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^{1}

It is known that the solutions of a second order linear differential equation with periodic coefficients are almost always analytically impossible to obtain and in order to study its properties we often require a computational approach. In this paper we compare graphically, using the Arnold Tongues, some sufficient criteria for the stability of periodic differential equations. We also present a brief explanation on how the authors, of each criterion, obtained them. And a comparison between four sufficient stability criteria and the stability zones found by perturbation methods is presented.

The second order differential equations are often encountered in engineering and physical problems, and they have been studied for more than a hundred years. The Hill equation is a particular and representative equation among the linear periodic equations, and it receives its name after the work of W. Hill on the lunar perigee. The Hill equation can be used to describe from the simples dynamical systems to the more complex systems; from a child playing on a swing or a spring mass system [

For any linear T-periodic ODE, the knowledge of the state transition matrix in one period; t ∈ [ 0, T ] , gives us sufficient information to know the solution in any t ∈ ℝ . The main problem with the determination of the stability of a periodic system solutions lies in the calculation of the state transition matrix, which is almost always impossible to obtain analytically. Hence it is desirable to find some criteria for determining the stability of solutions without the need of obtain them. Zhukovskii [

n 2 π 2 T 2 ≤ p ( t ) ≤ ( n + 1 ) 2 π 2 T 2 then the solutions of the periodic system are stable.

^{1}Let x 1 and x 2 be two linearly independent solutions of a periodic differential equation subject to the initial conditions x 1 ( t 0 ) = 1 , x ˙ 1 ( t 0 ) = 0 , x 2 ( t 0 ) = 0 and x ˙ 1 ( t 0 ) = 1 , then the discriminant is equal to x 1 ( T ) + x ˙ 2 ( T ) , T is the minimum period of the differential equation.

Lyapunov in his celebrated work “The general problem of the stability of motion” [^{1} of the periodic system (see section 2) and obtained a variety of sufficient stability conditions, being the most known the one here presented. Authors such as Borg [

The aim of the present work is to collect and graphically display some of the most known and efficient sufficient criteria for the stability of the periodic differential equation x ¨ + ( α + β p ( t ) ) x = 0 , p ( t + T ) = p ( t ) , known as Hill’s equation. There is a vast number of stability criteria, in the literature, they are based on different approaches, here we present an easy explanation of four of those approaches starting with: a) the approximation of the discriminant due to Lyapunov followed by; b) canonical forms (Hamiltonian structure); c) Sturm-Liouville equation properties and ending with d) another discriminant approximation due to Shi [

Mathieu x ¨ + ( α + β 2 c o s ( t ) ) x = 0 Meissner x ¨ + ( α + β s i g n ( c o s ( t ) ) ) x = 0 Lyapunov x ¨ + ( α + β 32 25 ( c o s ( t ) + 3 4 c o s ( 2 t ) ) ) x = 0

the scaling factor 2 and 32 25 of the excitation functions related to the Mathieu and Lyapunov equations were added so its L 2 [ 0,2 π ] norm be equal to 2 π that is, ∫ 0 2π | p ( t ) | 2 d t = 2 π , for each excitation function. The excitation function of the classical Mathieu equation is p ( t ) = cos ( t ) , if we want the excitation function to have L 2 [ 0,2 π ] = 2 π we must multiply the function cos ( t ) by a constant a, since ∫ 0 2π | a cos ( t ) | 2 d t = a 2 π then, a = 2 ; by doing a similar procedure for the classical Meissner ( p ( t ) = s i g n ( cos ( t ) ) ) and Lyapunov ( p ( t ) = cos ( t ) + 3 4 cos ( 2 t ) ) we get the equations given above.

This work is structured as follows: Section 2 is dedicated to introduce some basic concepts on periodic differential equations and its solutions; in section 3 we give the discriminant approximation made by Lyapunov and the first two criteria are presented; section 4 is devoted to the study of canonical (Hamiltonian) systems, some properties of such systems solutions are described and four criteria are presented; In section 5 two criteria based on Sturm-Liouville equation properties are exhibit, both criteria follows from solutions proposed by Hochstadt in [

Consider the linear differential equation with periodic coefficients

y ¨ + ( α + β p ( t ) ) y = 0 (1)

where p ( t + T ) = p ( t ) is a periodic piece-wise continuous real function with zero average, i.e. 1 T ∫ 0 T p ( t ) d t = 0 , and α , β are real constants; Equation (1) is known as Hill equation. One can prove that (1) is equivalent to the system

y ¨ + ( λ + q ( t ) ) y = 0 (2)

where q ( t ) is a T periodic real function and λ is a real constant. Throughout the document we will use system (1) or (2) indistinctly.

We say that a system is stable if and only if all its solutions are bounded in the whole real line. It is known that for some values of the parameters α and β (or λ ) the Equation (1) has bounded solutions and for some others the solutions grow without bounds. The plane of parameters α − β can be splited into stable zones (where all solutions are stable) and unstable zones (where at least one solution is unstable). Stable and unstable regions are separated by some curves known as transition curves, these transition curves are defined by having at least one periodic or anti-periodic solution [

By the usual change of variables, x 1 = y and x 2 = y ˙ , Equation (1) may be rewritten as

x ˙ = [ 0 1 − α − β p ( t ) 0 ] ︸ ≜ B ( t ) x (3)

where x = [ x 1 x 2 ] ; and B ( t ) = B ( t + T ) ∈ ℝ 2 × 2 . Notice that (3) may also be written as the Hamiltonian system

x ˙ = J H ( t ) x (4)

where H ( t ) is the symmetric and T periodic matrix H ( t ) = [ α + β p ( t ) 0 0 1 ] and J = [ 0 1 − 1 0 ] . The Equation (4) may be taken as a definition of Hamiltonian system and it will be used in section 4.

Let Φ ( t , t 0 ) be the state transition matrix of the system (3), from Floquet Theorem it is known that the matrix Φ ( t , t 0 ) may be written as a multiplication of three matrices, two of them are time dependent matrices and one constant matrix; one of the time independent matrices is bounded and periodic, and the other is an exponential one, the former gives us information about the phase of the solutions and the latter contains information about the growth of the solutions, see [

Φ ( t , t 0 ) = F − 1 ( t ) e R ( t − t 0 ) F ( t 0 ) (5)

where F ( t ) = F ( t + T ) is a real bounded matrix, and R is a 2 × 2 constant matrix not necessarily real. Factorization (5) is due to Floquet [

Φ ( T , 0 ) = M = e R T (6)

so, the growth of the system solutions are related with monodromy matrix M . For better understanding notice that for all t ≥ 0 , t = k T + τ , τ ∈ 0, T ) and for some positive integer k any solution of (3) can be written as

x ( t ) = Φ ( t , 0 ) x ( 0 ) = Φ ( k T + τ , k T ) Φ ( k T , ( k − 1 ) T ) ⋯ Φ ( T , 0 ) x ( 0 ) = Φ ( k T + τ , k T ) Φ k ( T , 0 ) x ( 0 ) = Φ ( τ , 0 ) M k ( T ) x ( 0 ) (7)

thus one can conclude the following

Theorem 2.1. Let μ i be the eigenvalues of the monodromy matrix M . The system (3) is:

a) Asymptotically stable if and only if all the eigenvalues μ i of the monodromy matrix are | μ i | < 1 ;

b) Stable if and only if all | μ i | ≤ 1 and if any μ i has modulo one then it must be a simple root of the minimal polynomial of M ; and

c) Unstable if and only if there is a μ i such that | μ i | > 1 or if all | μ i | ≤ 1 and there exists a μ j such that | μ i | = 1 and it is a multiple root of the minimal polynomial of M .

Since Equation (3) can be written as a Hamiltonian system, the matrix Φ ( t , t 0 ) is a symplectic matrix^{2} [ [

Let f ( t ) and g ( t ) be real solutions of (2) subject to the initial conditions

f ( 0 ) = 1 g ( 0 ) = 0 f ˙ ( 0 ) = 0 g ˙ ( 0 ) = 1 (8)

Then the state transition matrix associated to (2) is

Φ ( t , 0 ) = [ f ( t ) g ( t ) f ˙ ( t ) g ˙ ( t ) ] (9)

and the characteristic multipliers^{3} are solutions of the characteristic equation

det [ M − μ i I 2 ] = μ i 2 − 2 A ( λ ) μ i + 1 = 0 (10)

^{2}We say that a matrix B is symplectic if the condition B ′ J B = J is fulfilled. Symplectic matrices are of even order and they form a group [

^{3}The monodromy matrix eigenvalues μ i are usually called characteristic multipliers or just multipliers.

^{4}The fact that the independent term be equal to one follows from the symplectic matrix property that states that the determinant of a symplectic matrix is equal one.

where the independent term is equal to one because of the Liouville theorem^{4} [

A ( λ ) = 1 2 ( f ( T ) + g ˙ ( T ) ) = 1 2 T r a c e ( M ) (11)

is known as the characteristic constant or the discriminant associated to (2) and it plays a fundamental roll on the determination of the stability of the system, the Trace operator in (11) is the sum of the main diagonal entries of a square matrix. By Floquet theory, the condition for the solutions, f ( t ) and g ( t ) , to be stable may be restated as: If A 2 ( λ ) < 1 , both solutions are stable; if A 2 ( λ ) > 1 one solution is stable and one unstable; if A 2 ( λ ) = 1 then one solution is periodic when A ( λ ) = 1 , (or anti-periodic when A ( λ ) = − 1 ); and the second solution may or may not be periodic (anti-periodic).

The Haupt oscillation Theorem asserts that the λ real line can be split into alternating intervals known as stability and instability intervals, the former are characterized by | A ( λ ) | < 1 and the latter by | A ( λ ) | > 1 , the endpoints of the intervals | A ( λ ) | = 1 are characterized by values of λ for which the system (2) has at least one periodic solution. The Haupt oscillation Theorem [

Theorem 2.2. For the differential equation x ¨ + ( λ + q ( t ) ) x = 0 . There exists an infinite sequence

λ 0 < λ 1 ≤ λ 2 < λ 3 ≤ λ 4 < ⋯ (12)

such that A ( λ i ) = 1 . There exists a second infinite sequence

λ ′ 1 ≤ λ ′ 2 < λ ′ 3 ≤ λ ′ 4 < ⋯ (13)

such that A ( λ ′ i ) = − 1 . Both sequences do not have accumulation points, λ i → i → ∞ ∞ and λ ′ i → i → ∞ ∞ . These two sequences interlace such that

λ 0 < λ ′ 1 ≤ λ ′ 2 < λ 1 ≤ λ 2 < λ ′ 3 ≤ λ ′ 4 < λ 3 ≤ λ 4 < ⋯ (14)

whenever λ lies in one of the intervals

( λ 0 , λ ′ 1 ) , ( λ ′ 2 , λ 1 ) , ( λ 2 , λ ′ 3 ) , ⋯ , | A ( λ ) | < 1 (15)

if λ lies in

( − ∞ , λ 0 ) , ( λ ′ 1 , λ ′ 2 ) , ( λ 1 , λ 2 ) , ⋯ , | A ( λ ) | > 1 (16)

The proof of the Theorem is based on the fact that the functions A ( λ ) − 1 = 0 and A ( λ ) + 1 = 0 are entire functions of the real variable λ and its order of

growth for | λ | → ∞ is exactly 1 2 , see [

Notice that the Theorem 2.2 establishes conditions, in terms of λ , for the solutions of (2) to be stable; Theorem 2.2 is easily restated so that the stability conditions depend on the parameters α and β of the system (1). It is well known that the stability of solutions of any Hill equation, of the form (1), can be represented as a stability chart in the plane of parameters α − β , unstable zones are called Arnold tongues.

In [

A ( λ ) = A 0 − A 1 + A 2 + ⋯ + ( − 1 ) n A n + ⋯ (17)

where each coefficient is defined as a definite n-multiple integral, that is

A 0 = 1 , A 1 = T 2 ∫ 0 T p ¯ ( t ) d t

A 2 = 1 2 ∫ 0 T d t 1 ∫ 0 t 1 ( T − t 1 + t 2 ) ( t 1 − t 2 ) p ¯ ( t 1 ) p ¯ ( t 2 ) d t 2 (18)

⋮

A n = 1 2 ∫ 0 T d t 1 ∫ 0 t 1 d t 2 ⋯ ∫ 0 t n − 1 ( T − t 1 + t n ) ( t 1 − t 2 ) ⋯ ( t n − 1 − t n ) ⋅ p ¯ ( t 1 ) ⋯ p ¯ ( t n ) d t n (19)

notice that the sub-index of each coefficient A n is equal to the order of the n-multiple integral, that is, the coefficient A 3 requires a triple definite integral to be calculated, A 4 requires a forth order definite integral to be calculated, and so on.

In [

A n − 1 A 1 − n A n > 0 (20)

For details of the inequality (20) see appendix A. Using (20) Lyapunov got the following

Criterion 3.1. (Lyapunov 1). If p ¯ ( t ) ≥ 0 , and if it satisfies the condition

T ∫ 0 T p ¯ ( t ) d t ≤ 4 (21)

then the solutions of x ¨ + p ¯ ( t ) x = 0 are stable.

Remark 3.2. As far as knowledge of the authors, the only reference in which this criterion is graphically shown is [ [

Before proceeding, we shall call the interval ( − ∞ , λ 0 ) the zeroth instability zone, the intervals ( λ ′ 1 , λ ′ 2 ) and ( λ 1 , λ 2 ) the first and the second instability zone and so on. Similarly the intervals ( λ 0 , λ ′ 1 ) , ( λ ′ 2 , λ 1 ) , ( λ 2 , λ ′ 3 ) are called the zeroth, first and second stability zones respectively.

Next criterion can be proved by using the above described Lyapunov method [

Criterion 3.3 (Lyapunov 2). If

p ¯ ( t ) ≥ − a 2 (22)

and

∫ 0 T p ¯ ( t ) d t ≥ 0, ∫ 0 T ( p ¯ ( t ) + a 2 ) ≤ 2 a c o t h ( a T 2 ) (23)

where 0 ≤ a 2 ≤ π 2 T 2 . Then, p ¯ ( t ) belongs to the zero stability domain.

1 and Lyapunov 2 on Mathieu, Meissner and Lyapunov equations.

The blue and red areas are the stability zones found by criterion 0.3 and criterion 0.5 respectively. From the Lyapunov 2 criterion, one can see that the

parameter a must fulfill the inequality 0 < a < 1 2 . The stability area found by the

Criterion 0.5 is obtained by calculating the stability area defined by the equation (23) and 500 different values of a, equispaced in the interval [ 0,0.5 [ , and then, merging the 500 resulting areas.

From

c o s ( t ) + 3 4 c o s ( 2 t ) are all equal to zero, so both criteria depends only on the real constants α and a .

For more criterion based on Lyapunov method see [

Consider a second order system in canonical (Hamiltonian) form

x ˙ = J H ( t ) x (24)

where x = [ x 1 x 2 ] , H ( t ) is a symmetric real periodic matrix H ( t + T ) = H ( t ) = H T ( t ) and J is the skew symmetric matrix defined in section 2.

The state transition matrix of (24) may be expressed as

Φ ( t , 0 ) = F ( t ) e t K Φ ( 0 , 0 ) = I 2 (25)

^{5}In [

where F ( t + T ) = F ( t ) or F ( t + T ) = − F ( t ) , det ( F ( t ) ) = 1 , F ( t ) is a real matrix function, and K is a real matrix with T r ( K ) = 0 , see appendix B. It can be proved that matrix K could be defined as^{5}

K = 1 T ln ( ± M ) (26)

where the sign ± is chosen so that K be real. From the factorization (25) we can notice that the stability of the canonical system (24) depends on the exponential matrix e K T and therefore on the matrix K; it can be proved that the matrix K is similar to one of the three matrices shown in

^{6}Coexistence refers to the simultaneous existence of two linearly independent solutions of period T or 2T of (1) or (24).

with ϵ ≠ 0 , see ^{6}; and, if K ∈ O then both solutions are bounded (the characteristic multipliers are complex, lie on the unit circle and are distinct from ± 1 ). All of these properties are summarized in the

Remark 4.1. The subset Π defines the transition boundaries, i.e., Π defines lines on the α − β plane separating the stable zones from the unstable ones. Moreover if K ∈ Π and ϵ ≠ 0 then (24) has one periodic stable solution x ( t ) and one unstable solution of the form y ( t ) = y 1 ( t ) + t x ( t ) .

Let Ω be the set of real continuous function matrices F ( t ) with F ( t + T ) = ± F ( t ) and det ( F ( t ) ) = 1 . And, let x = F ( t ) a be a solution of Hamiltonian system (24), where a is a non-zero arbitrary vector and let φ x denote the rotation of the solution x in time T, since F ( T ) = ± I 2 it follows that φ x = n π . Let Ω n denote the set of matrices F ( t ) such that the rotation over a period T is φ x = n π , and Ω = ∪ n = − ∞ ∞ Ω n , each of the subsets Ω n are disjoint sets [

By the above mentioned properties, of the subsets Π , H , O and Ω n , we can say that the symmetric matrix H ( t ) in (24) belongs to one of the subsets H n ≜ Ω n × H , Π n ≜ Ω n × Π or O n ≜ Ω n × O if and only if the pair ( F ( t ) , K ) defined by the state transition matrix of (24) are: F ( t ) ∈ Ω n , and K belongs to H , Π or O respectively, that is, H ( t ) ∈ H n if and only if Φ ( t ,0 ) ∈ H n and so on.

Let H ( t ) ∈ O n , i.e. the system (24) is stable and F ( t ) ∈ Ω n and K ∈ O , then the rotation φ of all the solutions of (24) will be n π < φ < ( n + 1 ) π . The sketch of the proof is as follows, by assumption F ( t ) ∈ Ω n and K ∈ O , from (25) we know that any solution of (24) can be written as x = F ( t ) y where y ( t ) = e t K y ( 0 ) , from form (c) it follows

y ( t ) = S [ cos ( ϕ t ) − sin ( ϕ t ) sin ( ϕ t ) cos ( ϕ t ) ] S − 1 y ( 0 ) , 0 < ϕ < π T (27)

with b = S − 1 y ( 0 ) and det ( S ) = 1 . Thus S − 1 y ( t ) moves uniformly in a circle, describing and angle ϕ T in time T. Since 0 < ϕ T < π then 0 < φ y < π and finally n π < φ < ( n + 1 ) π .

Remark 4.2. Let x = F ( t ) a , y = F ( t ) b , be two linear independent solutions of (24), and Z = [ x , y ] then det ( Z ) = det F ( t ) det ( [ a , b ] ) = det ( [ a , b ] ) = ‖ a ‖ ‖ b ‖ sin ( θ ) so the area of the parallelogram defined by x and y , do not change do to the influence of F ( t ) and vectors x , y cannot overlap each other neither θ ≥ π . Then the rotation number of x and y must coincide.

Remark 4.3. The definition of the subsets Π n , H n and O n allows us to discriminate between stable (unstable) zones where the solutions rotation have similar properties, see appendix B.

Remember that every linear Hamiltonian system (24) may be defined as

p ˙ = − ∂ H ¯ ( p , q ) ∂ q q ˙ = ∂ H ¯ ( p , q ) ∂ p (28)

where H ¯ ( p , q ) is the quadratic form

H ¯ ( p , q ) = 1 2 x T H x (29)

the matrix H is the symmetric matrix associated to (24) and x ( t ) = [ q ( t ) p ( t ) ] is

a solution of the same equation. Defining ω x ( t ) as the argument of x ( t ) and deriving

ω x ( t ) = arctan ( q p ) d ω x ( t ) d t = q ˙ p − p ˙ q p 2 + q 2 = 2 H ¯ ( p , q ) p 2 + q 2 (30)

it follows from the multiplication [ p − q ] [ q ˙ p ˙ ] = [ q p ] H [ q p ] . Integrating (30) we get

ω x ( t ) = ω x ( 0 ) + ∫ 0 t 2 H ¯ ( p ( t 1 ) , q ( t 1 ) ) p 2 ( t 1 ) + q 2 ( t 1 ) d t 1 (31)

and then

ϕ = ∫ 0 T 2 H ¯ ( p ( t 1 ) , q ( t 1 ) ) p 2 ( t 1 ) + q 2 ( t 1 ) d t 1 (32)

and then ∫ 0 T h min ( t ) d t ≤ φ x ≤ ∫ 0 T h max ( t ) d t , where h m i n ( t ) and h m a x ( t ) be the smallest and largest eigenvalues H ( t ) . The latter procedure was obtained from [

Criterion 4.4. (Yakubovich 1). Let h m i n ( t ) and h m a x ( t ) be the smallest and largest characteristic values of the matrix H ( t ) in (24). And let

n π < ∫ 0 T h min ( t ) ≤ ∫ 0 T h max ( t ) < ( n + 1 ) π (33)

then the Hamiltonian system (24) belongs to the n-th stability region O n ( n = 0 , 1 , 2 , 3 , ⋯ ).

One must notice that Hill’s Equation (1) could be written as in (24), setting y 1 = x , y 2 = x ˙ and q ( t ) ≜ α + β p ( t ) , one gets

J ' [ y ˙ 1 y ˙ 2 ] = [ q ( t ) 0 0 1 ] ︸ H ( t ) [ y 1 y 2 ] (34)

thus Hill’s equation could be seen as a Hamiltonian system.

It is easy to verify that (34) may be written as

J ' [ c y ˙ 1 y ˙ 2 ] = [ 1 c q ( t ) 0 0 c ] [ c y 1 y 2 ] (35)

where c is a positive real constant. Then the smallest and largest eigenvalues of H ( t ) are

h min ( t ) = { 1 c q ( t ) if q ( t ) < c 2 c if c 2 < q ( t ) h max ( t ) = { 1 c q ( t ) if q ( t ) > c 2 c if c 2 > q ( t ) (36)

Notice that the Yakubovich 1 criterion inequalities could be reformulated as follow. For simplicity consider the Meissner equation, the integral over one period of the functions h m i n ( t ) and h m a x ( t ) are

∫ 0 2π h min ( t ) d t = { 2π c α if α + β s i g n ( cos ( t ) ) < c 2 2 π c if c 2 < α + β s i g n ( cos ( t ) ) (37)

∫ 0 2 π h max ( t ) d t = { 2π c α if α + β s i g n ( cos ( t ) ) > c 2 2 π c if c 2 > α + β s i g n ( cos ( t ) ) (38)

obviously the inequalities α + β s i g n ( cos ( t ) ) < c 2 and α + β s i g n ( cos ( t ) ) > c 2 are fulfilled if α + β < c 2 and α − β > c 2 respectively. So, for the α + β < c 2 case we get that the sufficient stability condition reduces to

n < 2 c α ≤ 2 c < 2 ( n + 1 ) (39)

and for the α − β > c 2 case we get

n < 2 c ≤ 2 c α < 2 ( n + 1 ) (40)

In fact the inequalities (39) and (40) are invariant as long as the periodic excitation function p ( t ) has zero mean, 1 T ∫ 0 T p ( t ) d t = 0 . Then the conditions for Mathieu, Meissner and Lyapunov equations to have stable solutions are:

ln < 2 c α ≤ 2 c < 2 ( n + 1 ) if α + β max t ∈ [ 0 , T ] p ( t ) < c 2 n < 2 c ≤ 2 c α < 2 ( n + 1 ) if α + β min t ∈ [ 0 , T ] p ( t ) > c 2 (41)

where the function p ( t ) is the excitation function of each periodic differential equation, see section 1.

criterion and the definitions of h m i n ( t ) and h m a x ( t ) given in (36) to the Mathieu, Meissner and Lyapunov equation.

For blue stable areas we have calculated the Yakubovich 1 stability criterion for 200 different values of 0 < c ≤ 3 and then we merge the resulting areas.

By Yakubovich 1 criterion, a sufficient condition for (35) to belong to O n with q ( t ) ≡ c 2 is

n π T < c < ( n + 1 ) π T (42)

Set

ρ ′ n ( c ) = inf ∫ 0 T | q ( t ) − c 2 | d t over all q ( t ) ∈ Π n ρ ″ n + 1 ( c ) = inf ∫ 0 T | q ( t ) − c 2 | d t over all q ( t ) ∈ Π n + 1 (43)

where Π n are the boundaries of instability areas H n , the boundary of O n are the sets Π n and Π n + 1 . Suppose that for some function q (t)

inf ∫ 0 T | q ( t ) − c 2 | d t < min [ ρ ′ n ( c ) , ρ ″ n + 1 ( c ) ] (44)

in words, this condition establishes that the distance from c 2 ∈ O n to the function q ( t ) is less than the distance from c 2 to the boundary of O n , so the function q ( t ) ∈ O n . Thus inequality (44) is a test for the stability of (24). This test requires the explicit expressions for ρ ′ n ( c ) and ρ ″ n ( c ) . The next three criteria use inequality (44) and the expressions of ρ ′ n ( c ) and ρ ″ n ( c ) to found sufficient conditions for the stability of the Hamiltonian system (24).

Now consider the differential equation

x ¨ + q ( t ) x = 0 (45)

where q ( t ) is a non-negative, non-identically equal to zero, piecewise continuous periodic function with period T. This and the following results were obtained by V. A. Yakubovich [

Remark 4.5. Notice that in these criterion, the regions which are guaranteed stable are not convex.

Criterion 4.6 (Yakubovich 2). Let

q ( t ) ≥ n 2 π 2 T 2 , n π T ≤ c < ( n + 1 ) π T (46)

if the following inequality holds,

∫ 0 T | q ( t ) − c 2 | d t ≤ 2 c ( n + 1 ) cot T c 2 ( n + 1 ) (47)

then the solution of equation (45) is stable and q ( t ) belongs to the n-th zone of stability, n = 0 , 1 , 2 , ⋯ .

Criterion 4.7 (Yakubovich 3). Let

q ( t ) ≤ ( n + 1 ) 2 π 2 T 2 , n π T ≤ c < ( n + 1 ) π T (48)

if for some n = 0 , 1 , 2 , ⋯ , we have the inequality

∫ 0 T | q ( t ) − c 2 | d t ≤ c ( T c − n π ) (49)

then the trivial solution of Equation (45) is stable and q ( t ) belongs to the n-th zone of stability.

For Meissner equation, the criteria 0.11 and 0.12 may be rewritten just as we did in the criterion 0.9.

From

∪ n 2 ≤ c < n + 1 2 P ( c ) ∩ T ( n ) ⊂ O n

From the

V n = { ( α , β ) | α − β ≥ n 2 4 and α + β ≤ ( n + 1 ) 2 4 } (50)

By doing a similar procedure for Yakubovich 3 criterion we obtain the

Notice that the cone V = { ( α , β ) | α − β ≥ 1 4 and α + β ≤ 1 } is inside the

stable zone obtained with the Yakubovich 3 criterion. It can be proved that the cones V n defined in (50) are inside of the stable zones (subsets of O n ) obtained by Yakubovich 3 criterion. Then, for the case of Meissner equation, we can say that the stable zones obtained by Yakubovich 2 criterion are contained in the zones defined by the Yakubovich 3 criterion. This is not the same for the cases of Mathieu and Lyapunov equations, see

Next criterion was developed by Borg [

Criterion 4.8 (Borg). Consider the system (45) and let

q a v = 1 T ∫ 0 T q ( t ) d t (51)

Suppose that for some integer n

n 2 π 2 T 2 < q a v < ( n + 1 ) 2 π 2 T 2 (52)

∫ 0 T | q ( t ) − q a v | d t < 2 q a v ( T q a v − n π ) , if n ≥ 1, (53)

∫ 0 T | q ( t ) − q a v | d t ≤ 4 q a v ( n + 1 ) cot ( T q a v 2 ( n + 1 ) ) (54)

Then all solutions of Equation (45) are bounded on ( − ∞ , + ∞ ) and the corresponding Hamiltonian in (24) is in the stability domain O n .

3 (green) and Borg (blue) criteria for the Mathieu, Meissner and Lyapunov equations.

It is worth to notice that Borg criterion uses almost the same statements of the criteria 0.11 and 0.12 but q a v is written instead of c 2 and the distance between the function q ( t ) and the constant c is divided by 2. Moreover for Mathieu, Meissner and Lyapunov equations the constant q a v is equal to α , so Borg criterion may be rewritten as:

The solutions of Mathieu, Meissner and Lyapunov equations belongs to the stability domain O n if for some integer n the inequalities

n 2 4 < α < ( n + 1 ) 2 4

β ∫ 0 2π | p ( t ) | d t < 2 α ( 2 α − n ) π , if n ≥ 1 ,

β ∫ 0 2π | p ( t ) | d t ≤ 4 α ( n + 1 ) cot ( π α ( n + 1 ) )

are fulfilled. The integral ∫ 0 2π | p ( t ) | d t for Mathieu, Meissner and Lyapunov equations are equal to 4 2 , 2 π and 5.40537 respectively.

^{7}These class of systems are called reversible by V. I. Arnold, see [

In [^{7}

x ¨ + ( λ + p ( t ) ) y = 0 , p ( t + T ) = p ( t ) , p ( t ) = p ( − t ) (55)

may, under suitable conditions, be solved by assuming solutions of the form

x 1 = A 1 ( t ) cos φ 1 ( t ) x 2 = A 2 ( t ) sin φ 2 ( t ) x ˙ 1 = − A 1 ( t ) λ + p ( t ) sin φ 1 ( t ) x ˙ 2 = A 2 ( t ) λ + p ( t ) cos φ 2 ( t ) } (56)

where

φ ˙ n ( t ) = λ + p ( t ) + ( − 1 ) n 1 4 p ˙ ( t ) λ + p ( t ) sin 2 φ n ( t ) , n = 1 , 2

A ˙ 1 ( t ) = − A 1 ( t ) p ˙ ( t ) 2 λ + 2 p ( t ) sin 2 φ 1 (t)

A ˙ 2 ( t ) = − A 2 ( t ) p ˙ ( t ) 2 λ + 2 p ( t ) cos 2 φ 2 (t)

A n ( 0 ) = 1 , φ n ( 0 ) = 0 , n = 1 , 2

notice that φ could be seen as a simile of the rotation of the solutions, see section 4.

We know that for the solutions of (55) to be stable the inequality | x 1 ( T ) + x ˙ 2 ( T ) | < 2 most be fulfilled. If p ( t ) = p ( − t ) it is possible to establish relations between x 1 , x ˙ 1 , x 2 and x ˙ 2 at t = T and x 1 , x ˙ 1 , x 2 , x ˙ 2 at

t = T 2 as follows

x 1 ( T ) = 2 x 1 ( T 2 ) x ˙ 2 ( T 2 ) − 1 = 1 + 2 x ˙ 1 ( T 2 ) x 2 ( T 2 ) x ˙ 1 ( T ) = 2 x 1 ( T 2 ) x ˙ 1 ( T 2 ) x 2 ( T ) = 2 x 2 ( T 2 ) x ˙ 2 ( T 2 ) x ˙ 2 ( T ) = x 1 ( T ) } (57)

since if x 1 ( t ) and x 2 ( t ) are linearly independent solutions of (55) then x 1 ( t + T ) and x 2 ( t + T ) are also solutions and can be written as

x 1 ( t + T ) = x 1 ( t ) x 1 ( T ) + x 2 ( t ) x ˙ 1 ( T ) x 2 ( t + T ) = x 1 ( t ) x 2 ( T ) + x 2 ( t ) x ˙ 2 ( T ) (58)

differentiating both sides

x ˙ 1 ( t + T ) = x ˙ 1 ( t ) x 1 ( T ) + x ˙ 2 ( t ) x ˙ 1 ( T ) x ˙ 2 ( t + T ) = x ˙ 1 ( t ) x 2 ( T ) + x ˙ 2 ( t ) x ˙ 2 ( T ) (59)

setting t = − T 2 and noticing that the solutions x 1 ( t ) and x 2 ( t ) are even

and odd respectively, that is, x 1 ( t ) = x 1 ( − t ) and x 2 ( t ) = − x 2 ( − t ) . Solving the equations for x 1 ( T ) , x ˙ 1 ( T ) , x 2 ( T ) and x ˙ 2 ( T ) we arrive to (57), see [

From (57) one can easily rewrite the stability inequality | x 1 ( T ) + x ˙ 2 ( T ) | < 2 as

| 4 x 1 ( T 2 ) x ˙ 2 ( T 2 ) − 2 | < 2 (60)

or

| 2 + 4 x ˙ 1 ( T 2 ) x 2 ( T 2 ) | < 2 (61)

The stability of the solution is then determined by the examination of the signs of x 1 ( T 2 ) , x 2 ( T 2 ) , x ˙ 1 ( T 2 ) and x ˙ 2 ( T 2 ) , and the number of zeros of x 1 ( t ) , x 2 ( t ) in the open interval ( 0, T 2 ) . This follows from the Sturm oscillation theorem. Moreover, if x 1 ( T 2 ) , x 2 ( T 2 ) and x ˙ 2 ( T 2 ) are positive and x ˙ 1 ( T 2 ) is negative and x 1 ( t ) , x 2 ( t ) have no zeros in t ∈ ( 0, T 2 ) then x 1 and x 2 belongs to the first stability zone [

| φ n ( T 2 ) | = | ∫ 0 T 2 [ λ + p ( t ) + ( − 1 ) n 1 4 p ˙ ( t ) λ + p ( t ) sin 2 φ ( t ) ] d t | ≤ ∫ 0 T 2 λ + p ( t ) d t + 1 4 ∫ 0 T 2 | p ˙ ( t ) λ + p ( t ) | d t ≤ π 2 (62)

and we get the following

Criterion 5.1 (Hochstadt 1). A sufficient condition for the boundedness of all solutions of the periodic differential equation x ¨ + q ( t ) x = 0 is

∫ 0 T 2 ( q ( t ) ) 1 2 d t + 1 4 ∫ 0 T 2 | q ˙ ( t ) q ( t ) | d t ≤ π 2 (63)

It is well known that, for q ( t ) = α + β p ( t ) , the transition curves are defined by points in the α − β plane for which there is at least one periodic (anti-periodic) solution of (55). In [

φ n ( T ) = 2 n π (64)

for some positive integer n, must be satisfied. And for anti-periodic solutions the condition

φ n ( T ) = ( 2 n + 1 ) π (65)

must be satisfied.

If the solutions of (55) are stable then, the condition

n π ≤ φ n ( T ) ≤ ( n + 1 ) π (66)

must be satisfied. By noticing the above Hochstadt generalized criterion 13 as follows [

Criterion 5.2 (Hochstadt 2). A sufficient condition for the boundedness of all solutions of the periodic differential equation x ¨ + q ( t ) x = 0 is

n π ≤ ∫ 0 T ( q ( t ) ) 1 2 d t − 1 4 ∫ 0 T | q ˙ ( t ) q ( t ) | d t ≤ ∫ 0 T ( q ( t ) ) 1 2 d t + 1 4 ∫ 0 T | q ˙ ( t ) q ( t ) | d t ≤ ( n + 1 ) π

Both criteria, Hochstadt 1 and Hochstadt 2, require the derivative of the function p ( t ) , obviously the functions c o s ( t ) and c o s ( t ) + 3 4 c o s ( 2 t ) do not

have any trouble, but the function s i g n ( c o s ( t ) ) does, i.e. both criteria can be used to obtain stability zones for Mathieu and Lyapunov equations but, we can not directly apply the criteria for Meissner equation.

For the Meissner case we must use the expansion on Fourier series of

s i g n ( s i n ( t ) ) = ∑ n = 0 ∞ 4 ( 2 n + 1 ) π s i n ( ( 2 n + 1 ) t ) (67)

instead of s i g n ( c o s ( t ) ) . The change of the excitation function is possible since: if Φ ( t ,0 ) is the transition matrix of x ¨ + ( α + β s i g n ( cos ( t ) ) ) x = 0 and Φ ¯ ( t ,0 ) is the transition matrix of x ¨ + ( α + β s i g n ( cos ( t ) ) ) x = 0 then, T r a c e ( Φ ( T , 0 ) ) = T r a c e ( Φ ¯ ( T , 0 ) ) . We take just the first 20 terms of the expansion.

In [

[ x ˙ 1 x ˙ 2 ] = [ 0 1 − λ q ( t ) q ˙ ( t ) 2 q ( t ) ] [ x 1 x 2 ] + [ 0 − q ˙ ( t ) 2 q ( t ) x 2 ] (68)

where q ( t ) is assumed to be positive and differentiable for all t, then it is

proved that the system

[ y ˙ 1 y ˙ 2 ] = [ 0 1 − λ q ( t ) q ˙ ( t ) 2 q ( t ) ] [ y 1 y 2 ] (69)

can be solved and its state transition matrix is

X ( t ) = [ cos ( w Φ ( t ) ) 1 w ϕ ( 0 ) sin ( w Φ ( t ) ) − w ϕ ( t ) sin ( w Φ ( t ) ) ϕ ( t ) ϕ ( 0 ) cos ( w Φ ( t ) ) ] (70)

where w = λ , ϕ = q ( t ) and Φ ( t ) = ∫ 0 t ϕ ( τ ) d τ . Then the solution of (68), with [ x 1 ( 0 ) x 2 ( 0 ) ] ′ = [ x 10 x 20 ] ′ , has the form

[ x 1 x 2 ] = X ( t ) [ x 10 x 20 ] + ∫ 0 t X ( t ) X − 1 ( τ ) [ 0 − q ˙ ( τ ) 2 q ( τ ) x 2 ( τ ) ] d τ (71)

suppose that [ x 1 x 2 ] and [ x ¯ 1 x ¯ 2 ] are solutions of (68) and they are subject to the initial conditions [ x 1 ( 0 ) x 2 ( 0 ) ] = [ 1 0 ] , [ x ¯ 1 ( 0 ) x ¯ 2 ( 0 ) ] = [ 0 1 ] then the discriminant (2) is

A ( λ ) = x 1 ( T ) + x ¯ 2 ( T ) and one can obtain 2 A ( λ ) by means of successive approximations. For details see [

The approximation given in [

A ( λ ) = 2 cos ∫ 0 T λ q ( t ) d t + ∑ n = 1 ∞ 1 2 4 n − 1 ∫ 0 T ∫ 0 t 1 ⋯ ∫ 0 t 2 n − 1 cos ψ ( t 1 t 2 ⋯ t 2 n ) ∏ i = 1 2 n q ˙ ( t i ) q ( t i ) d t 2 n ⋯ d t 1 = 2 cos ∫ 0 T λ q ( t ) d t + ∑ n = 1 ∞ Δ n (72)

where

ψ ( t 1 t 2 ⋯ t 2 n ) = ∫ 0 T λ q ( s ) d s − ∫ t 2 t 1 λ q ( s ) d s − ∫ t 4 t 3 λ q ( s ) d s − ⋯ − ∫ t 2 n t 2 n − 1 λ q ( s ) d s

Following [

∫ 0 T ∫ 0 t 1 ⋯ ∫ 0 t k − 1 f ( t 1 ) f ( t 2 ) ⋯ f ( t k ) d t k ⋯ d t 1 = 1 k ! ( ∫ 0 T f ( t ) d t ) k (73)

and

2 ∑ n = 1 ∞ x 2 n ( 2 n ) ! = e x + e − x − 2 (74)

the second term of the right hand side of (72) is

| ∑ n = 1 ∞ Δ n | ≤ e v 4 + e − v 4 − 2 (75)

where

ν ≜ ∫ 0 T | q ˙ ( t ) | q ( t ) d t (76)

then

| A ( λ ) | ≤ 2 | c o s ∫ 0 T λ q ( t ) d t | + e v 4 + e − v 4 − 2 (77)

for the solutions of (2) to be stable the inequality

2 | c o s ∫ 0 T λ q ( t ) d t | + e v 4 + e − v 4 − 2 < 2 (78)

must be fulfilled.

From (78) it is not so hard to prove the next

Criterion 6.1 (Xu). Suppose v = ∫ 0 π | q ˙ ( T π τ ) | q ( T π τ ) T π d τ and

λ ∈ ( ( n π + cos − 1 ( φ 0 ) ∫ 0 T q ( t ) d t ) 2 , ( ( n + 1 ) π − cos − 1 ( φ 0 ) ∫ 0 T q ( t ) d t ) 2 ) n = 0 , 1 , 2 , ⋯ (79)

where φ 0 = 1 2 ( 4 − e v / 4 − e v / 4 ) . Then (2) is stable.

Xu criterion, as the Hochstadt 1 and Hochstadt 2 criteria, needs the derivative of the function q ( t ) . In order to avoid troubles we will do the same as in the previous section, i.e. take the first 20 elements of the expansion of s i g n ( s i n ( t ) ) and then substitute them instead of s i g n ( c o s ( t ) ) .

Applying Xu criterion to Mathieu, Meissner and Lyapunov equations we obtain the

There is a vast number of sufficient conditions for the stability of periodic differential equations, we have just mentioned and explained some of them. For more stability criteria see for example: [

(Hamiltonian) systems.

We have reviewed some of the most known stability criteria for second order differential equations; these criteria were obtained by four different approaches; by an approximation of the discriminant made by Lyapunov; by properties of canonical (Hamiltonian) systems; by Sturm-Liouville equation properties and by a discriminant approximation due to Shi. We have given an easy explanation of each approach.

From the figures of the present work we can see, by simple inspection, that the best criterion among the ones presented is Hochstadt 2 (criterion 5.2) which is based on some properties of the solutions of the Sturm-Liouville equation and the rotation of the solutions, the second best is Xu’s criterion which is based on the approximation of the discriminant of the Hill’s equation made by Shi, and the discriminant approximation was obtained by means of successive approximations.

We have numerically calculated the percentage of the stability zones, in the ranges α ∈ [ 0,3 ] and β ∈ [ 0,1.5 ] , for each of the equations: Mathieu, x ¨ + ( α + β 2 cos ( t ) ) x = 0 ; Meissner, x ¨ + ( α + β s i g n ( cos ( t ) ) ) x = 0 ; and

Lyapunov, x ¨ + ( α + β 32 25 ( cos ( t ) + 3 4 cos ( 2 t ) ) ) x = 0 . See

From

The best two criteria for stability of the solution for the whole α − β plane ( α ∈ [ 0,3 ] and β ∈ [ 0,1.5 ] ) are Hochstadt 2 and Xu, both of them are based on an approximation of the discriminant of the Hill equation, the former approximation is based on 2 proposed linear independent solutions of the Sturm-Liouville problem, see section 5, and the later is obtained by means of successive approximations, see section 6.

Perturbation methods, such as strained parameters (Lindstedt-Poincare method) and multiple scales methods, are frequently used for analysing the stability of the periodic differential equations. They are based on the assumption that the variable-coefficient terms are small in some sense. The stability boundaries associated to a Hill equation may be determined by the strained parameters method, that is, assuming that β ≪ 1 , and then seeking the value of α such that the solutions be T or 2T periodic. The general solution and the coefficient α are written in terms of powers of β (perturbation expansion)

x ( t , β ) = x 0 ( t ) + β x 1 ( t ) + β 2 x 2 ( t ) + ⋯ (80)

α = α 0 + β α 1 + β 2 α 2 + ⋯ (81)

then, the series (80) and (81) are substituted into the Hill equation and by grouping terms of like powers of β , one obtains a set of recursive differential equations. The initial condition of α , i.e. α 0 , is the value of α where the

Arnold tongues rise, and depends on the excitation function period, α 0 = ( π n T ) 2 ,

n = 0 , 1 , 2 , ⋯ . The accuracy of the method depends on how many elements of the series (80) and (81) are obtained. Notice that the same procedure must be done for each transition curve, see [

In [

x ¨ + ( α + β cos ( t ) ) = 0 (82)

are obtained as

α = − 1 2 β 2 + O ( β 3 ) Transition curve associated to the 0^{th} tongue

α = 1 4 ± 1 2 β − 1 8 β 2 + O ( β 3 ) Transition curves associated to the 1^{st} tongue

α = 1 + 5 12 β 2 + O ( β 3 ) Transition curve associated to the 2^{th} tongue

α = 1 − 1 12 β 2 + O ( β 3 ) Transition curve associated to the 2^{th} tongu

Notice that the stability areas found with the strained parameters method are larger than the ones obtained with the sufficient stability criteria (Hochstadt 2, Xu, Yakubovich 1 and Borg).

Even though the stability areas found with strained parameters method are larger than the ones obtained with the stability criteria, for the Mathieu equation case (82), the complexities of the method and the set of differential equations that has to be solved, for obtaining the stable zones, make the strained parameters method less suitable than the simplicity of the criteria statements for the stability analysis of periodic differential equations. Besides, we must remember that the strained parameters method has the limitation that it just

work for small values of β , the sufficient stability criteria, here presented, do not have that limitation.

The first author, Carlos A. Franco acknowledges the financial support of CONACyT and CINVESTAV.

Franco, C.A. and Collado, J. (2017) Comparison on Sufficient Conditions for the Stability of Hill Equation: An Arnold’s Tongues Approach. Applied Mathematics, 8, 1481-1514. https://doi.org/10.4236/am.2017.810109

Let f ( t ) and g ( t ) be two linear independent solutions of

x ¨ + λ q ( t ) x = 0 , q ( t + T ) = q ( t ) , q ( t ) > 0 , ∀ t (83)

subject to the initial condition

f ( 0 ) = 1 g ( 0 ) = 0 f ′ ( 0 ) = 0 g ′ ( 0 ) = 1 (84)

The approximation of the characteristic constant A ( λ ) = 1 2 [ f ( T ) + g ′ ( T ) ]

associated to the periodic differential Equation (83), consists in write A ( λ ) and the linear independent solutions f ( t ) and g ( t ) as alternating series of powers of λ

A ( λ ) = A 0 − A 1 λ + A 2 λ 2 + ⋯ + ( − 1 ) n A n λ n + ⋯

f ( t ) = f 0 ( t ) − λ f 1 ( t ) + λ 2 f 2 ( t ) − ⋯ (85)

g ( t ) = g 0 ( t ) − λ g 1 ( t ) + λ 2 g 2 ( t ) − ⋯

where

A k = 1 2 ( f k ( T ) + g ˙ k ( T ) ) , k = 0 , 1 , 2 , ⋯

f k ( t ) = ∫ 0 t d t 1 ∫ 0 t 1 q ( t 2 ) f k − 1 ( t 2 ) d t 2 (86)

g k ( t ) = ∫ 0 t d t 1 ∫ 0 t 1 q ( t 2 ) g k − 1 ( t 2 ) d t 2

with f 0 ( t ) = 1 and g 0 ( t ) = t . And then solve the recurrence system of equations.

A 0 = 1 , A 1 = T 2 ∫ 0 T q ( t ) d t

A 2 = 1 2 ∫ 0 T d t 1 ∫ 0 t 1 ( T − t 1 + t 2 ) ( t 1 − t 2 ) q ( t 1 ) q ( t 2 ) d t 2

⋮ (87)

A n = 1 2 ∫ 0 T d t 1 ∫ 0 t 1 d t 2 ⋯ ∫ 0 t n − 1 ( T − t 1 + t n ) ( t 1 − t 2 ) ⋯ ( t n − 1 − t n ) ⋅ q ( t 1 ) ⋯ q ( t n ) d t n

In [

A n − 1 A 1 − n A n > 0 (88)

this can be demonstrated following [

( f n − 1 ( t ) + g ˙ n − 1 ( t ) ) t ∫ 0 t q ( t ) d t − 2 n ( f n ( t ) + g ˙ ( t ) ) > 0 (89)

for n > 2 . Defining S n ≜ ( f n − 1 ( t ) + g ˙ n − 1 ( t ) ) t ∫ 0 t q ( t ) d t − 2 n ( f n ( t ) + g ˙ ( t ) ) and noticing that it can be rewritten as

S n = ∫ 0 t ( F n + q ( t ) Φ n ) d t (90)

where

F n = t f ˙ n − 1 ( t ) ∫ 0 t q ( t ) d t + ( f n − 1 ( t ) + g ˙ n − 1 ( t ) ) ∫ 0 t q ( t ) d t − 2 n f ˙ n ( t ) Φ n = t g n − 2 ( t ) ∫ 0 t q ( t ) d t + ( f n − 1 ( t ) + g ˙ n − 1 ) t − 2 n g n − 1 ( t ) (91)

then (89) is fulfilled if all the coefficients F n and Φ n are positive, F n > 0 and Φ n > 0 . Rewriting F n and Φ n we have

F n = ∫ 0 t ( 2 f ˙ n − 1 ( t ) ∫ 0 t q ( t ) d t + q ( t ) u n ) d t Φ n = ∫ 0 t ( 2 q ( t ) t g ˙ n − 2 ( t ) + ν n ) d t (92)

where

u n = ∫ 0 t ( 2 q ( t ) ( g n − 2 ( t ) + t f n − 2 ( t ) ) + F n − 1 ) d t ν n = ∫ 0 t ( 2 f ˙ n − 1 ( t ) + 2 g ˙ n − 2 ( t ) ∫ 0 t q ( t ) d t + q ( t ) Φ n − 1 ( t ) ) d t (93)

since the excitation function q ( t ) is positive for all t, one can notice that f n ( t ) > 0 , g n ( t ) > 0 , f ˙ n ( t ) > 0 and g ˙ n ( t ) > 0 ∀ t > 0 , these follow from definition. So, u n and v n are positive which implies that F n and Φ n are positive and the inequality (89) is fulfilled. For details of the proof see [

Consider a second order differential equation in canonical form

x ˙ = J H ( t ) x (94)

where x = [ x 1 x 2 ] , H ( t ) is a symmetric real periodic matrix H ( t + T ) = H ( t ) = H T ( t ) and J is the skew symmetric matrix J = [ 0 1 − 1 0 ] .

The state transition matrix of (94) may be expressed as

Φ ( t , 0 ) = F ( t ) e t K Φ ( 0 , 0 ) = I 2 (95)

where F ( t + T ) = F ( t ) or F ( t + T ) = − F ( t ) , det ( F ( t ) ) = 1 , F ( t ) is a real matrix function, and K is a real matrix with T r ( K ) = 0 [

K = ln ( ± M ) (96)

where the sign ± is chosen so that K be real.

Remembering that, for periodic systems, Φ ( t + T , 0 ) = Φ ( t + T , T ) Φ ( T , 0 ) = β ∈ [ 0,1.5 ] and substituting (95) into Φ ( t + T ,0 ) we have

F ( t + T ) e ( t + T ) K = F ( t ) e t K F ( T ) e T K = F ( t ) e ( t + T ) K (97)

since F ( T ) = M e − T K = I 2 . Then F ( t + T ) = ± F ( t ) where the sign is the same as the one in the definition of K. From (95) one can easily see that det ( F ( t ) ) = α ∈ [ 0,3 ] , β ∈ [ 0,1.5 ] . . Since det ( F ( t + T ) ) = det ( F ( t ) ) and T r ( K ) is a real numbers, we can infer that that T r ( K ) = 0 and det ( F ( t ) ) = 1 .

It follows from matrix similarity that the K matrix may be brought by a similarity transformation to one of following forms:

(a) K = S [ λ 0 0 − λ ] S − 1

(b) K = S [ 0 ε 0 0 ] S − 1

(c) K = S [ 0 − ϕ ϕ 0 ] S − 1

where λ , ε and ϕ are real numbers and S is a real non-singular matrix. Following the nomenclature of [

From the Floquet factorization (95) one can say that the stability of the solutions of (94) depends on the exponential matrix e K T , and therefore on K i. e. if K ∈ H then (94) has one stable solution and one unstable solution (the characteristic multipliers are real and distinct from ± 1 ). If K ∈ Π with ϵ ≠ 0 then (94) has one periodic solution x 1 ( t ) and one unstable solution t x 2 ( t ) , if ϵ = 0 both solutions are stable and periodic (the characteristic multipliers are ± 1 ). And, if K ∈ O then both solutions, of (94), are bounded (the characteristic multipliers are complex, lie on the unit circle and are distinct from ± 1 ). All the latter properties are summarized in the

Let Ω be the set of real continuous function matrices F ( t ) with F ( t + T ) = ± F ( t ) and det ( F ( t ) ) = 1 . Now, let x = F ( t ) a be a solution of Hamiltonian system (94), where a is a non-zero arbitrary vector and let φ x denote the rotation of the solution x in time T, since F ( T ) = ± I 2 it follows that φ x = n π , where n is even if F ( T + t ) = F ( t ) and n is odd if F ( T + t ) = − F ( t ) . Let Ω n denote the set of matrices F ( t ) such that the rotation over a period T is φ = n π , Ω = ∪ n = − ∞ ∞ Ω n , each of these Ω n are disjoint sets [

Let L be the set of all symmetric matrices H ( t ) in (94). As we know each matrix H ( t ) ∈ L determines a unique state transition matrix Φ ( t ,0 ) . By (95)

the matrix Φ ( t ,0 ) determines the pair F ( t ) , K where F ( t ) ∈ Ω and K ∈ H ∪ Π ∪ O . So one can say that L = Ω × ( H ∪ Π ∪ O ) . As we have seen Ω = ∪ n = − ∞ ∞ Ω n then

L = ∪ ∀ n ( H n ∪ Π n ∪ O n ) (98)

where

H n = Ω n × H

Π n = Ω n × Π (99)

O n = Ω n × O

In words, the set L is divided into the subsets H n , Π n and O n where the matrices K and F ( t ) , which are defined by the state transition matrix of the Hamiltonian system (94), belong to one of the sets H , Π or O and F ( t ) ∈ Ω n . We say that H ( t ) ∈ H n if and only if Φ ( t ,0 ) ∈ H n and so on.

Let H ( t ) ∈ H n then F ( t ) ∈ Ω n and K ∈ H , let v + and v − be eigenvectors of K such that

K v + = λ v + , K v − = − λ v −

e t K v + = e t λ v + , e t K v − = e − t λ v −

we can notice that the rotation of the two linear independent solutions of (94), x 1 = e t λ F ( t ) v + and x 2 = e − t λ F ( t ) v − , is φ x 1 = φ x 2 = n π , these follows from the fact that the rotation of a solution doesn’t depend on the eigenvectors v + and v − but on the matrix F ( t ) . For any other solution the rotation is either n π < φ < ( n + 1 ) π or ( n − 1 ) π < φ < n π (See [

Similarly, one can prove that if H ( t ) ∈ O n then the rotation φ of all the solutions of (94) will be n π < φ < ( n + 1 ) π . The sketch of the proof is as follows, by assumption F ( t ) ∈ Ω n and K ∈ O , from (95) we know that any solution of (94) can be written as x = F ( t ) y where y ( t ) = e t K y ( 0 ) , from form (c) it follows

y ( t ) = S [ cos ( ϕ t ) − sin ( ϕ t ) sin ( ϕ t ) cos ( ϕ t ) ] S − 1 y ( 0 ) , 0 < ϕ < π T (100)

with b = S − 1 y ( 0 ) and det ( S ) = 1 . Thus S − 1 y ( t ) moves uniformly in a circle, describing and angle ϕ T in time T . Since 0 < ϕ T < π then 0 < φ y < π and finally n π < φ < ( n + 1 ) π .