The evidence indicates that, information is not immediately impounded in stock prices, and short-term stock returns may e predictale y using technical analysis. This is to the contrast of Hsieh [7], who shows strong evidence that no chaotic state exists in U.S. stock market. The deate remains unresolved as researchers examine capital markets around the world in search of chaotic characteristics [-,, 5]. This paper contriutes to the literature y ridging the gap etween prior empirical results and the existing nonlinear dynamic models. We are the first to present an analytical model that systematically predicts circumstances, in which stock prices exhiit chaotic ehavior. Following the coherent market hypothesis proposed y Vaga, we recognize that a market can e chaotic, random-walk like, or in transition, ut it cannot stay in any state permanently []. Specifically, market participants may react to information in a linear fashion under some conditions, and such state may last for certain periods. However, when other conditions are satisfied, market reactions can ecome nonlinear and a chaotic state may emerge. Based on our model, we demonstrate that the ehavior of Chinese stock prices is more likely to e chaotic.. Logistic Model We demonstrate the chaotic characteristics of stock prices under the following three assumptions.

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 55 Assumption. All information related to supply and demand of capital such as macroeconomic, political, financial, natural, and psychological factors is reflected in the market. In other words, the aove factors are captured y changes in supply and demand, which decide the stock price. In contrast to the EMH, market reactions can e linear or non-linear. Assumption. There are stochastic perturations in the stock market. The perturations can e periodic or sporadic. Assumption. The price change in the stock markets over time is a deterministic function of price, demand, and supply. Let the price of a stock e x, the demand of the stock e N d, and the supply e N s. If N d > N s, then the excessive capital, which could have ought the stock if there were more supply, is y ( N N ) x. Choosing ( x, y) as independent variales, we have the following differential equations to descrie the system d s f ( x y) + ε g ( x, y, ), (.), t dy f ( x y) + ε g ( x, y, ), (.), t where ε and ε are higher-order small quantities, t is time, and g ( x, y, t) and g ( x, y, t) are perturations. If N s > N d, the stock s supply surplus is N N s N d. Choosing ( x, N ) as independent variales, we have dn f ( x N ) + ε g ( x, N, ), (.), t f ( x N ) + ε g ( x, N, ), (.) 4, 4 4 t where ε and ε 4 are higher-order small quantities, and g ( x, N, t) and g 4 ( x, N, t) are perturations.

56 Recall that independent variales x ( t) and N ( t) are functions of time t. Suppose g ( x, N, t) and g 4 ( x, N, t) are periodic functions, in which the period is T for oth functions. Let τ e the integral multiple of T, we have τ g ( x, N, t) g( x, N, t), g4 ( x, N, t) g4 ( x, N, t). τ τ τ Defining x τ τ x() t, N N() t, τ τ we have knx. Considering ( kn ) + ( kn ) x, x kn x we otain ( kn ) x ( kn ) x Ax Bx, x x where A ( kn ), B ( kn ). x x Suppose the stock price is x n when t nτ, and x n+ when t ( n + ) τ. Let the unit of time e τ, then t ( n + ) τ nτ τ. Thus, we have n xn+ xn + n n Suppose µ A +, it can e shown t ( A + ) x Bx.

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 57 B xn+ µ xn ( xn ). A + Let un B B xn and u n+ xn+, and we otain A + A + un+ µ un ( un ). () This is a typical logistic equation. Prior literature has documented that, when µ > µ ( µ.56994567 ), u and x show chaotic characteristics []. Theorem. Chaos is not an inevitale characteristic of the stock price. However, the stock price can e chaotic, when certain conditions are satisfied.. Melnikov s Method Equations (.) and (.) can e written in vector form f ( x ) + εg( x, t ), (4) where x ( x, y), f ( x ) ( f ( x, y), f ( x, y)), g( x, t ) ( g( x, y, t), and ε ( ε ε ). Equation (4) descries a dynamic system., g ( x, y, )), t Let J. If there is a function H( x ) in a system such that f ( x ) J H( x ), the system is called as a generalized Hamiltonian system. If H( x ) does not exist for the system, it is called as a non- Hamiltonian system. When ε, the system is an unpertured one. If point x ( x, y ) is a solution to the algeraic equation f ( x ), then x ( x, y ) is called as a singular point. From t to t, if ( x, y) of x satisfies Equation (4), the curve descried y ( x, y) is called an orit of Equation (4). If an orit enters the same singular point, when t and t, then the orit is called a homoclinic orit. If an orit enters different singular points, respectively, when t and t, then the orit is called heteroclinic orit.

58 When ε, for an unpertured Hamiltonian system, we have [, 5, 6] J H( x ). (5) If the system has a homoclinic orit () t ( x ( t), y ( t)) T and a q hyperolic saddle point P, we define its Melnikov s function as M ( t ) f ( q ( t)) g( q () t, t + t ), (6) where stands for the exterior product. Recall that the Smale horseshoe map is a strong form of chaos in the mathematics of chaos theory, we have the following theorem. Theorem. If M i ( t ) has simple zero points, then Equation (4) is a chaotic system in the sense of Smale horseshoe. When ε, if there are n hyperolic saddle points p, p,, pn ( n > ) in Equation (5), and there is a heteroclinic orit q i () t connecting from p to p i ( i,,, ) and a heteroclinic orit q n () t i + n connecting from p to p, then we define the Melnikov s function for the n i-th heteroclinic orit as (similar to the homoclinic orit case). Mi ( t ) f ( q ( t)) g( q () t, t t ) i i +. (7) Theorem. If M i ( t ) has simple zero points, then Equation (4) is a chaotic system in the sense of Smale horseshoe. Next, we consider a non-hamiltonian system F ( x ) + εg( x, t ), (8)

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 59 where F ( x ) ( F ( x ), F ( x )) T is not the vectorial field produced y a T r Hamiltonian function, and G( x t ) ( G ( x, t ), G ( x, t )) C ( r ) is, k a periodic function of t, in which the period is T, and C is a continuous function, that is, k-th order differentiale. When ε, if Equation (8) has a hyperolic saddle point P and a homoclinic orit () t through the saddle point, then we have the following theorem. Theorem 4. If Melnikov s function t t Q passing M ( t ) F ( Q ( t t )) G( Q ( t t ), t ) exp[ Tr DF ( Q () τ dτ)] has simple zero points, Equation (8) is a chaotic system in the sense of Smale horseshoe. To see the applications of Melnikov s method, consider the following example of the dynamical system α y, α >, (9.) dy α x αx + ε( γ cos ωt δy), α >, α >, (9.) where α α, α,, and δ are real constants., γ The unpertured system has three singular points (, ) and α ( ±, ). The derivative matrix of the singular points (, ) is α α Df (, ). α It has two characteristic values λ αα and λ αα. Because λ > > λ, the singular point is a saddle point. It is easy to prove that,

6 α the characteristic values of points ( ±, ) are a pair of conjugate α α imaginary numers. Therefore, points ( ±, ) are two centers. α Let s find its Hamiltonian function. From the partial differential equations, H y α y, H x αx αx, we have H 4 αy αx + αx. 4 When ε, the system has a conserved quantity 4 α y αx + αx c. 4 Because the orit passes through the origin (, ), we have c. It can e shown y α α ± x x ± ax x, () α α α α where a and. From Equation (), there are two α α homoclinic orits. From Equation (9.), we have ± ax x, (a) where a α a α. α Solving Equation (a) and y using the initial conditions, we otain the two homoclinic orits: (see Appendix A)

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 6 q+ a [ x (), ()] [ sech ( ), + t y+ t at sech ( at) tanh( at)], a [ () ()] [ ( ) q x t, y t sech at, + sech ( at) tanh( at)]. Clearly, we have T f ( x, y) ( α y, α x α x ), T g( x, y) (, γ cos ωt δy). According to Equation (6), the Melnikov s function for the homoclinic orit q + is ( t ) f ( q ()) t g( q () t, t + t ) M + α y+ ()[ t γ cos ω( t + t ) δy+ ( t )] a [ α γ sech ( at) tanh( at) cos ω( t + t ) a δ sech ( at) ( at)] tanh α a δ + γ sech ( ) sin ωt a a I + Φ sin ωt, where I αa δ, Φ γ sech( ). a a If M + ( t ), then we have sin ωt I. Φ (a)

6 Only when I < Φ, Equation (a) has solutions. If Equation (a), then We have M + t ( t ) I + Φ sin ω. t is a solution to d M + t ( t ) Φω cos ω. So, t is a simple zero point. Clearly, from Theorem, when the following condition γ α ω sech ( ) > δ α α π α () is satisfied, the system is chaotic in the sense of Smale horseshoe. γ α Let Φ ω sech ( ). It can e proven that Φ is an α αα increasing function of α and α and when Equation () is satisfied, Φ > α. In fact, when Φ α γω sech ( ) >, α α α α Φ α γω α α α sech ( ) α α π γ ω α sech ( ) tanh( ) >, 4 α α α α α α Φ α γω α α [ α sech ( )] α α

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 6 When γω α sech ( ) [ tanh( ) ]. 5 α α α α α α α α α tanh( ) > 6, α α () Φ we have >. α Therefore, the chaotic condition, Equation (), is more likely to hold with larger values of α and α. 4. Application to Chinese Stock Markets In contrast to stock markets in the developed countries and areas, Chinese stock markets have the following characteristics. 4.. A large numer of individual investors and an extensive amount of capital Chinese stock markets are dominated y individual investors, who have a huge amount of capital in hand, and are generally unsophisticated and short-term oriented. As a result, there are extreme movements in the demand for stocks and thus high volatility in Chinese stock markets. The statistics from China Securities Depository and Clearing Corporation shows that as of May 8, 7, Chinese investors opened 87.5 million investment accounts, the majority of which is held y individual investors [5]. One of the official wesites, www.cnstock.com, reports that individual investor s trading account for as high as 6.% of the volume in Shanghai Stock Exchanges for the first three months of 7; 64.7% of market shares in Shenzhen Stock Exchanges is held y individual investors [6]. These individual investors hold an extensive amount of capital. According to the People s Bank of China, the total saving of Chinese households was over US$.5 trillion at the end of 7, which was twice the total market worth of A-share [7]. The dominance of Chinese individual investors and their unsophistication highlight the speculative nature of Chinese stock markets, which is likely to cause large movements in the demand of stock shares.

64 4.. Different classes of stock and non-circulating shares Companies listed in Chinese stock markets have different classes of stocks: state-owned shares, legal person shares, domestic listed shares (A-shares are traded in Chinese Yuan and are availale to domestic investors, and B-shares are traded in foreign currencies and are availale only to foreign investors until Feruary 9, ), and shares listed aroad (e.g., H shares are listed in Hong Kong, and N shares are listed in New York). The state-owned shares and legal person shares account for aout two thirds of the total shares. The two classes of shares are noncirculating meaning that, they are not pulicly traded except y special approval of the central and local governments. Generally, there are two types of non-circulating shares: large non-circulating shares (LN) refer to the amount of shares that are more than 5% of the total shares of the listed company, and small non-circulating shares (SN) refer to the amount of shares that are less than 5% of the total shares. Shanghai Stock Exchange and Shenzhen Stock Exchange started the stock reform in 5. LN and SN are permitted to decrease their holding shares step y step according to a define rule. Non-circulating shareholders took the advantage of the stock reform, and sold their stock ownership for easy capital gains. The trading of SN and LN on Chinese stock markets has greatly increased the amount of shares availale in the markets. The speculative incentives of non-circulating shareholders are likely to lead to extreme movements in the supply of stocks, and thus high volatility in Chinese stock markets. 4.. High sensitivity to government regulations Chinese stock markets are very sensitive to government regulations. Su and Fleisher find that large volatility hikes in Chinese stock markets are associated with exogenous changes in government regulations []. For example, China Securities Regulatory Commission (hereafter CSRC) removed a 5% limit on daily stock price movements on May, 99, when the index of Shanghai Stock Exchange A-share douled in one day. In order to restrain volatility, CSRC re-imposed a % daily price-change limit on individual stocks on Decemer, 996. Not surprisingly, the regulation change was followed y 5 consecutive days of decrease in the

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 65 Shanghai Stock Exchange index. In addition, CSRC relaxed the restriction on the trading of non-circulating shares in August 6, which was widely elieved to contriute to the massive amount of the index decline starting from Novemer 7. Chinese stock markets have a relatively short history-they were only re-opened in early 99s after eing closed for nearly half a century. Therefore, the Chinese government is still going through the learning process and lack of continuous and consistent policies. This again may lead to high volatility in Chinese stock markets. 4.4. The speculative incentives of domestic investors The speculative incentives of domestic investors create a speculative component of stock prices, especially for A-share. Prior studies document that severe misevaluation of A-shares in Chinese stock markets is in part due to the joint effect of short-sale prohiition, and investors heterogeneous eliefs on stock value [4,,, 4]. Specifically, stock prices will e iased upward, if there is a sufficient amount of divergent opinions, since the marginal uyer of shares is likely among optimists (pessimists, who are prevented from short-selling, simply sit out of the market). Because of the upward ias in stock prices, investors could e willing to pay more than their expected stock value, anticipating that they could sell the shares to other investors, who pay even more in the future. The existence of the speculative component of stock prices may result in extreme movements of equity values in Chinese stock markets. The aove characteristics shed lights on the applications of our model to Chinese stock markets. In the logistic model, we have x t Ax Bx, µ A +, () where A ( kn ). When N >, we have N N s N < and d N s d A >. If k is ig enough, it can e true that µ > µ. In other words, a large A suggests that the increase in stock price over time is more sensitive to the current price, when there is increasing uying pressure due to demand surplus. The quadratic term Bx, which is a necessary

66 condition for a chaotic system, indicates that the change in stock price over time is a nonlinear function of the current price, and the decrease in price is greater, if selling pressure due to supply surplus is greater. Given that there are extreme movements in the supply and demand of stocks, and Chinese stock markets are highly sensitive to government regulations and full of speculative investors, the changes in stock price over time is likely nonlinear, and the market may e chaotic. As for the Melnikov s method, there is a quadratic term α x in Equation (9.). Because Φ is an increasing function of α, α, and α, when Equation () is satisfied, large values of α, α, and α are likely to result in chaos. The larger the values of α, α, and α are, the higher the sensitivity of the market s reactions to the current price. There is a periodic perturation term γ cos ωt in Equation (9.). Given that stock prices go up and down, the change of stock price may generate periodicity to a certain extent. Besides the periodic perturation term, there may e a non-periodic term in g ( x, y, t), and we have also included a correction term δ y. Considering the unique features of Chinese stock markets earlier discussed, the aove conditions are easily satisfied, and thus, the markets are more likely to e chaotic. As a result, our model can descrie the chaotic characteristics of Chinese stock markets. The predictions of our model are in line with previous empirical evidence on Chinese stock markets. Peters [8] points out two characteristics are essential for a chaotic system: the fractal dimension and the sensitivity to initial conditions, and provides evidence that most of the capital markets are fractal. The chaotic ehavior of Chinese stock markets has een documented y a group of papers. For example, y using the high-frequency data of the Shanghai Stock Exchange Composite Index (hereafter SSECI) in, Wang et al. analyze its nonlinear characteristics and found that the Lyapunov index is positive. This is consistent with the chaotic ehavior of the SSECI []. Using three different methods of multifractal analysis, Du and Ning [4] examine the variation in the SSECI and found evidence that the index has

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 67 multifractal features. More importantly, they show that, as the order of the partition function increases, the multifractal strength increases, the singular spectrums ecome tougher, and the general Hurst exponents decrease. In addition, Zhuang et al. [4] show that, the indices of oth Shanghai Stock Exchange and Shenzhen Stock Exchange do not follow the normal distriution, ut rather exhiit fractal time series. To illustrate that the multifractal features of Chinese stock markets y using our model, take SSECI, for example. The ase value of the SSECI is, which is for the ase day Decemer 9, 99. The SSECI for day t is calculated as Index t nt i n, () i x x t, i, i N N st, i s, i where x, i and N, i are the price and the numer of shares issued for the stock i on the ase day, respectively, x t, i and t i N, are the price and the numer of shares issued for the stock i on day t, and n and n t are the numer of stocks on the ase day and day t. The time series of prices for each stock has its own scaling characteristics, i.e., fractal dimension, and different time series of stock prices can have different scaling characteristics. Since the time series of the SSECI summarizes price fluctuations of all n stocks, it is more appropriate to descrie the index y using a multi-scale fractal structure. In terms of our model, we have constructed a spectrum f ( d) y using an infinite sequence of different fractal dimensions d i. f ( d), which is called a singular spectrum, has een employed y prior studies to descrie the characteristics of a multi-scale fractal structure. We allow f ( x y), f ( x, y), g ( x, y, ), and g ( x, y, t), t in Equations (.) and (.) to take different forms for different stocks. By doing so, we elieve that our model have properly accounted for the multifractal characteristics of the SSECI.

68 5. Conclusion Based on our model as well as previous empirical evidence on Chinese stock markets, we draw following conclusions. () By adjusting f ( x ) and g ( x, t ) according to the status of stock markets, Equation (4) can e used to descrie stock price changes. If certain conditions are satisfied, then the stock market is chaotic; if the conditions are not satisfied, the stock market is not chaotic. In other words, a stock market can fall into or rise out of chaos. An important characteristic of chaos is high sensitivity to initial conditions. Since a stock market is not always sensitive to initial conditions, the market can e efficient for a certain period. () Generally speaking, our model is descriptive of all stock markets, ut it is especially so for Chinese stock markets, due to the unique features of the markets. Said differently, Chinese stock markets can easily fall into chaos. () The different forms of f ( x ) and g ( x, t ) can affect the status of a stock market to a great extent. For example, if α and < in Equation (9.), the equation ecomes < α α y, (4.) dy x + x + ε( f cos ωt δ ), (4.) y where > and >. When ε, the unpertured equations are α y, (5.) dy x x +. (5.)

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 69 The unpertured system has three singular points (, ) and ( ±, ). It is easy to show that point (, ) is a center and points ( ±, ) are saddle points. There are two heteroclinic orits connecting the two saddle points. It can e proven that, when f ω csch ( ) > δ, α π (6) the system is chaotic in the sense of Smale horseshoe ( see Appendix B). f Let Φ ω csch ( ). α (6a) It can e shown that Φ is an increasing function of α and. When certain conditions are satisfied, Φ is also an increasing function of. Therefore, large values of α and are likely to result in chaos. The aove is consistent with the notion that high sensitivity is a necessary condition of chaos. (4) Our model can e capale of descriing the highly speculative nature of Chinese stock markets. Because f ( x ) and g ( x, t ) are affected y multiple factors, the two functions can take various forms, and thus, Chinese stock markets are more likely to have large movements unexplained y a linear paradigm. For example, the SSECI was down 7.8% to,7 as of Octoer 9, 8 from its record close of 6,9 on Octoer 6, 7. The decline in the SSECI was quite dramatic, comparing to the country s officially recorded GDP growth rates of.4% in 7 versus 9% in 8. Thus, such turulent movements in stock prices, which might e largely caused y speculations, manifest the legitimacy of application the chaotic model to Chinese stock prices.

7 From Equation (a), we have Appendix A t ±. (A) ax x Upon integration of Equation (A), we otain t ± [ log( x ) log( + x )]. (A) a From Equation (A), we otain x sech ( at) + c. (A) If t ±, then x, and thus, we have c (see Figure ). Sustituting Equation (A) into Equation (), we have q a y ± sech ( at) tanh( at) + c. q + q q + Figure. Two homoclinic orits.

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 7 When t ±, y, so c. From Figure, when t is in the interval (, ), y > for q, so we have + a [ () ()] [ ( ) q+ x+ t, y+ t sech at, sech ( at) tanh( at)], a q [ (), ()] [ sech ( ), x t y t at + sech ( at) tanh( at)]. M ( t ) I +, I I αa δ α sech ( ) tanh ( ) a δ at at. a Noticing cos ω ( t + t ) cos ωt cos ωt sin ωt sin ωt, we have aγ α sech ( at) tanh( at) cos ω( t + t ) I αa γ sin ωt sech ( at) tanh( at) sin ωt αa γ sin ωt sech ( at) tanh( at) sin ωt γ sin ωt sin ω td sech ( at) γ sin ωt sech ( at) d sin ωt γ ωπ sin ωt sech ( ). (A4) a a

7 Appendix B From Equations (5.) and (5.), we otain the derivation matrix of the singular point (, ), which is α Df (, ). The characteristic values are λ ± α. This is a pair of conjugate imaginary numers, so point (, ) is a center. The derivation matrix of the singular point ( ±, ) is ( α Df ±, ). The characteristic values are λ ± α. Because λ α > > λ ( α, ±, ) are saddle points. From Equations (5.) and (5.), we otain the Hamiltonian function of the system, which is H 4 αy + x x. 4 When ε, the system has a conserved quantity 4 α y + x x c. (A5) 4 The orits pass through points ( ±, ), so we have c. 4

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 7 q + q Figure. Two heteroclinic orits. From Equation (A5), we otain y ± a( x ), (A6) where a and. α From Equations (5.) and (A6), we otain ± a ( x ), (A7) α where a. From Equation (A7), we have ± a ( x ). (A8) Upon integrating of Equation (A8), we otain

74 t Arc tanh( x ). (A9) a From Equation (A9), we otain x tanh( a t ) + c4. (A) For q +, when t ±, x, so we have c 4. Sustituting Equation (A) into Equation (A6), we have y ± a sech ( a t ). (A) When t is in the interval (, + ), y > for otain two heteroclinic orits + q + (see Figure ). We q + [ x+ () t, y + ()] t [ tanh( a t ), a sech ( a t )], q [ x () t, y ()] t [ ± tanh( a t ), a sech ( a t )]. From Equation (7), we have M + ( t ) f ( q+ ()) t g( q+ () t, t + t ) y () t α + x () t [ x ()] t δy () t + f ω( t + t ) + + + + cos [ δy () t + f ω( t + t )] α y () t + cos + 4 αaδ sech ( a t )

THE ANALYTICAL CHAOTIC MODEL OF STOCK PRICES 75 + αf a sech ( a t ) cos ω( t + t ) I + I4. 4 αa δ sech ( a t ) I αδa 4 4 α sech ( ) a t δ. a a Noticing cos ω ( t + t ) cos ωt cos ωt sin ωt sin ωt, we have I 4 αf a cos ωt sech ( a t ) cos ωt αf a ωt cos ωt sech ( t ) cos a a αf a cos ωt csch ( ). a a The condition for M + ( t ) is f ω aa 4 csch ( ) > δ. a π f From Φ ωcsch ( ), α (6a) we have Φ f ωcsch ( ) >, α

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