Stability of Periodic Motions in Satellite Dynamics - Stability Theory … · 2014-12-25 · Introduction Lyapunov’s Stability Theory Stability of Hamiltonian Systems Contents 1

IntroductionLyapunov’s Stability Theory

Stability of Hamiltonian Systems

Stability of Periodic Motions in SatelliteDynamics

Stability Theory for Hamiltonian Systems

Boris S. Bardin

Moscow Aviation Institute (Technical University)Faculty of Applied Mathematics and Physics

Department of Theoretical Mechanics

Course of Computer Algebra and Differential Equations

Boris Bardin



Contents

1 IntroductionStability ConceptSome Problems of Satellite Dynamics

2 Lyapunov’s Stability TheoryBasic definitions and theoremsSecond (direct) Liapunov MethodStability in First Approximation

3 Stability of Hamiltonian SystemsStability in the First ApproximationKAM theory and Nonlinear Stability of HamiltonianSystemsStability of Cylindric Precession

Boris Bardin



Stability ConceptSome Problems of Satellite Dynamics

Contents




Boris Bardin




What does stability of motion mean?

Roughly speaking, it’s a property of dynamical system to keepits motion near a certain trajectory under a small initialperturbation.

How to study stability?choose a certain (unperturbed) trajectory of motionconsider perturbed trajectories starting in vicinity of theunperturbed onecompare unperturbed and perturbed trajectoriesif any perturbed trajectory is close to the unperturbed onethen we say that unperturbed trajectory is stable

Boris Bardin







Boris Bardin






How to study stability?

choose a certain (unperturbed) trajectory of motionconsider perturbed trajectories starting in vicinity of theunperturbed onecompare unperturbed and perturbed trajectoriesif any perturbed trajectory is close to the unperturbed onethen we say that unperturbed trajectory is stable

Boris Bardin






How to study stability?choose a certain (unperturbed) trajectory of motion

consider perturbed trajectories starting in vicinity of theunperturbed onecompare unperturbed and perturbed trajectoriesif any perturbed trajectory is close to the unperturbed onethen we say that unperturbed trajectory is stable

Boris Bardin






How to study stability?choose a certain (unperturbed) trajectory of motionconsider perturbed trajectories starting in vicinity of theunperturbed one

compare unperturbed and perturbed trajectoriesif any perturbed trajectory is close to the unperturbed onethen we say that unperturbed trajectory is stable

Boris Bardin






How to study stability?choose a certain (unperturbed) trajectory of motionconsider perturbed trajectories starting in vicinity of theunperturbed onecompare unperturbed and perturbed trajectories

if any perturbed trajectory is close to the unperturbed onethen we say that unperturbed trajectory is stable

Boris Bardin







Boris Bardin







Boris Bardin




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Boris Bardin




Motion of a Satellite with respect to its Center of Mass

Main assumptions:Satellite is a rigid bodyMotion of mass center does not depend on motion withrespect to center of massCenter of mass moves in circular or elliptic orbit

Reference framesorbital frame (axes directed along the radius vector of thecenter of mass O, the transversal and normal of the orbit)principal axes frame Oxyz

We use the Euler angles ψ, θ, ϕ to describe the position of thethe satellite in orbital system

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Euler angles

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Relative Equilibrium Positions of Satellite in a CircularOrbit

Orbit of the mass center n

v

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Regular Precession of Symmetric Satellite in aCircular Orbit

x

x y ϕv

z

R

orbit of the mass center

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Satellite Planar Motions in a Circular Orbit

x

y

ϕ

v

z

R


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Basic definitions and theoremsSecond (direct) Liapunov MethodStability in First Approximation

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Basic DefinitionsEquations of motion

We characterize the state of a dynamical system by the finiteset of variables

y1, y2, . . . , yn.

The motion of the system is described by ordinary differentialequations

dy1

dt= Y1(y1, . . . , yn),

. . .

dyn

dt= Yn(y1, . . . , yn).

(1)

Boris Bardin




Basic DefinitionsUnperturbed motion

The unperturbed motion corresponds to the partial solution ofthe system (1)

y1 = f1(t), y2 = f2(t), . . . , yn = fn(t)

such that at t = t0 the initial conditions

y1 = f1(t0), y2 = f2(t0), . . . , yn = fn(t0)

are satisfied.

Boris Bardin




Basic DefinitionsPerturbed motion. Variations

Perturbed motion corresponds to the perturbed initial conditions

y1 = f1(t0) + µ1, y2 = f2(t0) + µ2, . . . , yn = fn(t0) + µn.

The values µ1, µ2, . . . , µn are called perturbations.Let us introduce the variables

x1 = y1 − f1(t), x2 = y2 − f2(t), . . . , xn = yn − fn(t),

that are called deviations or variations of y1, y2 . . . , yn.At t = t0 variations satisfy the initial conditions xi(t0) = µi

Boris Bardin




Stability in the Sense of Lyapunov

If for any positive value ε, however small, one can find a positivevalue δ such that at t = t0, for all perturbations µi satisfying

|µi | < δ (2)

the inequality|xi(t)| < ε . . . (i = 1, . . . ,n), (3)

is valid, then the unperturbed motion is stable; otherwise it isunstable.

Boris Bardin




Stability in the Sense of Lyapunov

ε

δ

x1

x2

Boris Bardin




Definition of asymptotic stability

If the unperturbed motion is stable, and at the same time, forsmall enough initial perturbations, any perturbed motioncoverges to the unperturbed motion, i.e. if

limt→∞|xi(t)| = 0 (i = 1, . . . ,n), (4)

then the unperturbed motion is called asymptotically stable.

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Boris Bardin




Autonomous systems

Let us consider autonomous systems. The equations of theperturbed motion

dx1

dt= X1(x1, . . . , xn),

. . .

dxn

dt= Xn(x1, . . . , xn).

(5)

Boris Bardin




Lyapunov function

Let us consider the real functions V (x) = V (x1, . . . , xn) definedin the domain

|xi | < η, (6)

where η is a positive constant.It is assumed that V (x) is single-valued, continuous andV (0) = 0.The full derivative of V (x) evaluated by virtue of equations (5)

dVdt

=n∑

i=1

∂V∂xi

dxi

dt=

n∑i=1

∂V∂xi

Xi . (7)

The functions V defined in shuch manner are called Liapunovfunctions.

Boris Bardin




Some definitions

Def1: If in domain (6) the function V has only one sign but canbecome zero at some point other than origin, then V iscalled correspondingly positive semidefinite ornegative semidefinite.

Def2: If semidefenite function vanishes only at the originx1 = · · · = xn = 0, then V is called definite function(correspondingly positive definite or negative defenite).

Def3: Functions that take both positive and negative values arecalled indefinite functions.

Boris Bardin




Liapunov’s Theorem of Stability

If for the differential equations of a perturbed motion we canfind a definite function V such that by virtue of the given

equations its derivativedVdt

is either identically equal to zero oris semidefinite with the opposite sign of V , then theunperturbed motion is stable.

Boris Bardin




Liapunov’s Theorem of Asymptotic Stability

If one can find a definite function V for the differential equations

of a perturbed motion such that its derivativedVdt

is also adefinite function but with the opposite sign of V , then theunperturbed motion is asymptotically stable.

Boris Bardin




Chetaev’s Theorem of Instability

If for the differential equations of a perturbed motion we canfind a function V such that there exists a domain V > 0 in anarbitrarily small vicinity of zero and if the derivative

dVdt

of Vevaluated by virtue of these equations is positive at all points ofthe domain V > 0, then the unperturbed motion is unstable.

Boris Bardin




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Boris Bardin




The equations of the perturbed motion

dx1

dt= a11x1 + · · ·+ a1nxn + X ∗1 (x1, . . . , xn),

. . .

dxn

dt= an1x1 + · · ·+ annxn + X ∗n (x1, . . . , xn),

(8)

where X ∗i are terms of order higher than one in x1, . . . , xn.

Boris Bardin




The equations of the first approximation

dx1

dt= a11x1 + · · ·+ a1nxn,

. . .

dxn

dt= an1x1 + · · ·+ annxn.

(9)

When can we solve the stability problem on the basis ofthe equations of the first approximation?

Boris Bardin




Stability in First Approximation

Theorem of Stability in the First ApproximationIf all roots of the characteristic equation of a first approximationhave negative real parts, then irrespective of terms of orderhigher than one, the unperturbed motion is asymptoticallystable.

Theorem of Instability in the First ApproximationIf at least one of the roots of the characteristic equation has apositive real part, then irrespective of terms of order higher thanone, the unperturbed motion is unstable.

Boris Bardin



Stability in the First ApproximationKAM theory and Nonlinear Stability of Hamiltonian SystemsStability of Cylindric Precession

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Boris Bardin




Autonomous Hamiltonian Systems

Let us suppose that the perturbed motion is described byHamiltonian system of equations

dqdt

=∂H∂p

,dpdt

= −∂H∂q

, (10)

where the vector q = (q1, . . . ,qk ) and p = (p1, . . . ,pk ) arecalled positions and momentum vectors, respectively.

The Hamiltonian H is analytic in a small neighborhood of theorigin q = p = 0.

H = H2 + H3 + · · ·+ Hm + . . . (11)

When can we solve the stability problem for Hamiltoniansystem in the first approximation?

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Theorem of Instability in the First Approximation

If at least one of the roots of the characteristic equation of a firstapproximation has a non-zero real part, then irrespective ofterms of order higher than one, the unperturbed motion isunstable.

Unperturbed motion of the Hamiltonian system cannot beasymptotically stable!

Boris Bardin




Theorem of Stability in the First Approximation

If the quadratic part H2 of the Hamiltonian is positive (ornegative) definite, then unperturbed motion is stable in thesense of Liapunov.

If the Hamiltonian is not definite then it is necessary to considernonlinear system to solve the stability problem.

Boris Bardin




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Boris Bardin




Integrable Hamiltonian system. Action-angle variables

Let us consider an integrable Hamiltonian system. Its motioncan be described by action-angle variables

ϕ = (ϕ1, . . . , ϕk ),

I = (I1, . . . , Ik )

The Hamiltonian H0(I) of such a system does not depends onangle variables.

Thus, we have

Ii = Ii0, ϕi = ωi(I)t + ϕi0.

and the phase space is foliated by invariant tori I = const.

Boris Bardin




Boris Bardin




Nondegeneracy Conditions

The motion on each torus is conditionally-periodic with

frequency vector ω(I) =∂H∂I

.

A torus on which the frequency are rationally independent issaid to be nonresonant.

The unperturbed system is called nondegenerate if thefrequencies are functionally independent:

det(∂ω

∂I

)= det

(∂2H0

∂I2

)6= 0 (12)

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Isoenergetically nondegeneracy Conditions

The unperturbed system is called isoenergeticallynondegenerate if:

det

(∂2H0

∂I2

) (∂H0

∂I

)(∂H0

∂I

)0

6= 0. (13)

Boris Bardin




Main results of the KAM theory

Let us consider the perturbed system

H(I,ϕ) = H0(I) + εH1(I,ϕ) (14)

If the unperturbed system is nondegenerate or isoenergeticallynondegenerate, then

For a sufficiently small Hamiltonian perturbation mostnonresonant invariant tori do not vanish but are onlyslightly deformed , so that in the phase space of theperturbed system there are invariant tori densely filled withconditionally-periodic phase curves winding around themwith a number of independent frequencies equal to thenumber of degrees of freedom;

Boris Bardin




Main results of the KAM theory

These invariant tori form a majority in the sense that themeasure of the complement of their union is small whenthe perturbation is small;In the case of isoenergetic nondegeneracy the invarianttori form a majority on each level manifold of the energy.

Boris Bardin




System with two degrees of freedom

The Theorem of Arnold - MoserLet us consider the Hamiltonian system with two degrees offreedom Hamiltonian reads

H = ω1r1 − ω2r2 + a11r21 + 2a12r1r2 + a22r2

2 + O((r1 + r2)52 ).

If the Hamiltonian satisfied the following conditions:1. m1ω1 + m2ω2 6= 0,

where integer numbers m1,m2 such that0 < |m1|+ |m2| ≤ 4;

2. a11ω22 + 2a12ω1ω2 + a22ω

22 6= 0,

then the Hamiltonian system is stable in the sense of Liapunov.

Boris Bardin




Third order resonance ω1 = 2ω2.

The Hamiltonian takes the form

H = 2ω2r1 − ω2r2 + a11r21 + 2a12r1r2 + a22r2

2 +

(15)+ ar2

√r1 cos(ϕ1 + 2ϕ2) + H∗(ϕ1, r1, ϕ2, r2),

Markeev’s Theorem. If a 6= 0 of Hamiltonian (15) is not equalto zero, then the Hamiltonian system is unstable in the sense ofLiapunov. If a = 0 and Hamiltonian (16) satisfies the secondconditions of the Theorem of Arnold - Moser, then theHamiltonian system is unstable.

Boris Bardin




Fourth order resonance ω1 = 3ω2.

The Hamiltonian takes the form

H = 3ω2r1 − ω2r2 + a11r21 + 2a12r1r2 + a22r2

2 +

(16)+ dr2

√r1r2 cos(ϕ1 + 3ϕ2) + H∗(ϕ1, r1, ϕ2, r2),

Markeev’s Theorem. If the coefficients of Hamiltonian (16)satisfy the following inequality

|a11 + 6a12 + 9a22| > 3√

3d ,

then the Hamiltonian system is stable in the sense of Liapunov.If the coefficients of Hamiltonian (16) satisfy the followinginequality

|a11 + 6a12 + 9a22| < 3√

3d ,

then the Hamiltonian system is unstable.Boris Bardin




Low order resonances ω1 = ω2, ω1 = 0, or ω2 = 0

In general position the Jordan canonical form of the matrix ofthe linear system is not diagonal.Let us consider resonance ω1 = 0. The Hamiltonian takes theform

H =δ

2η2

1+12ω(ξ2

2+η22)+

M∑m=3

[m2 ]∑

l=0

am−2l,2l(ξ22+η2

2)lξm−2L1 +H, (17)

Sokolskii’s Theorem. Let us suppose that ak ,0 = 0 fork = 1, . . . ,M − 1 and aM,0 6= 0. If M is odd, then the equilibriumis unstable. If M is even, then the equilibrium is unstable forδaM,0 < 0 and stable for δaM,0 > 0.

Boris Bardin




Transcendental cases

M = 3:

H =δ

2η2

1 +12ω(ξ2

2 + η22) + a3,0ξ

31 + a1,2ξ1(ξ2

2 + η22) + H, (18)

M = 4:

H =δ

2η2

1+12ω(ξ2

2+η22)+a4,0ξ

41+a2,2ξ

21(ξ2

2+η22)+a0,4(ξ2

2+η22)2+H,

(19)Definition. If for any integer m the coefficients of the normalform am,0 = 0 for any integer m, then we shall say that atranscendental case takes place.

Boris Bardin




Transcendental cases

When transcendental cases can appear?Is equilibrium position of system stable or unstable intranscendental case?

Boris Bardin




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Boris Bardin




Cylindric Precession of Symmetric Satellite in aCircular Orbit

Reference frames:– The orbital frame OXYZ . The axes OX , OY are directed

along the transversal and normal of the orbit respectively.The axis OZ is directed along the radius vector of thecenter of mass O.

– The principal axes frame Oxyz, fixed in the satellite. Theaxis Oz is directed along the symmetry axis of the inertiaellipsoid.

Variables: Euler angles ψ, θ, ϕ and corresponding momenta pψ,pθ, pϕ

Boris Bardin




Cylindric Precession of Symmetric Satellite in aCircular Orbit

x

y ϕ

X

Y (z)

R


Z

Cylindrical Precession:

ψ=π, θ=π/2

Boris Bardin




The Hamiltonian of the Problem

H =p2ψ

2 sin2 θ+

p2θ

2− pψ cot θ cosψ − αβpψ

cos θsin2 θ

−

− pθ sinψ + αβcosψsin θ

+α2β2

2 sin2 θ+

32

(α− 1) cos2 θ .

(20)

Parameters: α = C/A and β = Ω0/ω0where ω0 and Ω0 are the orbital angular velocity and theprojection of the angular velocity of the satellite onto itssymmetry axis respectively.α ∈ [0;2], β ∈ R

Boris Bardin




Algorithm of Stability Study

Hamiltonian of the perturbed motionLinear problem of stability. Instability domains.Normalization of quadratic part of the Hamiltonian.Nonlinear normalization up to term of fourth order.Conclusions of stability in the sense of Lyapunov.Nonresonant case and resonant cases.Do open questions on stability still remain?

Boris Bardin




Stability Diagram

I

Γ

II

0

–4

4

β

2

α

Boris Bardin




Stability Diagram

Boris Bardin

Stability of Periodic Motions in Satellite Dynamics - Stability Theory … · 2014-12-25 · Introduction Lyapunov’s Stability Theory Stability of Hamiltonian Systems Contents 1

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