Fejer and Suffridge polynomials in the delayed feedback control theory
Dmitriy Dmitrishin, Anna Khamitova, Anatolii Korenovskyi, Alex Stokolos
aa r X i v : . [ m a t h . D S ] A ug FEJ ´ER AND SUFFRIDGE POLYNOMIALS IN THE DELAYEDFEEDBACK CONTROL THEORY
D. DMITRISHIN, A. KHAMITOVA, A. KORENOVSKYI AND A. STOKOLOS
Abstract.
A remarkable connection between optimal delayed feedback control(DFC) and complex polynomial mappings of the unit disc is established. The ex-plicit form of extremal polynomials turns out to be related with the Fej´er poly-nomials. The constructed DFC can be used to stabilize cycles of one-dimensionalnon-linear discrete systems. Introduction
Non-negative trigonometric polynomials appear in many problems of harmonicanalysis, univalent mappings, approximation theory, orthogonal polynomials on torus,number theory and in other branches of mathematics. One of the most natural andbeautiful application of the properties of non-negative trigonometric polynomials oc-curs solving extremal problems.For non-negative trigonometric polynomials P nj =0 a j cos jt, a = 1 , the followingFej´er inequality holds [6] (see also [12], 6.7, problem 52): | a | ≤ πn + 2 . This inequality is sharp and the extremizer2 n + 2 sin πn + 2 Φ (1) n ( t )is unique, where Φ (1) n ( t ) = cos n +22 t cos t − cos πn +2 ! . In 1900 L.Fej´er showed [7] that trigonometric polynomial1 + 2 n X j =1 (cid:18) − jn + 1 (cid:19) cos jt is nonnegative by proving the multiplicative presentation1 + 2 n X j =1 (cid:18) − jn + 1 (cid:19) cos jt = 1 n + 1 Φ (2) n ( t ) , where Φ (2) n ( t ) = (cid:18) sin n +12 t sin t (cid:19) . In [8] (see also [12], 6.7, problem 50) Fej´er proved the extremal property of thefunction Φ (2) n ( t ) , that the maximal value of the non-negative trigonometric polynomial P nj =0 a j cos jt, a = 1 does not exceed n + 1 . Moreover, the equality happens only forthe polynomial n +1 Φ (2) n ( t ) and only at points 2 πk, k ∈ Z . The functions Φ ( i ) n ( t ) , i = 1 , Optimal Stabilization and extreme properties of polynomialmappings
Impact on the problem of optimal chaotic regime is fundamental in nonlinear dy-namics. The purpose of these actions is synchronization of chaotic motions, or con-versely, randomization of regular motions. Moreover, the allowed values of the controlare small which nonetheless completely change the character of the movement. In thispaper we consider the problem of optimal stabilization of cycles in families of discreteautonomous systems with delayed feedback control (DFC) methods [1].We are given a scalar nonlinear discrete open-loop system(1) x k +1 = f ( x k ) , x k ∈ R , n = 1 , , . . . , with one or more unstable T - cycles ( η , ..., η T ) , where all the numbers η , ..., η T are distinct and η j +1 = f ( η j ) , j = 1 , ..., T − , η = f ( η T ) . It is assumed that themultipliers µ = T Y j =1 f ′ ( η j )of the considered unstable cycles are negative.It is required to stabilize all (or at least some) T - cycles by the control(2) u k = − n − X j =1 ε j ( f ( x k − jT + T ) − f ( x k − jT )) , < ε j < , j = 1 , . . . , n − , in a way so that the depth of used prehistory N ∗ = ( n − T would be minimal.Note that for state synchronization x k = x k − T the control (2) vanishes, i.e. closedsystem takes the same form as in the absence of control. This means that T - cyclesof the open-loop and closed-loop systems coincide. EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 3
The closed-loop system x k +1 = f ( x k ) + u k can be written as(3) x k +1 = n X j =1 a j f ( x k − jT + T ) , where a = 1 − ε , a j = ε j − − ε j , j = 2 , ..., n − , a n = ε n − . It is clear that P nj =1 a j = 1 . Let apply the following schedule of linearization for constructing of Jacoby matrixof the system (3)eqref and its characteristic equation. It is clear that x n +1 = a f ( x n ) + a f ( x n − T ) ... + a N f ( x n − T ( N − ) . . . . . .x n + T = a f ( x n + T − ) + a f ( x n − ) ... + a N f ( x n − T ( N − − )(4)Solution to the system (4) can be written in the form x T n = η + u n . . . . . .x T n + T − = η T + u Tn (5)Substitute solutions (5) in (4) assuming that in a neigborhood of a cycle the quantities u n , . . . , u Tn are small.Let n = T m.
Then x n +1 = x T m +1 = η + u m , x n +2 = x T m +2 = η + u m , . . . , x n + T = x T ( m +1) = η + u m Extracting linear part and taking into account that η = f ( η ) , . . . , η t = f ( η ) we get u m = f ′ ( η )( a u m + ... + a N u m − N +1 ) u m = f ′ ( η )( a u m + ... + a N u m − N +1 ) . . . . . .u Tm = f ′ ( η T − )( a u T − m + ... + a N u T − m − N +1 ) u m +1 = f ′ ( η T )( a u Tm + ... + a N u Tm − N +1 )(6)Since this is a linear system it’s solution can be written as u m . . .u Tm = c . . .c T λ m , which after substitution to the system (6) leads to a system (7) − f ′ ( η ) · p (cid:0) λ − (cid:1) . . . − f ′ ( η ) · p (cid:0) λ − (cid:1) . . . . . . . . . . . . . . . . . . . . . . . . − f ′ ( η T − ) · p (cid:0) λ − (cid:1) λ . . . − f ′ ( η T ) · p (cid:0) λ − (cid:1) c c . . .c T − c T = . . . D. DMITRISHIN, A. KHAMITOVA, A. KORENOVSKYI AND A. STOKOLOS where p ( λ − ) = ( α + α λ − + ... + α n λ − n +1 ). Standard technique to study thestability of T - cycle is check the location of all the zeros of the determinant of theJacobian matrix of the system (7) in the unit disc D of complex plane. In this case,the determinant is equal to( − T − λ + T Y j =1 ( − f ′ ( η j ) p ( λ − )) , where Q Tj =1 f ′ ( η j ) = µ. Since we need to consider the stability of not one, but severalcycles we should considered a family of characteristic equations(8) n λ − µ · (cid:0) p (cid:0) λ − (cid:1)(cid:1) T = 0 , µ ∈ ( − µ ∗ , − o , where µ ∗ is the lower bound of T - cycles multipliers, and p (1) = 1 . For a sufficiently small value | µ | , all the roots of λ + | µ | · ( p ( λ − ))) T = 0 lie in aunit disc. With the increasing of | µ | , the roots come out to the boundary of the unitdisc. For each coefficient vector ( a , ....a n ) , the minimum value of | µ | that keeps theroots in the unit disc will be denoted by µ o ( a , ..., a n ) . If there are coefficients ˜ a , ...., ˜ a n , ˜ a + .... + ˜ a n = 1 such that µ o (˜ a , ...., ˜ a n ) > µ ∗ , then all equations in the family (8) corresponding to these coefficients have roots onlyin the unit disc and therefore the cycles of the system (1) can be stabilized by thecontrol (2). In this regard a number of problems pope out. Problem 1.
Demonstrate that for any n > T, thefunction µ o ( a , ..., a n ) defined on the hyperplane a + .... + a n = 1 is semicontinuousfrom below and is bounded. Find the value µ n ( T ) = sup a + .... + a n =1 µ o ( a , ..., a n ) . Problem 2.
Show that for any µ ∗ > T, there is aninteger n and the coefficients ˜ a , ...., ˜ a n , ˜ a + .... + ˜ a n = 1 such that µ o (˜ a , ...., ˜ a n ) > µ ∗ , i.e. µ n ( T ) → ∞ as n → ∞ . Problem 3. (Dual to Problem 2). For a fixed T and a fixed µ ∗ , find the minimalpositive integer n and strength coefficients ε , . . . , ε n − of the controller (2) such thatall T - cycles for the system (1) closed by this control with multipliers µ ∈ ( − µ ∗ , n ∗ to the Problem 3 is related to the solution µ n ( T ) of the Problem2 by the relations µ n ∗ ( T ) > µ ∗ , µ n ∗ − ( T ) ≤ µ ∗ . Note that T - cycle stability by itself does not mean practical realization of tra-jectories of this cycle for a given control. The trajectories start being attracted tothe cycle when the previous coordinate values fall into the basin of attraction of thecycle in the space of initial data. This basin of attraction may be so small that no EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 5 segment of trajectories of an open-loop or closed-loop system would be inside thisregion. Thus, another problems arises.
Problem 4.
For a closed system (3) evaluate the domain of attraction of T -cyclesin the space of initial data. Problem 5.
For a closed system (3) with stable T - cycles determine all stablecycles of length different from T .Below we provide solution to the Problems 1, 2 and 3 for T = 1 , . Construction of the objective function
The equation (8) implies that 1 µ = 1 λ (cid:18) p (cid:18) λ (cid:19)(cid:19) T , where λ ∈ D . Therefore the value1 µ is in the image of the exterior of the unit disc under the mapping z ( p ( z )) T . Theboundary of this image consists entirely of the points in the set (cid:8) e iω ( p ( e iω )) T : ω ∈ [0 , π ) (cid:9) but not necessarily from all those points. It is assumed that µ is real and negative,therefore 1 | µ | is equal to maximal distance from zero to the the boundary of the imageof the exterior of the unit disc in the direction of negative real semi-axis :max ω ∈ [0 , π ) n(cid:12)(cid:12) p ( e iω ) (cid:12)(cid:12) T : arg( e iω ( p ( e iω )) T ) = π o = − min ω ∈ [0 , π ) (cid:8) ℜ (cid:0) e iω p ( e iω )) T (cid:1) : ℑ (cid:0) e iω p ( e iω )) T (cid:1) = 0 (cid:9) . The problem 1 is to minimize the distance:1 µ n ( T ) = − sup p ( z ): p (1)=1 (cid:26) min ω ∈ [0 , π ) (cid:8) ℜ ( e iω ( p ( e iω )) T ) : ℑ ( e iω ( p ( e iω )) T ) = 0 (cid:9)(cid:27) =inf p ( z ): p (1)=1 (cid:26) max ω ∈ [0 , π ) n(cid:12)(cid:12) p ( e iω ) (cid:12)(cid:12) T arg( e iω ( p ( e iω )) T ) = π o(cid:27) . Remark . Since ¯ z ( p (¯ z )) T = z ( p ( z )) T , then in the above formulas ω can be consid-ered in the interval [0 , π/T ] . By making the change ω = T t, the formula for µ n ( T ) can be written as(9) 1 µ n ( T ) = " inf p ( z ): p (1)=1 ( max t ∈ [ , πT ) n(cid:12)(cid:12) p ( e iT t ) (cid:12)(cid:12) : arg (cid:0) e it (cid:0) p ( e iT t ) (cid:1)(cid:1) = πT o ) T . Since ( ze i πT ) · p (cid:18)(cid:16) ze i πT (cid:17) T (cid:19) = e i πT z · p ( z T ) , the polynomial z · p ( z T ) has T -symmetry. D. DMITRISHIN, A. KHAMITOVA, A. KORENOVSKYI AND A. STOKOLOS
Formula (9) for T = 1 and 2 can be conveniently written as(10) 1 µ n (1) = − sup P nj =1 a j =1 ( min ω ∈ [0 ,π ] ( n X j =1 a j cos jt : n X j =1 a j sin jt = 0 ) ) , (11) 1 µ n (2) = " inf P nj =1 a j ( max ω ∈ [ , π ] ( n X j =1 a j sin(2 j − t : n X j =1 a j cos(2 j − t = 0 ) ) . Surprisingly, the values of µ n (1) and µ n (2) admit a simple explicit expressionthrough n, and the coefficients of limiting extreme polynomials z · p (1)0 ( z ) , z · ( p (2)0 ( z )) can be easily expressed in terms of the Fejer kernel Φ (1) n ( t ) and Φ (2) n ( t ) . Auxiliary results
The main idea of solving the above problems is to determine necessary conditionsfor the mappings of the form z · p ( z T ) maximally reduce the class of possible mappingsThe self-intersection of the circle image corresponds to a factorization of the con-jugate trigonometric polynomials. The factorization theorem can be regarded as realanalogue of Bezout’s theorem Theorem 1.
Let (12) C ( t ) = n X j =1 a j cos jt, S ( t ) = n X j =1 a j sin jt be a pair of conjugate trigonometric polynomials with real coefficients. And let theequalities (13) S ( t ) = . . . = S ( t m ) = 0 , C ( t ) = . . . = C ( t m ) = γ, are valid where the numbers t ..., t m belong to the interval (0 , π ) , and m ≤ n . Thentrigonometric polynomials (12) admit a presentation (14) C ( t ) = γ + m Y j =1 (cos t − cos t j ) n − m X k = m α k cos kt, S ( t ) = m Y j =1 (cos t − cos t j ) n − m X k = m α k sin kt, where α m = − m γ and the coefficients α m , ....α n − m can be uniquely expressed in termsof γ, a , ...., a n . Proof.
Consider the algebraic polynomial F ( z ) = − γ + n X j =1 α j z j . Then C ( t ) = γ + ℜ (cid:8) F ( e it ) (cid:9) , S ( t ) = ℑ (cid:8) F ( e it ) (cid:9) . EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 7
By (13) F ( e it j ) = 0 and F ( e − it j ) = 0 , j = 1 , ...., m. The fundamental theorem ofalgebra implies the existence of the numbers β , ...., β n − m such that F ( z ) = m Y j =1 ( z − e it j )( z − e − it j ) ! − γ + n − m X k =1 β k z k ! . Let us modify the product m Y j =1 (cid:0) z − e it j (cid:1) (cid:0) z − e − it j (cid:1) = m Y j =1 (cid:0) z − z cos t j + 1 (cid:1) = 2 m z m m Y j =1 (cid:18) (cid:18) z + 1 z (cid:19) − cos t j (cid:19) . Therefore, F ( z ) = m Y j =1 (cid:18) (cid:18) z + 1 z (cid:19) − cos t j (cid:19) − m γz m + 2 m n − m X k =1 β k z m + k ! . Then ℜ (cid:8) F (cid:0) e it (cid:1)(cid:9) = m Y j =1 (cos t − cos t j ) − m γ cos mt + 2 m n − m X k = m +1 β k − m cos kt ! , ℑ (cid:8) F (cid:0) e it (cid:1)(cid:9) = m Y j =1 (cos t − cos t j ) − m γ sin mt + 2 m n − m X k = m +1 β k − m sin kt ! , which implies (14) with α m = − m γ, α m + k = 2 m β k , k = 1 , ..., n − m. (cid:3) Formula (14) allows us to generalize well-known identities, and to obtain new.
Examples.
1. The following multiplicative representations of conjugate Dirichlet kernels arewell known m X j =1 sin 2 jt = sin mt sin t sin( m + 1) t, m X j =1 cos 2 jt = sin mt sin t cos( m + 1) t. Formula (14) gives another multiplicative representations of Dirichlet kernels m X j =1 sin 2 jt = 2 m m Y j =1 (cos t − cos πjm + 1 ) sin mt, m X j =1 cos 2 jt = − m m Y j =1 (cos t − cos πjm + 1 ) cos mt. Letting t = 0 , we obtain the identity m + 12 m = m Y j =1 (cid:18) − cos πjm + 1 (cid:19) .
2. In a similar way, one can get multiplicative formulas for the Dirichlet kernels
D. DMITRISHIN, A. KHAMITOVA, A. KORENOVSKYI AND A. STOKOLOS m X j =1 sin jt = 2 m m Y j =1 (cos t − cos 2 πj m + 1 ) sin mt, m X j =1 cos jt = − m m Y j =1 (cos t − cos 2 πj m + 1 ) cos mt, m +1 X j =1 sin jt = 2 m m Y j =1 (cos t − cos πjm + 1 )(sin mt + sin( m + 1) t ) , m +1 X j =1 cos jt = − m m Y j =1 (cos t − cos πjm + 1 )(cos mt + cos( m + 1) t ) . e − it ) m = 1 + m X j =1 (cid:18) mj (cid:19) e − jt , (1 + e − it ) m = 2 m e − i mt e − i t + e i t ! m , we have multiplicative representations m X j =1 (cid:18) mj (cid:19) sin 2 jt = 2 m cos m t sin mt, m X j =1 (cid:18) mj (cid:19) cos 2 jt = − m cos m t cos mt, m X j =1 ( − j + m (cid:18) mj (cid:19) sin jt = 2 m (1 − cos t ) m sin mt, m X j =0 ( − j + m (cid:18) mj (cid:19) cos jt = 2 m (1 − cos t ) m cos mt. As a consequence, we obtain the following representations for the Chebyshev poly-nomials T m ( x ) = 12 m x m m X j =0 (cid:18) mj (cid:19) T j ( x ) ,T m ( x ) = 12 m (1 − x ) m m X j =0 ( − j + m (cid:18) mj (cid:19) T j ( x ) . In the future, we will need a property of nonlocal separation from zero for theimage of the unit disc. A good example of such claim could is a famous K¨obe QuarterTheorem [9]. Unfortunately, it is unclear how to apply it for two reasons. First, thereis a different normalization of the mappings: 1 mps 1. Second, considered mappingsare not necessarily univalent.We can prove a weaker statement which is enough for our needs.
EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 9
Lemma 1.
Let F ( z ) = a z + ... + a n z n , ( a j ∈ C , j = 1 , ..., n ) , and D = {| z | < } . Then the set F ( D ) contains a disc with the center at the origin and radius n n X j =1 | a j | . Proof.
Let γ be an exceptional value for the polynomial F ( z ) for all z ∈ D . Since F ( z ) ≤ P nj =1 | a j | , such values do exist. Then the polynomial F ( z ) − γ does not haveroots inside D and therefore the inversion produces a polynomial z n ( − γ + F (1 /z ))that has all zeros in D , i.e. Schur stable. It can be written as − γz n + a z n − + · · · + a n = − γ ( z n − a γ z n − − · · · − a n γ ) . Applying Vieta’s theorem to the polynomial in parenthesis, we get the estimate (cid:12)(cid:12)(cid:12) a j γ (cid:12)(cid:12)(cid:12) ≤ (cid:0) nj (cid:1) , j = 1 , . . . , n, which implies P nj =1 (cid:12)(cid:12)(cid:12) a j γ (cid:12)(cid:12)(cid:12) ≤ n − , and finally | γ | ≥ n − P nj =1 | a j | . Note that the above estimate cannot be improved as the example of F ( z ) = ( z +1) n − (cid:3) Solution of optimization problems
T=1 case.Theorem 2.
Let C ( t ) and S ( t ) be a pair of conjugated trigonometric polynomials C ( t ) = n X j =1 a j cos jt, S ( t ) = n X j =1 a j sin jt, normalized by the conditions n P j =1 a j = 1 . Let J be a solution to the extremal problem sup a ,...,a n min t { C ( t ) : S ( t ) = 0 } . Then J = − tan π n + 1) . Proof.
Note that S ( π ) = 0 for any choice of the coefficients a , . . . , a n . Denote ρ ( a , . . . , a n ) = min t ∈ [0 ,π ] { C ( t ) : S ( t ) = 0 } . The quantity sup { ρ ( a , . . . , a n ) } will be evaluated along the set A R = ( ( a , . . . , a n ) : n X j =1 a j = 1 , n X j =1 | a j | ≤ R ) , where R is a large enough positive number. Together with ρ ( a , . . . , a n ) we consider a function ρ ( a , . . . , a n ) = min t ∈ [0 ,π ] { C ( t ) : t ∈ T ∪ { π }} , where T is a subset of (0 , π ), such that the function S ( t ) changes sign. Lemma 1implies that the curve ( C ( t ) , S ( t )) enclosed zero, therefore there is a negative valueof the variable t where the trigonometric polynomial S ( t ) changes the sign, hence ρ ( a , . . . , a n ) < a , . . . , a n . Lemma 2.
There is a pair of trigonometric polynomials { C ( t ) , S ( t ) } such that sup ( a ,...,a n ) { ρ ( a , . . . , a n ) } = min t ∈ [0 ,π ] (cid:8) C ( t ) : t ∈ T ∪ { π } (cid:9) where T is a subset of the interval (0 , π ) , such that the function S ( t ) changes thesign.Proof. The function ρ ( a , . . . , a n ) is upper semi-continuous on the set A R , besidespossibly the points ( a , . . . , a n ) for which the minimal value of C ( t ) is achieved inthose zeros of the function S ( t ) , where S ( t ) does not change the sign. An upperlimit of the function ρ ( a , . . . , a n ) is equal the value of the function at the points ofdiscontinuity which means that the function ρ ( a , . . . , a n ) is semi-continuous fromabove. Therefore, it achieves the maximum value on the set A R ρ = max ( a ,...,a n ) ∈ A R { ρ ( a , . . . , a n ) } . It is clear that ρ ≥ − , therefore | ρ | < a , . . . , a n we have the estimate P nj =1 | a j | ≤ n . Thus, for
R > n the maximum isachieved in an internal point of the region A R and the optimal pair { C ( t ) , S ( t ) } that provides the maximum is independent on R if R > n . (cid:3) The set
T ∪ { π } is a subset of all zeros of the function S ( t ) , therefore ρ ≤ ρ , where ρ = sup ( a , ... .,a n ) ∈ A R { ρ ( a , . . . , a n ) } . A pair of trigonometric polynomials { C ( t ) , S ( t ) } , where the value ρ is achieved will be called optimal. Lemma 3.
If polynomial S ( t ) has zero in (0 , π ) it cannot be optimal.Proof. Let for the optimal polynomial S ( t ) the set T = { t , . . . , t q } , ≤ q ≤ n − , be nonempty. And let min (cid:8) C ( t ) , . . . , C ( t q ) (cid:9) = C ( t ) , assuming additionally that C ( t ) = C ( t j ) , j = 1 , . . . , m (1 ≤ m ≤ q )and C ( t ) < C ( t j ) , j = m + 1 , . . . , q. EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 11
Then three cases are possible: either C ( t ) < C ( π ) , or C ( π ) ≤ C ( t ) ≤ , or C ( t ) > . Case 1.
By Theorem 1, the trigonometrical polynomials S ( t ), C ( t ) can be writtenas S ( t ) = m Y j =1 (cos t − cos t j ) n − m X k = m α k sin kt,C ( t ) = − α m m + m Y j =1 (cos t − cos t j ) n − m X k = m α k cos kt. Since C ( t ) = − α m m ≤ α m >
0. Moreover, C (0) = 1, therefore − α m m + m Y j =1 (1 − cos t j ) n − m X k = m α k = 1 , n − m X k = m α k = 1 + α m m m Q j =1 (1 − cos t j ) > . Let us define the following auxiliary polynomials S ( θ , . . . , θ m ; t ) = N ( θ , . . . θ m ) ·· m Y j =1 (cos t − cos θ j ) n − m X k = m α k sin kt,C ( θ , . . . , θ m ; t ) = N ( θ , . . . θ m ) ·· − α m m + m Y j =1 (cos t − cos θ j ) n − m X k = m α k cos kt ! , where the normalizing factor N ( θ , . . . , θ m ) guaranties the sums of the coefficientsof each polynomials S ( θ , . . . , θ m ; t ) and C ( θ , . . . , θ m ; t ) to be 1. For the polyno-mial S ( θ , . . . , θ m ; t ) , the set of sign changes will be T θ = { θ , . . . , θ m , t m +1 , . . . , t q } .It is clear that S ( t , . . . , t m ; t ) ≡ S ( t ) and C ( t , . . . , t m ; t ) ≡ C ( t ). The factor N ( θ , . . . , θ m ) is determined by the condition C ( θ , . . . , θ m ; 0) = 1, i.e. N ( θ , . . . , θ m ) = 1 − α m m + m Q j =1 (1 − cos θ j ) n − m P k = m α k . So, the polynomials S ( θ , . . . , θ m ; t ) and C ( θ , . . . , θ m ; t ) could be defined by theexpressions S ( θ , . . . , θ m ; t ) = m Q j =1 (cos t − cos θ j ) n − m P k = m α k sin kt − α m m + m Q j =1 (1 − cos θ j ) n − m P k = m α k , C ( θ , . . . , θ m ; t ) = − α m m + m Q j =1 (cos t − cos θ j ) n − m P k = m α k cos kt − α m m + m Q j =1 (1 − cos θ j ) n − m P k = m α k . Note that the coefficients α m , . . . , α n − m are independent of the choice of parameters θ , . . . , θ m . Let us show that for some θ , . . . , θ m the value of ρ for the pair { C ( θ , . . . , θ m ; t ) , S ( θ , . . . , θ m ; t ) } is bigger then for the pair { C ( t ) , S ( t ) , } , i.e. the pair { C ( t ) , S ( t ) , } cannot beoptimal.By the defintion of C ( θ , . . . , θ m ; θ j ) we get C ( θ , . . . , θ m ; θ ) = · · · = C ( θ , . . . , θ m ; θ m ) = − α m m − α m m + m Q j =1 (1 − cos θ j ) n − m P j = m α j . Since α m > n − m P j = m α j > C ( θ , . . . , θ m ; θ ), j = 1 , . . . , m , is an increasingfunction of the parameters θ , . . . , θ m .Let 0 < θ j − t j < ε, j = 1 , . . . , m. From the continuity of trigonometric polynomialson t and an all coefficients the following inequalities hold C ( θ , . . . , θ m ; θ j ) > C ( t j ) , j = 1 , . . . , m (cid:12)(cid:12) C ( θ , . . . , θ m ; t j ) − C ( θ j ) (cid:12)(cid:12) < δ, j = m + 1 , . . . , q, (cid:12)(cid:12) C ( θ , . . . , θ m ; π ) − C ( π ) (cid:12)(cid:12) < δ for any δ with appropriate choice of ε. The above inequalities mean that the valuemin { C ( θ , . . . , θ m ; θ ) , . . . , C ( θ , . . . , θ m ; θ m ) , C ( θ , . . . , θ m ; t m +1 ) , . . . ,C ( θ , . . . , θ m ; t q ) , C ( θ , . . . , θ m ; π ) } is larger then min { C ( t ) , . . . , C ( t q ) , C ( π ) } at least for small enough positive θ j − t j , j = 1 , . . . , m , i.e. the pair { C ( t ) , S ( t ) , } is not an optimal. Case 2.
Let compute C ( π ) = − α m m − m Y j =1 (1 + cos t j ) n − m X j = m ( − j + m − α j ,C ( θ , . . . , θ m ; π ) = − α m m + m Q j =1 (1 + cos θ j ) n − m P j = m ( − j + m − α j − α m m + m Q j =1 (1 − cos θ j ) n − m P j = m α j , EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 13 and C ( t , . . . , t m ; π ) = C ( π ) . Since we assume that C ( π ) ≤ − α m m , then n − m P j = m ( − j + m − α j ≥
0, and the quantity α m m + m Q j =1 (1 + cos θ j ) n − m P j = m ( − j + m − α j is decreasing with respect to each parameter θ , . . . , θ m .Since by the assumption C ( t ) = − α m m ≤ n − m X j = m α j = 1 + α m m m Q j =1 (1 − cos θ j ) > . For small increments of θ j − t j , j = 1 , . . . , m the value of − α m m + m Q j =1 (1 − cos θ j ) n − m P j = m α j is close to 1, and is increasing with respect to each parameter θ , . . . , θ m . Therefore C ( θ , . . . , θ m ; π ) is increasing with respect to each parameters θ , . . . , θ m . At the sametime C ( θ , . . . , θ m ; θ j ), j = 1 , . . . , m are equal and are increasing with respect to eachparameters θ , . . . , θ m too. Therefore in this case the pair { C ( t ) , S ( t ) } cannot bean optimal either. Case 3.
By the Theorem 1, the optimal polynomials can be written in the form C ( t ) = − β t − cos t ) n − X j =1 β j cos jt,S ( t ) = (cos t − cos t ) n − X j =1 β j sin jt where − β − cos t ) n − X j =1 β j = 1 , C ( t ) = − β > C ( π ) = − β − (1 + cos t ) n − X j =1 ( − j β j < . Therefore n − X j =1 β j = β + 11 − cos t < , n − X j =1 ( − j β j > . Consider a family C ( θ, t ) = − β + (cos t − cos θ ) P n − j = m β j cos jt − β + (1 − cos θ ) P n − j =1 β j , S ( θ, t ) = (cos t − cos θ ) P n − j =1 β j sin jt − β + (1 − cos θ ) P n − j =1 β j . It is clear that C ( t , t ) ≡ C ( t ) and S ( t , t ) ≡ S ( t ) . Let compute C ( θ, π ) = − β − (1 + cos θ ) P n − j =1 ( − j β j − β + (1 − cos θ ) P n − j =1 β j . The function C ( θ, π ) is either monotonic on (0 , π ) as a rational function of cos θ or isidentically constant.If it is monotonic, then there exists θ such that C ( θ , π ) > C ( t , π ) = C ( π ) . Thus,the pair { S ( t ) , C ( t ) } is not the optimal.Let now C ( θ, π ) be a constant. Since C ( t , π ) = C ( π ) < C ( θ, π ) < . Therefore lim θ → π C ( θ, π ) = − β − β + 2 P n − j =1 β j < . Since − β > , the function − β +(1 − cos θ ) P n − j =1 β j is positive at zero and negativefor θ close to π, because the denominator of the above fraction is a value of the functionat π. Therefore, there is zero value and then there exists θ such that C ( θ , θ ) = − β − β + (1 − cos θ ) P n − j =1 β j > n . Since the absolute value of a trigonometric polynomial does not exceed the sum ofthe absolute values of the coefficients, then the sum of the absolute values of thecoefficients of the polynomial C ( θ , t ) is bigger then 2 n . This means that either someof the functions C ( θ ; t j ) , j = 2 , . . . , q, become less then one for certain θ ∈ ( t , θ )being positive, or Lemma 1 implies that C ( θ , π ) < − . In any of those cases thequantities C ( θ , π ) = C ( t , π ) = C ( π ) cannot be extremal.Thus, it is shown that the set T is empty, therefore for the optimal pair { C ( t ) , S ( t ) } we have S ( t ) > t ∈ (0 , π ) . The lemma is proved. (cid:3)
Trigonometric polynomial S ( t ) can be written as S ( t ) = sin t · (cid:0) γ + 2 γ cos t + · · · + 2 γ n cos( n − t (cid:1) , where γ s = P a j and the summation runs over indices s ≤ j ≤ n of same par-ity with s, s = 1 , ..., n. There is a bijection between a , . . . , a n and γ , . . . , γ n . Thenormalization a + ... + a n = 1 implies γ + γ = 1 . Since T = ∅ then ρ = max ( a ,...,a n ) ∈ A R { C ( π ) : S ( t ) > , t ∈ (0 , π ) } =max ( a ,...,a n ) ∈ A R {− a + a − a + ... : S ( t ) > , t ∈ (0 , π ) } . Note that − a + a − a + . . . = − γ + γ . EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 15
Since S ( t ) / sin t is non-negative, the well-known Fej´er inequality for non-negativepolynomials [6] (see also [12] 6.7, Problem 52) implies that | γ | ≤ cos πn + 1 · | γ | . From here we get ρ ≤ ρ := max γ ,γ (cid:26) − γ + γ : γ + γ = 1 , | γ | ≤ cos πn + 1 · | γ | (cid:27) . The conditional maximum is achieved for γ = 11 + cos πn +1 , γ = cos πn +1 πn +1 , and is equal to ρ = − − cos πn +1 πn +1 = − tan π n + 1) . The polynomial S ( t ) / sin t = γ +2 γ cos t + ... +2 γ n cos( n − t is a non-nagative Fej´erpolynomial and its coefficients a , . . . , a n are determined in a unique way: a = γ − γ , a = γ − γ , a = γ − γ . . . . Since a j are positive then P nj =1 (cid:12)(cid:12) a j (cid:12)(cid:12) = P nj =1 a j = 1which is independent of R. Therefore for any a , . . . , a n such that P nj =1 | a j | = 1 , the following inequalities are valid ρ ( a , . . . , a n ) ≤ ρ , ρ ( a , . . . , a n ) ≤ ρ ≤ ρ . To prove that the function ρ ( a , . . . , a n ) the supremum is achieved and is equal to ρ , let consider one-parametric family of trigonometric polynomials S ε ( t ) = a + ε ε sin t + a ε sin 2 t + · · · + a n ε sin nt. It is clear that a + ε ε + a ε + · · · + a n ε = 1 and S ε ( t ) = S ( t )1+ ε + ε ε sin t , C ε ( t ) = C ( t )1+ ε + ε ε cos t . Now, for all t ∈ (0 , π ) and ε > S ε ( t ) >
0. Since C ε ( π ) = ρ ε + ε ε then C ε ( π ) < ρ and C ε ( π ) → ρ as ε → . Therefore ρ ≤ ρ , and thus ρ = ρ . These relations plus independence of the coefficients from R means that J = ρ = ρ = sup P a j =1 { ρ ( a , . . . , a n ) } = − tan π n + 1) . The theorem is proved. (cid:3)
Formula (10) implies that µ n (1) = cot π n + 1) . Corollary . Let conjugate trigonometric polynomials (12) be normalized by thecondition P nj =1 a j = 1 . And let T be the set of sign changes for the function S ( t ) on(0 , π ). Then max a ,...,a n min { C ( t ) : t ∈ T ∪ { π } } = − tan π n + 1) , and the coefficients of the extremal pairs of the trigonometric polynomials are defineduniquely by the formulas(15) a j = 2 · tan π n + 1) · (1 − jn + 1 ) · sin πjn + 1 , j = 1 , . . . , n. To prove the corollary it is enough to find the coefficients a , . . . , a n . The polyno-mial S ( t )sin t is proportional to the Fej´er polynomial Φ (1) n ( t ): S ( t )sin t = 11 + cos πn +1 + 2 cos πn +1 πn +1 cos t + ... =1 − cos πn +1 n + 1 · n +12 t (cos t − cos πn +1 ) = γ + 2 γ cos t + ... . From here the coefficients γ , . . . , γ n , a , . . . , a n are determined by the formulas γ j = ( n − j + 3) sin π jn +1 − ( n − j + 1) sin π ( j − n +1 n + 1) sin πn +1 · (1 + cos πn +1 ) ,a j = γ j − γ j +2 = 2 · tan π n + 1) · (1 − jn + 1 ) · sin π jn + 1 , j = 1 , . . . , n. It is assumed that γ n +1 = γ n +2 = 0. Example
For n = 5 by the formula (15) we get a = · tan π , a = √ · tan π , a = tan π , a = √ · tan π , a = · tan π . The graph and a fragment of the imageof the unit circle is displayed on the Figures 1 and 2. Figure 1.
Graphof the curve x ( t ) = ℜ (cid:16)P nj =1 a j e − ijt (cid:17) , y ( t ) = ℑ (cid:16)P nj =1 a j e − ijt (cid:17) for n = 5 t ∈ [0 , π ]. Figure 2.
Graphof the curve x ( t ) = ℜ (cid:16)P nj =1 a j e − ijt (cid:17) , y ( t ) = ℑ (cid:16)P nj =1 a j e − ijt (cid:17) for n = 5 , t ∈ [1 . , . z · p (1)0 ( z ) = 2 · tan π n + 1) · n X j =1 (1 − jn + 1 ) · sin π jn + 1 · z j , EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 17 and ℑ n e it · p (1)0 (cid:0) e it (cid:1)o = 2 (cid:0) − cos πn +1 (cid:1) n + 1 · sin t · Φ (1) n − ( t ) . Note that the polynomials (16) are not new. Suffridge [15] used polynomials(17) s k,n ( z ) = z + 1sin kπn +1 · n X j =2 n − j + 1 n · sin kπ jn + 1 · z j to obtain sufficient conditions of the univalency for the polynomials with the firstcoefficient 1 and with the leading coefficient a n = 1 /n. D.Dimitrov [3] established arelation between the polynomials (17) for k = 1 and Fej´er kernels Φ (1) n ( t ) . From therefor the univalent polynomials of the type q ( z ) = z + . . . he derived the sharp estimatesfor the ratio q ( − /q (1) . Let us note that Dimitrov’s estimate can be obtained fromTheorem 2, however the Theorem 2 does not follow from the results of [3] because theinitially considered mappings are not necessary univalent. Moreover, as it is provenin Lemma 3, a necessary condition of optimality - the image of the upper semi-disclies in the upper half plain - does not implies the univalency. One can judge about theunivalency only at the end of the proof when the optimal polynomial is constructed.The univalency of the polynomials (17) was established in [15] and since zp (1)0 ( z ) = 4 nn + 1 · sin π n + 1) · s ,n ( z )then the polynomial (16) is univalent.5.2. T=2 case.Theorem 3.
Let C ( t ) and S ( t ) be a pair of conjugate trigonometric polynomials (18) C ( t ) = n X j =1 a j cos(2 j − t, S ( t ) = n X j =1 a j sin(2 j − t, normalized by the condition n P j =1 a j = 1 . Consider the extremal problem J = inf n P j =1 a j =1 " max t ∈ [ , π ] {| S ( t ) | : C ( t ) = 0 } . Then J = 1 n . The function C ( t ) turns to be zero for t = π and any a , . . . , a n . Denote ρ ( a , . . . , a n ) = max t ∈ [ , π ] { S ( t ) : C ( t ) = 0 , S ( t ) > } . It is clear that J ≥ inf ( a , ... , a n ) { ρ ( a , . . . , a n ) } . The quantity inf { ρ ( a , . . . , a n ) } will be minimized on the set A R = ( ( a , . . . , a n ) : n X j =1 a j = 1 , n X j =1 | a j | ≤ R ) , where R is a sufficiently large number.For t ∈ (cid:0) , π (cid:1) the following inequalities C ( t ) =
12 sin t P nj =1 ˆ a j sin 2 jt ,S ( t ) =
12 sin t (cid:16)P nj =1 ˆ a j − P nj =1 ˆ a j cos 2 jt (cid:17) , are valid, where ˆ a j = a j − a j +1 , j = 1 , . . . , n (consider a n +1 = 0). It is easy to seethat P nj =1 ˆ a j = a . Since trigonometric polynomials 2 sin t · C ( t ) and P nj =1 ˆ a j − t · S ( t ) are conjugatewe can apply Theorem 1 and prove the following lemma Lemma 4.
Let C ( t ) = 0 , S ( t ) = γ , where t ∈ (cid:0) , π (cid:1) . Then trigonometricpolynomials (18) admit the presentation C ( t ) = 12 sin t · (cos 2 t − cos 2 t ) n − X j =1 α j sin 2 jt,S ( t ) = 12 sin t a + α − (cos 2 t − cos 2 t ) n − X j =1 α j cos 2 jt ! , where α j , j = 1 , . . . , n − are uniquely determined by the coefficients a , . . . , a n andparameter γ , moreover α γ sin t − a . Together with the function ρ ( a , . . . , a n ) we will consider the function ρ ( a , . . . , a n ) = max t ∈ [ , π ] n S ( t ) : t ∈ T ∪ n π o o , where T is the set of the points from the interval ( 0 , π ) where the function C ( t )changes the sign and the function S ( t ) is positive. Lemma 5.
There exists a pair of trigonometric polynomials { C ( t ) , S ( t ) } , such that ¯ ρ := inf ( a , ... , a n ) { ρ ( a , . . . , a n ) } = max t ∈ [ , π ] n S ( t ) : t ∈ T ∪ n π o o , where T is an intersection of the sets of sign changes of function C ( t ) over theinterval (0 , π ) , and the set of positivity of the function S ( t ) .Proof. Let A R = ( ( a , . . . , a n ) : n X j =1 a j = 1 , n X j =1 | a j | ≤ R ) . EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 19
The function ρ ( a , . . . , a n ) is continuous on the set A R , beside ( a , . . . , a n ), forwhich the maximal value of S ( t ) is achieved at zeros of the function C ( t ) where itdoes not change the sign. The lower limit of the function ρ ( a , . . . , a n ) at the pointof discontinuity is equal to the value of the function. So, it does achieve a minimalvalue ρ on the set A R , i.e. ρ = min ( a , ... ,a n ) ∈ A R { ρ ( a , . . . , a n ) } .It is clear that 0 ≤ ρ ≤ R > n − the maximum isachieved in the internal point of the set A R . The optimal pair { C ( t ) , S ( t ) } , wherethis maximum is achieved, is independent of R . Lemma 5 is proved. (cid:3) Since
T ∪ (cid:8) π (cid:9) is a subset of all zeros of the functions C ( t ), then ρ ≥ ρ , where ρ = inf ( a , ... .,a n ) ∈ A R { ρ ( a , . . . , a n ) } . Lemma 6.
If a polynomial C ( t ) has zero in (0 , π/ , then it cannot be optimal.Proof. Suppose that for the optimal polynomial C ( t ) the set T = { t , . . . , t q } , where0 ≤ q ≤ n − (cid:8) S ( t ) , ..., S ( t q ) (cid:9) = S ( t ) ,S ( t ) = S ( t j ) , j = 1 , ..., m (1 ≤ m ≤ q ) , and S ( t ) > S ( t j ) , j = m + 1 , . . . , q. The following three cases are possible: S ( t ) > S ( π , S ( π ≥ S ( t ) ≥ , S ( t ) < . Case 1.
Accordingly to Theorem 1 the trigonometric polynomials S ( t ) and C ( t )have the form C ( t ) = 12 sin t (cos 2 t − cos 2 t ) n − X j =1 α j sin 2 jt,S ( t ) = 12 sin t α + α − (cos 2 t − cos 2 t ) n − X j =1 α j cos 2 jt ! . Since S ( t ) =
12 sin t ( α + α ) , then α + α > . And since C (0) = 1 then(1 − cos 2 t ) n − X j =1 jα j = 1 , therefore n − X j =1 jα j > . Let us construct the auxiliary trigonometric polynomials S ( t ) and C ( t ) C ( θ, t ) = N ( θ ) 12 sin t (cos 2 t − cos 2 θ ) n − X j =1 α j sin 2 jt,S ( θ, t ) = N ( θ ) 12 sin t α + α − (cos 2 t − cos 2 θ ) n − X j =1 α j cos 2 jt ! , where the normalization factor N ( θ ) provides the sums of the coefficients for each ofthe polynomials C ( θ, t ) and S ( θ, t ) to be 1. For the polynomial C ( θ, t ) the set of signchanges is T θ = { θ, t , . . . , t q } . It is clear that C ( t ; t ) ≡ C ( t ), S ( t ; t ) ≡ S ( t ).The normalizing factor N ( θ ) is determined by the condition C ( θ,
0) = 1 , i.e. N ( θ ) = 1(1 − cos 2 θ ) P n − j =1 jα j . The polynomials C ( θ ; t ) and S ( θ ; t ) can be written in the following form C ( θ ; t ) = 1(1 − cos 2 θ ) P n − j =1 jα j ·
12 sin t (cos 2 t − cos 2 θ ) n − X j =1 α j sin 2 jt,S ( θ ; t ) = a + α − (cos 2 t − cos 2 θ ) P n − j =1 α j cos 2 jt (cid:16) (1 − cos 2 θ ) P n − j =1 jα j (cid:17) t . Let us show that for some θ the value ρ ( a , . . . , a n ) for the pair { C ( θ ; t ) , S ( θ ; t ) } isless then for the pair { C ( t ) , S ( t ) } , i.e. the pair { C ( t ) , S ( t ) } is not an optimal.Let compute S ( θ ; θ ) = 1(1 − cos 2 θ ) P n − j =1 jα j ·
12 sin θ (cid:16) a + α (cid:17) S ( θ ; t k ) = a + α − (cos 2 t k − cos 2 θ ) P n − j =1 α j cos 2 jt k (cid:16) (1 − cos 2 θ ) P n − j =1 jα j (cid:17) t k , k = 2 , . . . , m. Since S ( t ) = S ( t k ) , k = 2 , ..., m , then n − X j =1 α j cos 2 jt k = a + α sin t · sin t − sin t k cos 2 t k − cos 2 t > . Hence, the functions S ( θ ; θ ), S ( θ ; t k ), k = 2 , ..., m are decreasing with respect to theparameter θ .Now, let 0 < θ − t < ε . Because of the continuity of trigonometric polynomials by t and by all coefficients we have the following inequalities valid S ( θ ; θ ) < S ( t ) ,S ( θ ; t j ) < S ( t j ) , j = 2 , . . . , m , (cid:12)(cid:12) S ( θ ; t j ) − S ( t j ) (cid:12)(cid:12) < δ, j = m + 1 , . . . , q, (cid:12)(cid:12)(cid:12) S ( θ ; π − S ( π (cid:12)(cid:12)(cid:12) < δ for any δ with a proper choice of ε . These inequalities mean that the quantitymax n S ( θ ; θ ) , S ( θ ; t ) , . . . , S ( θ ; t q ) , S ( θ ; π o is less then max n S ( t ) , ..., S ( t q ) , S ( π o EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 21 at least for sufficiently small positive θ − t , i.e. the pair { C ( t ) , S ( t ) } is not anoptimal. Case 2.
Let evaluate S ( π a + α t ) n − X j =1 ( − j α j ! ,S ( θ ; π a + α + (1 + cos 2 θ ) P n − j =1 ( − j α j − cos 2 θ ) P n − j =1 jα j , and S ( t ; π ) = S ( π ). Since by the assimption S ( π ) ≥ S ( t ) =
12 sin t (cid:0) a + α (cid:1) ≥ P n − j =1 ( − j · α j ≥
0. Then the functions S ( θ ; π ), S ( θ ; θ ), S ( θ ; t k ), k = 2 , ..., m ,are decreasing by the parameter θ. Therefore, in this case the pair { C ( t ) , S ( t ) } cannot be an optimal. Case 3. If S ( t ) <
0, then a + α <
0. But S ( π ) >
0, therefore P n − j = m ( − j · α j ≥
0. Thus, in this case the function S ( θ ; π ) is decreasing by θ too, and the pair { C ( t ) , S ( t ) } cannot be an optimal. The lemma is proved and we can proceed withthe proof of Theorem 3. (cid:3) So, T = ∅ and therefore¯ ρ = min a + ... + a n =1 max t ∈ [0 ,π/ n | S ( t ) | : C ( t ) > , t ∈ (cid:16) , π (cid:17) , C ( t ) = 0 o =min a + ... + a n =1 n(cid:12)(cid:12)(cid:12) S (cid:16) π (cid:17)(cid:12)(cid:12)(cid:12) : C ( t ) > , t ∈ (cid:16) , π (cid:17)o The polynomial C ( t ) can be written as C ( t ) = cos t · ( γ + 2 γ cos 2 t + .. + 2 γ n cos 2( n − t ) , where the coefficients a , . . . , a n and γ , . . . , γ n are connected by a bijective rela-tions: γ s = n X j = s ( − s + j a j , s = 1 , . . . , n. It is clear that γ + 2 P nj =2 γ j = P nj =1 a j = 1 and S ( π n X j =1 ( − j +1 γ j = γ . Let ρ = min a + ... + a n =1 n(cid:12)(cid:12)(cid:12) S (cid:16) π (cid:17)(cid:12)(cid:12)(cid:12) : C ( t ) ≥ , t ∈ (0 , π o . Then ρ = min γ ,...,γ n ( | γ | : γ + 2 n X j =1 γ j = 1 , C ( t ) ≥ , t ∈ (0 , π ) . From the properties of Fej´er kernels ([2], 6.7, problem 50), it follows that the valueof a non-negative trigonometric polynomial of degree n − n . Moreover, if it is an even polynomial then the extremal values areachieved only at the points 2 kπ, k ∈ Z , and polynomial coefficients are determineduniquely. Fej´er condition implies that the inequality for the extremal polynomial C ( t ) γ ≤ n . Therefore, γ = C (0) n = n and ρ = n .Let us find ¯ ρ . To do that, consider one-parameter family of trigonometric polyno-mials C ε ( t ) = n X j =1 a εj cos(2 j − t, S ε ( t ) = n X j =1 a εj sin(2 j − t, where a ε = a + ε ε , a εj = a j ε , j = 2 , . . . , n. It is clear that n X j =1 a εj = a + ε ε + a ε + . . . + a n ε = 1 , and C ε ( t ) = C ( t )1 + ε + ε ε cos t, S ε ( t ) = S ( t )1 + ε + ε ε sin t. Note that C ε ( t ) > t ∈ (0 , π ) and ε > . Therefore,¯ ρ ≤ S ε ( π S ( π )1 + ε + ε ε . If ε → ρ ≤ S ( π ) = ρ . Since ¯ ρ ≥ ρ , then ¯ ρ = ρ = n .Let us find the coefficients a , . . . , a n . Using Fej´er kernel we obtain C ( t ) = (cid:18) sin ntn sin t (cid:19) cos t = cos t n + 2 n X j =2 n − j + 1 n cos 2( j − t ! . Since a j = γ j + γ j +1 , j = 1 , . . . , n, assuming γ n +1 = 0, then(19) a j = 2( n − j ) + 1 n , j = 1 , . . . , n. Let compute max t { | S ε ( t ) | : C ε ( t ) = 0 } = 11 + ε (cid:18) n + ε (cid:19) . From here J = 1 n .The proof of the theorem is completed.The formula (11) implies µ n (2) = n . EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 23
Corollary
Let C ( t ) and S ( t ) be a pair of trigonometric polynomials (18), normal-ized by the condition P nj =1 a j = 1. Let T be a set of sign changes of the function C ( t ). Then min ( a , ... , a n ) max t { | S ( t ) | : t ∈ T } = 1 n . Moreover, the coefficients of the extremal pair of the trigonometric polynomials aredefined in a unique way.Accordingly to Theorem 3 the optimal polynomial mapping for T = 2 is z · (cid:16) p (2)0 ( z ) (cid:17) = z · n X j =1 n − j ) + 1 n · z j − ! , and the coefficients of the polynomial p ( z ) are connected with the Fej´er kernel Φ (2) n ( t )by the relation ℜ n e it · p (2)0 (cid:0) e it (cid:1)o = 1 n · cos t · Φ (2) n − (2 t ) . Figure 3.
The graphof the curve x ( t ) = ℜ (cid:16)P nj =1 a j e − i (2 j − t (cid:17) ,y ( t ) = ℑ (cid:16)P nj =1 a j e − i (2 j − t (cid:17) for n = 5 , t ∈ [0 , π ]. Figure 4.
The graphof the curve x ( t ) = ℜ (cid:16)P nj =1 a j e − i (2 j − t (cid:17) ,y ( t ) = ℑ (cid:16)P nj =1 a j e − i (2 j − t (cid:17) for n = 5 , t ∈ [0 . , . Theorem 4.
The mapping zp (2)0 ( z ) is univalent in D . The proof follows from the univalency of the polynomial s n, n − ( iz ) [15] and theidentity zp (2)0 ( z ) = − i n n − · s n, n − ( iz ) . Corollary
The mapping z (cid:16) p (2)0 ( z ) (cid:17) is univalent in D . This follows from the univalency of zp (2)0 ( z ) . Figure 5.
The graphof the curve x ( t ) = ℜ (cid:18) e − it (cid:16)P nj =1 a j e − i ( j − t (cid:17) (cid:19) ,y ( t ) = ℑ (cid:18) e − it (cid:16)P nj =1 a j e − i ( j − t (cid:17) (cid:19) for n = 5 , t ∈ [0 , π ]. Figure 6.
The graphof the curve x ( t ) = ℜ (cid:18) e − it (cid:16)P nj =1 a j e − i ( j − t (cid:17) (cid:19) ,y ( t ) = ℑ (cid:18) e − it (cid:16)P nj =1 a j e − i ( j − t (cid:17) (cid:19) for n = 5 , t ∈ [1 . , . Figure 7.
The graph from the Fig. 5 (the outer) and from the Fig.1 (the inner). It is quite remarkable that not only the values µ n (1) and µ n (2) are very close but also the whole optimal images.6. Optimal stabilization of chaos T = 1 . For the one-parameter logistic mapping(20) f : [0 , → [0 , ,f ( x ) = h · x · (1 − x ) , ≤ h ≤ x ∗ = 1 − h of the open-loop system x k +1 = f ( x k ) , x k ∈ R , k = 1 , , ... is unstable for h ∈ (3 , . The multiplier µ ∈ [ − , − µ (1) = cot π < µ ∗ , µ (1) = cot π > µ ∗ . Therefore n ∗ = 1 and the minimal depth of the prehistory in the delayed feedbackis N ∗ = 1 . The optimal strength coefficient is ǫ = 1 / u = −
13 ( f ( x k ) − f ( x k − )) . EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 25
Figure 8.
Quasi-stochastic dynamicsof the solutions to thelogistic equations for h = 4 . Figure 9.
Dynam-ics of the solutions tothe logistic equations for h = 4 closed by sta-bilizing control u = − ( f ( x k ) − f ( x k − )) . Figure 10.
The control u = − ( f ( x k ) − f ( x k − )) . T = 2 . If h > , the equilibrium of the logistic equation lost stability and 2-periodic cycle ( η , η ) emerges, where η = 1 + h − √ h − h − h , η = 1 + h + √ h − h − h . The cycle multiplier is µ = − h + 2 h + 4 . For h ∈ (3 , √
6) the multiplier decreasesfrom 1 to −
1. For h = 1 + √ h ∈ (1 + √ ,
4] then µ ∈ [ − , − , i.e. µ ∗ = 4 . We want to stabilize 2-cycle for all µ ∈ [ − , − . Since µ (2) = 4 = µ ∗ and µ (2) = 9 > µ ∗ then n ∗ = 3 , N ∗ = ( n ∗ − T = 4 . The optimal strength coefficients are ǫ = a + a = 49 , ǫ = a = 19 . The requiredstabilizing control is u k = −
49 ( f ( x k ) − f ( x k − )) −
19 ( f ( x k − ) − f ( x k − )) . Searching for cycles of arbitrary length
Algorithm.
In the previous section to stabilize cycles of lengths 1 and 2 weused direct DFC regulators (2). Same regulators can be applied to stabilize cyclesof the arbitrary length. Indeed, each vale of T -cycle is a fixed point for T -folded f Figure 11.
Dynam-ics of the solutions tothe logistic equations for h = 4 closed by sta-bilizing control u k = − ( f ( x k ) − f ( x k − )) − ( f ( x k − ) − f ( x k − )) . Figure 12.
Thecontrol u k = − ( f ( x k ) − f ( x k − )) − ( f ( x k − ) − f ( x k − )) . mapping: x k +1 = f ( T ) ( x k ) , f ( T ) ( x k ) = f ( f ( T − ( x k )) , f (0) ( x k ) = f ( x k ) . The question of cycle stability is reducing to the question of stability for the fixedpoints of the mapping f ( T ) , that containes cycles of the length T. The value of the cyclemultiplier is independent of the choice of the fixed point belonging to the consideringcycle.Indeed, µ = ddη f ( T ) ( η j ) = ddη f ( f ( T − ( η j )) = dfdη ( η j − ) ddη f ( T − ( η j ) = ...dfdη ( η j − ) · dfdη ( η j − ) · . . . · dfdη ( η j − T ) = dfdη ( η ) · dfdη ( η ) · . . . · dfdη ( η T )Note that fixed points of the mapping f ( T ) are also the fixed points of the mapping f ( T ) , if T = mT ( m - is an integer number).Thus, the problem of stabilizing T -cycles (1), (2) can be reduced to the problem ofthe stabilizing the equilibrium for the system(21) x k +1 = f ( T ) ( x k ) , x k ∈ R , k = 1 , , ... by the control(22) u k = − n − X j =1 ε j (cid:0) f ( T ) ( x k − j +2 ) − f ( T ) ( x k − j ) (cid:1) , < ε < , j = 1 , . . . , n − . The solution to the problems (21), (22) is given by Theorem 2: ε j = 1 − P ji =1 a i , where a i ( i = 1 , . . . , n −
1) are defined by (15).
EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 27 If T is even, then the problem of stabilization of T -cycle (1), (2) can be reduced toto the problem of stabilizing for the 2-cycle of the system(23) x k +1 = f ( T/ ( x k ) , x k ∈ R , k = 1 , , ... by the control(24) u k = − n − X j =1 ε j (cid:0) f ( T/ ( x k − j +1 ) − f ( T/ ( x k − j ) (cid:1) , < ε < , j = 1 , . . . , n − . The solution to the problems (23), (24) is given by the Theorem 3: ε j = 1 − P ji =1 a i , where a i are defined by (19).In general case, when T = mT ( m - is an integer number) the problem of the T -cycle stabilization is reducing to the problem of the stabilizing of the cycle of thelength T of the auxiliary system(25) x k +1 = f ( m ) ( x k ) , x k ∈ R , k = 1 , , ... by the control(26) u k = − n − X j =1 ε j (cid:0) f ( m ) ( x k − T j +1 ) − f ( m ) ( x k − T j ) (cid:1) , < ε j < , j = 1 , . . . , n − . Let us note the important difference between the suggested method of stabilizationin this article and majority of currently known ones (in particular, from famous OGYmethod [11]). Namely, the control is applying at any instance of time, not necessaryin a neighborhood of the desired cycle. It is not necessary to know the cycle a priori.Moreover, one control allows to stabilize at once all cycles of the given length withnegative multipliers (with a proper choice of the number n of strength coefficients).To stabilize the specific cycle it is sufficient for the sequence of the initial points tofall into basin of attraction that corresponds to the fixed point of the mapping f ( T ) .The problems (1), (2) and (21), (22) seem to be equivalent at the first glance. Butthis is not the case. The basins of attractions of the investigated equilibriums orcycles in this problems, generally speaking, are different. This is why we need to havea collection of controls that stabilize cycles of the lengths 1 , , ..., T for all n. With thehelp of this controls it can be possible to stabilize the cycles of the arbitrary length.By varying parameters m and T ∈ { , , ..., ˆ T } it might be possible to increase thebasin of attraction for the investigating cycles.One of the possible applications of the suggested methods could be verificationof the existence of the periodic orbits of a given nonlinear mapping. If such orbitsare non-stable and their multipliers are negative, then such orbits can be detected bystabilization. It is possible to find all unstable periodic orbits with negative multiplierswith sufficiently large basins of attraction.Let the system contains s cycles with unknown multipliers { µ , . . . , µ s } ⊂ [ − µ, − , and we don’t know the value of s. However, it is not difficult to get an estimate forthe value µ ∗ from the properties of the derivative of the initial mapping or from the T iteration of this mapping. Then the number n in the stabilizing control, whichdepends on µ ∗ can be computed from the condition m n ( T ) > µ ∗ , m n − ( T ) ≤ µ ∗ . (The functions µ n (1) and µ n (2) are obtained in the theorems 2 and 3. ) On practice,there is no need to find the value n - it might be consecutively increasing until newstable cycles stop to appear.7.2. Examples.
It is known that the logistic mapping (20) for h = 4 has two 3-cycles: { η ≈ . , η ≈ . , η ≈ . } and { η ≈ . , η ≈ . , η ≈ . } . The first cycle has a positive multiplier and the second one has a negative one. Letus use the suggested methodology to find the second 3-cycle. To do that, considerthe system(27) x k +1 = f (3) ( x k ) , x k ∈ R , k = 1 , , ... closed by the control(28) u k = − n − X j =1 ε j (cid:0) f (3) ( x k − j +1 ) − f (3) ( x k − j ) (cid:1) , < ε j < , j = 1 , ..., n − . For n = 4 the system (27), (28) demonstrates three stable equilibriums x ∗ ≈ . , x ∗ ≈ . , x ∗ ≈ .
75 (see Fig. 13) The first and the second one correspond to the non-stable 3-cycle of the logistic mapping with negative multiplier, and the third onecorresponds to the unstable equilibrium. The stabilizing control vanish with increas-ing of the number of iterations k (see Fig. 14). Figure13.
Stabilization ofthe 3-cycle of the lo-gistic equations bystabilizing the equilib-rium (27) by the control(28).
Figure 14.
The con-trol (28).Same way was conducted to search for the cycles of the length 7. It should bementioned that it is not always true that the trajectory quickly arrived at the basinof attraction of the fixed point of the mapping (see Fig 15) and not for every initialpoint the stabilizing control start to decrease rapidly (see Fig. 16).A little bit different situation can be observed in the stabilizing of the cycles ofthe length 8. Besides the equilibriums of the auxiliary systems that corresponds
EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 29
Figure15.
Stabilization ofthe 7-cycle of the lo-gistic equations bystabilizing the equilib-rium of the auxiliaryequation.
Figure 16.
The con-trol stabilizing the cy-cles of the length 7of the logistic equation( n = 22) . to the actual cycles of the lengths 8, one can see a stabilization of the equilibriacorresponding to the cycles of the lengths 1,2,4 (see Fig. 17). It can be concludedfrom the analysis of the behavior of the graph for the corresponding control (se Fig.18). Namely, not for every initial value the control is vanishing. For some initialvalues the control approaching the 2-cycles and 4-cycles. Figure 17.
Stabi-lization of the 8-cycleof the logistic equationsby stabilizing the equi-librium of the auxiliaryequation.
Figure 18.
The con-trol stabilizing the cy-cles of the length 8of the logistic equation( n = 40) . For the stabilization of 8-cycles the following auxiliary systems turns out to bemore effective.(29) x k +1 = f (4) ( x k ) , x k ∈ R , k = 1 , , ... closed by the control(30) u k = − n − X j =1 ε j (cid:0) f (4) ( x k − j +2 ) − f (4) ( x k − j ) (cid:1) , < ε j < , j = 1 , ..., n − . To stabilize the 2-cycles for the system (29), (30) it is enough to take n = 19 (seeFig. 19). In this case the stabilizing control does vanish (see Fig. 20). Figure 19.
Stabilization of the 8-cycle of the logistic equations bystabilizing the 2-cycle of the auxiliary equations (29) by the control(30).
Figure 20.
The control (30).One of the traditional methods of cycle stabilization in non-linear discrete systems ispredictive control schedule [13]. In [17] the predictive control was used for stabilizationof the cycles of the length from 1 to 6 in the system with sudden occurrence of chaos,abbreviated SOC: f ( x ) = ha − ha (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12) + x, − ≤ ha ≤ √ . Below we will illustrate the application of the DFG method for the stabilization of3-cycle (Fig 21,22) and 7-cycles (Fig 23, 24) in a system with SOC.7.3.
Practical aspects.
Let us mention several advantages of the proposed modifiedDFC method in the article compare to the predictive control. The system withpredictive control in fact is a system with advancing argument. In such systems ithappen very often that cause-effect relations are broken, that can lead to various non-controlling effect. A system with modified DFC is a system with delay, the controluses not predicted values rather real. Moreover, instead of the values of the previousinstances of time one needs to know just the dispersion of the values of the functionin a prior cycle. As a result, the rate of the convergence is increasing and the totalnumber of computations is decreasing. For instance, to stabilize the 7-cycle of thelogistic equation the predictive control requires 10000 iterations while MDFC just 700.Predictive control has limitations on the length of stabilizing cycles, which is caused
EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 31
Figure 21.
Stabi-lization of 3-cycle forSOC system by stabi-lizing the equilibrium ofthe auxiliary equationby the DFC (n=7).
Figure 22.
The con-trol stabilizing 3-cyclefor SOC system.
Figure23.
Stabilization of7-cycle for SOC systemby stabilizing the equi-librium of the auxiliaryequation by the DFC(n=14).
Figure 24.
The con-trol stabilizing 7-cyclefor SOC system.by the accumulation of the rounding errors of the computational procedures. MDFCis robust to the parameters of the systems and to the cycle multipliers. Moreover,the rate of robustness can be improved.Let us explain this moment more precisely. Consider the control (2) where thestrength coefficients are defined by ε j = 1 − P ji =1 a i , and a i , i = 1 , ..., n − D under the polynomial mapping(31) F ( z ) = n X j =1 a j z j , i.e. (cid:26) µ : 1 µ ∈ F ( ¯ C \ D ) (cid:27) . Here ¯ C is the extended complex plane. The image of the exterior of the unit disc andthe inverse sets are displayed on the Fig. 25. Figure 25.
The image and the exterior of the unit disc by the poly-nomial mapping (31) and the inversion with respect to the unit circle(with n = 5)The inverse image in the neighborhood of some critical points could be tangent tothe negative real axis, which might negatively affect the robust properties of the con-trol if the values of multipliers are close to these critical points. Since the multipliersusually are not known in advance then such situation cannot be excluded. Therefore,it is necessary to separate this set from the real line. It can be done by applyingthe following procedure which we call ε -trick. Namely, define new coefficients of thecontrol ε j = 1 − P ji =1 a εi , where(32) a ε = a + ε ε , a ε = a + ε ε , . . . , a εn = a n + ε ε , ε > . It is clear that the normalizing conditions P ni =1 a εi = 1 is still valid and the control stillstabilize the system. In this way the boundary of the region of location of admissiblemultipliers is shifted off the real axis. On the other hand, the general linear size ofthe region decreases in this case. The Figure 26 displays the exterior of the imagesof the unit disc under the mappings 1 P j =1 a j z j and 1 P j =1 a ǫj z j where ε = 0 .
005 (Fig.26).
Figure 26.
The illustration of the ε -trick (with n = 5 , ε = 0 . EJ´ER AND SUFFRIDGE POLYNOMIALS IN THE DFC THEORY 33
It makes sense to use the formulas (32) of the variation of the extremal coefficientsin the case of 2-cycle stabilization as well.8.
Conclusion
In this article, the actual problem of stabilization of not known in advance unstableperiodic orbits of discrete chaotic systems is considered. The approach for solving thestabilization problem is based on the use of nonlinear delayed feedback control (DFC),proposed in [19] to stabilize an equilibrium and modified in [10] for cycles (there forthe case n = 2). Using non-linear (non-linearity must be related to the originalsystem ) DFC is the only way to avoid restrictions on the size of the area of possiblelocalization of multipliers. For example, in [18] it is shown that the linear DFC whith n = 2 is applicable only in the case of localization of multipliers in the region [ − , n does not provide substantial advantages. Namely,if the diameter of the areas of possible localization of multipliers are greater than 16in the general case, or greater than 4 in the case of a simply connected region, thenthe linear DFC does not solve stabilization problem.As it is shown in this article, the problem of limitation of the size of the regionof multipliers location can be circumvented by using a nonlinear DFC with multipledelays , i.e., when n > . For T = 1 , − µ ∗ ,
1) where the valueof µ ∗ depends on the number n of coefficients in delayed feedback loop. Theorems 2and 3 implies the asymptotic dependence on the linear size of the region of locationof multiples from n at T = 1 , µ ∗ ∼ n . A. Solyanik [14] claims that this asymptotic dependence is still valid for an arbi-trary T. He also claim an algorithm for constructing the coefficients of direct control,stabilizing cycles of arbitrary length.Other generalizations of the proposed scheme are possible. In particular, the pointof interest is a construction of direct stabilizing controls for multidimensional map-pings. Preliminary investigations indicates that the strength coefficients in these tasksare also associated with Fej´er polynomial coefficients .9.
Acknowledgement
The authors would like to thank Alexey Solyanik for valuable comments and deepinsight into considered problems and to Paul Hagelstein for interesting discussionsand for the help in preparation of manuscript.
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