Autoresonance in oscillating systems with combined excitation and weak dissipation
aa r X i v : . [ m a t h - ph ] S e p AUTORESONANCE IN OSCILLATING SYSTEMS WITH COMBINEDEXCITATION AND WEAK DISSIPATION
OSKAR A. SULTANOV
Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, 112,Chernyshevsky str., Ufa 450008 Russia.
Abstract.
A mathematical model describing the initial stage of the capture into au-toresonance for nonlinear oscillating systems with combined parametric and externalexcitation is considered. The solutions with unboundedly growing amplitude and lim-ited phase mismatch correspond to the autoresonant capture. The paper investigatesthe existence, stability and bifurcations of such solutions in the presence of a weakdissipation in the system. Our technique is based on the study of particular solutionswith power-law asymptotics at infinity and the construction of suitable Lyapunovfunctions.
Keywords: nonlinear oscillations, autoresonance, stability, Lyapunov function
Mathematics Subject Classification:
Introduction
Autoresonance is a phenomenon that occurs in nonlinear systems with slowly varying oscillatingperturbations. Under certain conditions, the system automatically adjusts to the disturbances andholds this state for a sufficiently long period of time. As a result, the energy of the system can increasesignificantly [1]. The autoresonance was first studied in the problems associated with the accelerationof particles [2, 3] and planetary dynamics [4, 5]. Nowadays, it is considered as a universal phenomenonwith a wide range of applications [6–13]. The study of the corresponding mathematical models leadsto new and challenging problems in the field of nonlinear dynamics [14–16].Mathematical models associated with the autoresonance have been studied in many papers. See,for instance, [17–21], where the systems with external driving were analyzed, and [22–26], wherethe models of parametric autoresonance were investigated. The effect of a combined external andparametric excitation on the autoresonant capture in nonlinear systems was first studied in [27,28]. Inthis paper, the autoresonance model with the combined excitation in the presence of a weak dissipationis considered, and the existence and stability of different autoresonant modes are discussed.The paper is organized as follows. In section 1, the mathematical formulation of the problem isgiven. In section 2, the particular autoresonant solutions are described and the partition of a parameterspace is constructed. The stability of particular solutions and asymptotics for general autoresonantsolutions are discussed in section 3. A discussion of the results obtained is contained in section 4.1.
Problem statement
Consider the non-autonomous system of two differential equations: dρdτ + γ ( τ ) ρ = α ( τ ) sin ψ − β ( τ ) ρ sin(2 ψ + ν ) ,ρ h dψdτ − ρ + λτ i = α ( τ ) cos ψ − β ( τ ) ρ cos(2 ψ + ν ) , (1)with the parameters λ = 0 and ν ∈ [0 , π ). Smooth given functions α ( τ ) β ( τ ) correspondto the amplitude of an external and a parametric driving, a positive function γ ( τ ) is associated E-mail address : [email protected] . with a dissipation. This system arises in the study of the autoresonance phenomena in a class ofnonlinear oscillatory systems with a combined chirped-frequency excitation and a weak dissipation.The functions ρ ( τ ), ψ ( τ ) describe the evolution of the amplitude and the phase mismatch of theoscillators. The solutions with ρ ( τ ) ∼ √ λτ and ψ ( τ ) ∼ σ , σ = const as τ → ∞ are associated withthe phase-locking phenomenon and the capture into autoresonance. Note that system (1) also hasnon-autoresonant solutions with a bounded amplitude, but such solutions are not considered in thepresent paper.The combined effect of parametric and external excitations is determined by the behaviour of theratio f ( τ ) ≡ β ( τ ) /α ( τ ) as τ → ∞ . Indeed, if f ( τ ) ∼ f τ − / − κ , κ > f = const = 0, theparametric pumping is insignificant and system (1) corresponds to a perturbation of the model withthe external driving. If f ( τ ) ∼ f τ − / κ , the impact of external driving becomes inconsiderable andthe system takes the form of a perturbed model of parametric autoresonance. The parametric andexternal excitations are comparable when f ( τ ) ∼ f τ − / . Note also that the existence of autoresonantsolutions in systems with a dissipation depends on the behaviour of the function g ( τ ) ≡ γ ( τ ) /α ( τ ) as τ → ∞ . From the first equation in (1) it follows that the necessary condition is g ( τ ) ∼ g τ − / − κ with κ ≥ g = const = 0. Thus, in this paper it is assumed that α ( τ ) = τ ∞ X k =0 α k τ − k , β ( τ ) = ∞ X k =0 β k τ − k , γ ( τ ) = ∞ X k =0 γ k τ − k , τ → ∞ , α k , β k , γ k = const . Without loss of generality, we assume that α = 1 and γ = 0.Note that system (1) appears after averaging of perturbed oscillatory nonlinear systems and de-scribes a long term evolution of solutions. For a system with one degree of freedom, the example isgiven by the following equation: d xdt + ǫC ( ǫt ) dxdt + (cid:16) ǫB ( ǫt ) cos (cid:0) ζ ( t ) − ν (cid:1)(cid:17) U ′ ( x ) = ǫA ( ǫt ) cos ζ ( t ) , (2)where ζ ( t ) = t − ϑt , U ( x ) = x / − ǫx /
24 + O ( ǫ ), 0 < ǫ, ϑ ≪
1. We see that equation (2) with ǫ = 0has a stable trivial solution x ( t ) ≡
0, ˙ x ( t ) ≡
0. Solutions of the perturbed equation with small enoughinitial data ( x (0) , x ′ (0)), whose the energy E ( t ) ≡ U ( x ( t ))+ ( x ′ ( t )) / t ) is synchronised with the pumping such that Φ( t ) − ζ ( t ) = O (1), correspond to thecapture into autoresonance. The approximation of such solutions is constructed by using the methodof two scales with slow and fast variables: τ = ǫt/ ζ = ζ ( t ). The substitution x ( t ) = 2 ρ ( τ ) cos (cid:0) ζ + ψ ( τ ) (cid:1) + O ( ǫ )into equation (2) and the averaging over the fast variable lead to system (1) for the slowly varyingfunctions ρ ( τ ) and ψ ( τ ) with λ = 16 ϑǫ − , α ( τ ) = A ( ǫt ), β ( τ ) = B ( ǫt ), γ ( τ ) = 2 C ( ǫt ). Likewise,system (1) is derived in many other nonlinear problems related to autoresonance, including infinite-dimensional systems (see [15].In this paper, the conditions for the existence and stability of autoresonant solutions to system(1) are discussed. Our technique is based on the analysis of particular solutions with power-lawasymptotics at infinity. In the first step, such solutions are constructed and the conditions for theirexistence specify the partition of the parameter space. Then, the Lyapunov stability of the particularsolutions is investigated. Since the considered system is non-autonomous, the use of linear stabilityanalysis is limited and nonlinear terms of equations must be taken into account. In this case, thestability can be justified with the Lyapunov function method. The presence of stability will ensurethe existence of a family of autoresonant solutions. For such solutions, the asymptotic estimates atinfinity are obtained at the last step from the properties of the constructed Lyapunov functions.2. Particular autoresonant solutions
Consider the particular autoresonant solutions having the following asymptotics: ρ ∗ ( τ ) = ρ − √ τ + ρ + ∞ X k =1 ρ k τ − k , ψ ∗ ( τ ) = ψ + ∞ X k =1 ψ k τ − k , τ → ∞ . (3) UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 3
Substituting these series into system (1) and grouping the terms of the same power of τ yield ρ − = √ λ , ρ = 0, and ψ = σ , where σ satisfies the equation P ( σ ; δ, ν, κ ) ≡ δ sin(2 σ + ν ) − sin σ + κ = 0 , δ = β √ λ, κ = γ √ λ. (4)Note that the number of roots to equation depends on the values of the parameters ( δ, ν, κ ). If,in addition, the inequality P ′ ( σ ; δ, ν, κ ) = 0 holds, the remaining coefficients ρ k , ψ k as k ≥ √ λρ k = A k ( ρ − , . . . , ρ k − , σ, ψ , . . . , ψ k − ) , P ′ ( σ ; δ, ν, κ ) ψ k = B k ( ρ − , . . . , ρ k − , σ, ψ , . . . , ψ k − ) , where A = 1 √ λ ( δ cos(2 σ + ν ) − cos σ ) , A = − ψ √ λ (2 δ sin(2 σ + ν ) − sin σ ) , A = − ρ + β cos(2 σ + ν ) − (cid:16) α − ρ √ λ (cid:17) cos σ √ λ − ψ √ λ (2 δ sin(2 σ + ν ) − sin σ ) − ψ √ λ (4 δ cos(2 σ + ν ) − cos σ ) , B = 0 , B = −P ′′ ( σ ; δ, ν, κ ) ψ (cid:16) α − ρ √ λ (cid:17) sin σ − β √ λ sin(2 σ + ν ) − (1 + 2 γ ) ρ − , B = − ψ ψ P ′′ ( σ ; δ, ν, κ ) − ρ √ λ sin σ − ψ P ′′′ ( σ ; δ, ν, κ ) + α ψ cos σ − ψ ( β √ λ + β ρ ) cos(2 σ + ν ) , etc. In particular, ψ = 0 , ψ = θ, θ := B ( σ ) P ′ ( σ ; δ, ν, κ ) . Note that the pair of equations P ( σ ; δ, ν, κ ) = 0 and P ′ ( σ ; δ, ν, κ ) = 0 defines a bifurcation surface S = S ∪ S in the parameter space ( δ, ν, κ ), where S j := { ( δ, ν, κ ) ∈ R × [0 , π ) × R : sin ν = p j ( δ, κ ) } ,p j ( δ, κ ) := δ − (cid:16) κ (2 sin ς j − − sin ς j (cid:17) , sin ς j = z j ( δ, κ ) ,z j ( δ, κ ) := 13 (cid:16) κ + ( − j p κ + 12 δ − (cid:17) , j ∈ { , } . For every κ >
0, the bifurcation set is determined by the properties of p ( δ, κ ) and p ( δ, κ ) (see Fig. 1).In particular, if 0 < κ < /
4, there are two curves s + := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s − := { ( δ, ν ) ∈ [ n , δ ∗ ] × [0 , π ) : sin ν = p ( δ, κ ) } ∪ { ( δ, ν ) ∈ [ m , δ ∗ ] × [0 , π ) : sin ν = p ( δ, κ ) } , dividing the parameter plane ( δ, ν ) into tree parts (see Fig. 2,a):Ω + : = { ( δ, ν ) ∈ R × [0 , π ) : δ > s + } , Ω − : = { ( δ, ν ) ∈ R × [0 , π ) : δ < s − } , Ω : = { ( δ, ν ) ∈ R × [0 , π ) : s − < δ < s + } , where δ ∗ = − p (3 − κ ) / n < n are the roots of the equation p ( n, κ ) = 0, m = κ − m = κ + 1. In this case, the equation P ( σ ; δ, ν, κ ) = 0 has four different roots on the interval [0 , π )if ( δ, ν ) ∈ Ω − ∪ Ω + . If ( δ, ν ) ∈ Ω , there are only two different roots (see Fig. 3, a). AUTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION - - ∆ p H ∆ * , p * L H n ,0 LH n ,0 LH m ,1 L H m ,1 L (a) κ = 0 . - - ∆ p H n ,0 LH n ,0 LH m ,1 L H m ,1 L H m ,1 L (b) κ = 0 . - - ∆ p H n ,0 LH n ,0 LH n ,0 LH m ,1 L H m ,1 L (c) κ = 1 - - - ∆ p H n ,0 L H n ,0 L H n ,0 L H n ,0 LH m ,1 L H m ,1 L H m ,1 L (d) κ = 1 . Figure 1.
Graphs of p ( δ, κ ) (black curves) and p ( δ, κ ) (gray curves) as functionsof the parameter δ with fixed κ .If 3 / ≤ κ <
1, there are three curves s + := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s − := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s := { ( δ, ν ) ∈ [ m , m ] × [0 , π ) : sin ν = p ( δ, κ ) } , dividing the parameter plane ( δ, ν ) into four parts (see Fig. 2,b):Ω + : = { ( δ, ν ) ∈ R × [0 , π ) : δ > s + } , Ω − : = { ( δ, ν ) ∈ R × [0 , π ) : δ < s − } , Ω ∗ := { δ ∈ [ m , m ] , arcsin p ( δ, κ ) < ν < π − arcsin p ( δ, κ ) } , Ω := R × [0 , π ) \ (cid:0) Ω + ∪ Ω − ∪ Ω ∗ (cid:1) , where n < n are the roots of the equation p ( n, κ ) = 0, m = − ( √ κ + √ κ − / √ m = κ − m = κ + 1 are the roots of the equation p ( m, κ ) = 1.If κ = 1, the equation p ( δ, κ ) = 0 has three different roots n < n < n . In this case, theparameter plane ( δ, ν ) is divided into the following parts (see Fig. 2,c):Ω + : = { ( δ, ν ) ∈ R × [0 , π ) : δ > s + } , Ω − : = { ( δ, ν ) ∈ R × [0 , π ) : δ < s − } , Ω ∗ := { δ ∈ [ m , n ] : arcsin p ( δ, κ ) < ν < π − arcsin p ( δ, κ ) } , Ω := R × [0 , π ) \ (cid:0) Ω + ∪ Ω − ∪ Ω ∗ (cid:1) , by the curves s + := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s − := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s := { ( δ, ν ) ∈ [ m , n ] × [0 , π ) : sin ν = p ( δ, κ ) } ∪ { δ = 0 , ν ∈ [0 , π ) } , UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 5 - - ∆ Ν s + s - W + W - W (a) κ = 0 . - - ∆ Ν s + s - s W + W - W (b) κ = 0 . - - ∆ Ν s + s - s s s W + W - W W W (c) κ = 1 - - - ∆ Ν s + s - s s s W + W - W W W (d) κ = 1 . Figure 2.
Partition of the parameter plane ( δ, ν ). - - ∆ Σ s - s + (a) κ = 0 . ν = π - - - ∆ Σ s + s - s (b) κ = 0 . ν = π - - - ∆ Σ s + s - s s (c) κ = 1 . ν = π Figure 3.
The roots to equation (4) as functions of the parameter δ . The verticaldotted lines correspond to s − , s and s + .where m = − (1 + √ / √ m = 2, p ( m , , κ ) ≡ κ >
1, the equation p ( δ, κ ) = 0 has four different roots n < n < n < n and the parameterplane ( δ, ν ) is divided by the curves s + := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s − := { ( δ, ν ) ∈ [ n , m ] × [0 , π ) : sin ν = p ( δ, κ ) } ,s := { δ ∈ [ m , n ] ∪ [ m , n ] , ν ∈ [0 , π ) : sin ν = p ( δ, κ ) } AUTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION - - ∆ Κ p = p = p = p = p = p = Figure 4.
Existence domain (shaded area) of multiple roots to equation (4).into four parts (see Fig. 2,d):Ω + : = { ( δ, ν ) ∈ R × [0 , π ) : δ > s + } , Ω − : = { ( δ, ν ) ∈ R × [0 , π ) : δ < s − } , Ω ∗ := { δ ∈ [ m , n ] , arcsin p ( δ, κ ) < ν < π − arcsin p ( δ, κ ) } ∪ { ( δ, ν ) ∈ [ n , m ] × [0 , π ) }∪{ δ ∈ [ m , n ] , ≤ ν < arcsin p ( δ, κ ) } ∪ { δ ∈ [ m , n ] , π − arcsin p ( δ, κ ) < ν < π } , Ω := R × [0 , π ) \ (cid:0) Ω + ∪ Ω − ∪ Ω ∗ (cid:1) , where m = − ( √ κ + √ κ − / √ m = κ + 1, p ( m , , κ ) ≡ δ, ν ) ∈ Ω − ∪ Ω + , the equation P ( σ ; δ, ν, κ ) = 0 has four different roots on the interval[0 , π ). If ( δ, ν ) ∈ Ω , there are only two different roots. If ( δ, ν ) ∈ Ω ∗ , the equation has no solutions(see Fig. 3, b,c,d).Thus we have Theorem 1. If ( δ, ν ) ∈ Ω + ∪ Ω − and κ > , system (1) has four different solutions with asymptoticexpansion in the form of a series (3) . If κ > and ( δ, ν ) ∈ Ω , system (1) has 2 different solutionswith asymptotic expansion in the form of a series (3) . The existence of solutions ρ ∗ ( τ ), ψ ∗ ( τ ) with the asymptotics (3) as τ ≥ τ ∗ follows from [29,30]. Thecomparison theorems [31] applied to system (1) guarantees that the solutions can be extended to thesemi-axis.2.1. The roots of multiplicity 2.
If ( δ, ν ) ∈ s − ∪ s + ∪ s , there exists σ such that P ( σ ; δ, ν, κ ) = 0and P ′ ( σ ; δ, ν, κ ) = 0. It can easily be checked that σ ∈ { ς : sin ς = z , ( δ, κ ) } . The multiple roots existif ( δ, κ ) ∈ D m , where D m := (cid:0) { ≤ p ( δ, κ ) ≤ } ∪ { ≤ p ( δ, κ ) ≤ } (cid:1) ∩ (cid:0) {| z ( δ, κ ) | ≤ } ∪ {| z ( δ, κ ) | ≤ } (cid:1) ∩ { κ + 3 δ ≥ / } (see Fig. 4). In addition, suppose that P ′′ ( σ ; δ, ν, κ ) ≡ − σ + 4 κ = 0, then σ is the root of multiplicity 2. In this case, ψ is determined from the equation: P ′′ ( σ ; δ, ν, κ ) ψ C ( σ ) , (5) C ( σ ) ≡ sin 2 σ − λ λ √ λ − (cid:16) β √ λ sin(2 σ + ν ) − α sin σ + γ √ λ (cid:17) . It follows that the asymptotic solution in the form (3) does not exist when P ′′ ( σ ; δ, ν, κ ) C ( σ ) <
0. If P ′′ ( σ ; δ, ν, κ ) C ( σ ) >
0, equation (5) has two different roots: ψ = ± φ, φ := s C ( σ ) P ′′ ( σ ; δ, ν, κ ) . UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 7
Note that if α = β = γ = 0 and λ > /
2, then C ( σ ) < σ ∈ R and a suitable root is σ suchthat sin σ = z ( δ, κ ). The remaining coefficients ρ k , ψ k are determined from the following recurrentsystem of equations: 2 √ λρ k = A k ( ρ − , . . . , ρ k − , σ, ψ , . . . , ψ k − ) , P ′′ ( σ ; δ, ν, κ ) ψ ψ k + sin σ √ λ ρ k = C k ( ρ − , . . . , ρ k − , σ, ψ , . . . , ψ k − ) , where C = − ψ P ′′′ ( σ ; δ, ν, κ ) + α ψ cos σ − ψ ( β √ λ + β ρ ) cos(2 σ + ν ) , C = ρ − γ ρ − γ √ λ − ψ P ′′ ( σ ; δ, ν, κ ) − ψ ψ P ′′′ ( σ ; δ, ν, κ ) − ψ P (4) ( σ ; δ, ν, κ ) − α (cid:16) ψ cos σ − ψ sin σ (cid:17) − (cid:0) ψ ( β √ λ + β ρ ) + 2 ψ β ρ (cid:1) cos(2 σ + ν )+ (cid:0) ψ ( β √ λ + β ρ ) + α − β √ λ − β ρ (cid:1) sin(2 σ + ν ) , etc.2.2. The roots of multiplicity 3.
Now suppose ( δ, ν ) ∈ s − ∪ s + ∪ s and σ is the root of equation(4) such that P ′ ( σ ; δ, ν, κ ) = 0 and P ′′ ( σ ; δ, ν, κ ) = 0. These imply that P ′′′ ( σ ; δ, ν, κ ) ≡ − σ , σ ∈ { ς : sin ς = 4 κ/ } , κ + 3 δ = 3 /
4, and sin ν = − κδ − (1 + 32 δ ) /
9. Let P ′′′ ( σ ; δ, ν, κ ) = 0. In thiscase, the asymptotic solutions are constructed in the following form: ρ ( τ ) = ρ − √ τ + ρ + ∞ X k =1 ρ k τ − k , ψ ( τ ) = σ + ∞ X k =1 ψ k τ − k , τ → ∞ . (6)It can easily be checked that ρ − = √ λ, ρ = ρ = ρ = 0 , ρ = − cos σ λ , ψ = 0 , ψ = χ,χ := h N ( σ ) P ′′′ ( σ ; δ, ν, κ ) i , N ( σ ) ≡ σ − λ )4 λ √ λ − (cid:16) β √ λ sin(2 σ + ν ) − α sin σ + γ √ λ (cid:17) . The remaining coefficient ρ k +1 , ψ k as k ≥ √ λρ k +1 = M k +1 ( ρ − , . . . , ρ k , σ, ψ , . . . , ψ k − ) , P ′′′ ( σ ; δ, ν ) ψ ψ k σ √ λ ρ k +1 = N k ( ρ − , . . . , ρ k , σ, ψ , . . . , ψ k − ) , where M = 0 , M = ψ sin σ √ λ , M = ψ sin σ √ λ , M = ψ sin σ − ψ cos σ √ λ , N = 0 , N = − ψ ( β √ λ + β ρ − α ) cos σ − P ′′′ ( σ ; δ, ν, κ ) ψ ψ − P (4) ( σ ; δ, ν, κ ) ψ , N = − ψ ( β √ λ + β ρ − α ) cos σ − P ′′′ ( σ ; δ, ν, κ ) 6 ψ ψ ψ + ψ − P (4) ( σ ; δ, ν, κ ) ψ ψ , etc. It can easily be checked that this system is solvable whenever N ( σ ) = 0. Note that N ( σ ) = 0 forany σ ∈ R if α = β = γ = 0 and λ > / The roots of multiplicity 4.
Let ( δ, ν ) ∈ s − ∪ s + ∪ s and σ be the root of equation (4) suchthat P ′ ( σ ; δ, ν, κ ) = 0, P ′′ ( σ ; δ, ν, κ ) = 0 and P ′′′ ( σ ; δ, ν, κ ) = 0. In this case, κ = 3 / δ = − / ν = π/ σ = π/ P (4) ( σ ; δ, ν, κ ) = 3. The asymptotic solutions are constructed in the followingform: ρ ( τ ) = ρ − √ τ + ρ + ∞ X k =1 ρ k τ − k , ψ ( τ ) = σ + ∞ X k =1 ψ k τ − k , τ → ∞ . (7) AUTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION
It follows easily that ρ − = √ λ, ρ = ρ = ρ = 0 , ψ = Q , Q ≡ α + 4 √ λ (2 β − γ − . If Q >
0, the last equation has two real, distinct roots: ψ = ± ξ , ξ := Q / . The remaining coefficient ρ k +1 , ψ k as k ≥ √ λρ k +1 = T k +1 ( ρ − , . . . , ρ k , ψ , . . . , ψ k − ) ,ψ ψ k + 2 √ λ ρ k +1 = Q k ( ρ − , . . . , ρ k , ψ , . . . , ψ k − ) , where T = ψ √ λ , T = ψ √ λ , T = ψ + 3 ψ √ λ , T = ψ ψ + ψ − √ λρ √ λ , Q = 0 , Q = − α ψ − β √ λψ + ψ − ψ ψ ψ ρ √ λ , Q = − α ψ ψ − β √ λψ ψ + ψ ψ − ψ ψ − ψ ψ ψ − ψ ψ ρ √ λ + ψ ρ √ λ , etc. If Q ≤
0, the asymptotic solution in the form (7) does not exists.Thus we have the following
Theorem 2.
Let ( δ, ν ) ∈ s − ∪ s − ∪ s , κ > and σ be a root of equation (4) . • If P ′ ( σ ; δ, ν, κ ) = 0 , then there exists a solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotic expansion in theform (3) with ψ = σ , ψ = 0 , ψ = θ . • If P ′ ( σ ; δ, ν, κ ) = 0 and P ′′ ( σ ; δ, ν, κ ) C ( σ ) < , then system (1) has two solutions ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotic expansion in the form (3) with ψ = σ , ψ = ± φ . • If P ′ ( σ ; δ, ν, κ ) = 0 , P ′′ ( σ ; δ, ν, κ ) = 0 and N ( σ ) = 0 , then system (1) has solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotic expansion in the form (6) with ψ = 0 , ψ = χ . • If P ′ ( σ ; δ, ν, κ ) = 0 , P ′′ ( σ ; δ, ν, κ ) = 0 , P ′′′ ( σ ; δ, ν, κ ) = 0 and Q > , then system (1) has twosolutions ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotic expansion in the form (7) with ψ = ± ξ . The proof is the same as that of Theorem 1.3.
Stability analysis
Linear analysis.
Let ρ ∗ ( τ ), ψ ∗ ( τ ) be one of the particular autoresonant solutions with asymp-totics (3), (6) or (7). The substitution ρ ( τ ) = ρ ∗ ( τ ) + R ( τ ), ψ ( τ ) = ψ ∗ ( τ ) + Ψ( τ ) into (1) gives thefollowing system with a fixed point at (0 , dRdτ + γ ( τ ) R = α ( τ ) (cid:16) sin( ψ ∗ + Ψ) − sin ψ ∗ (cid:17) − β ( τ ) (cid:16) ( ρ ∗ + R ) sin(2 ψ ∗ + 2Ψ + ν ) − ρ ∗ sin(2 ψ ∗ + ν ) (cid:17) ,d Ψ dτ = 2 ρ ∗ R + R + α ( τ ) (cid:16) cos( ψ ∗ + Ψ) ρ ∗ + R − cos ψ ∗ ρ ∗ (cid:17) − β ( τ ) (cid:16) cos(2 ψ ∗ + 2Ψ + ν ) − cos(2 ψ ∗ + ν ) (cid:17) . (8)Consider the linearized system: ddτ (cid:18) R Ψ (cid:19) = Λ ( τ ) (cid:18) R Ψ (cid:19) , Λ ( τ ) := − γ ( τ ) − β ( τ ) sin(2 ψ ∗ + ν ) α ( τ ) cos ψ ∗ − β ( τ ) ρ ∗ cos(2 ψ ∗ + ν )2 ρ ∗ − α ( τ ) cos ψ ∗ ρ ∗ − α ( τ ) sin ψ ∗ ρ ∗ + 2 β ( τ ) sin(2 ψ ∗ + ν ) . Define ˆ ρ ( τ ) := ρ ∗ ( τ ) √ λτ − , ˆ ψ ( τ ) := ψ ∗ ( τ ) − σ, for τ ≥
0, where σ is one of the roots to equation (4). Then the functions ˆ ρ ( τ ) and ˆ ψ ( τ ) have thefollowing asymptotics as τ → ∞ : ˆ ρ ( τ ) = O ( τ − ) and • ˆ ψ ( τ ) = θτ − + O ( τ − ) if σ is the simple root, • ˆ ψ ( τ ) = ± φτ − + O ( τ − ) if σ is the root of multiplicity 2, UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 9 • ˆ ψ ( τ ) = χτ − + O ( τ − ) if σ is the root of multiplicity 3, • ˆ ψ ( τ ) = ± ξτ − + O ( τ − ) if σ is the root of multiplicity 4.Then the roots of the corresponding characteristic equation | Λ ( τ ) − z I | = 0 can be represented in theform z ± ( τ ) = 12 (cid:16) tr Λ ( τ ) ± p D ( τ ) (cid:17) , where tr Λ ( τ ) = 1 √ λ (cid:16) − κ + P ( ψ ∗ ; δ, ν, κ ) (cid:17) + O ( τ − ) ,D ( τ ) := (cid:0) tr Λ ( τ ) (cid:1) − Λ ( τ ) = − τ √ λ P ′ ( ψ ∗ ; δ, ν, κ ) + O (1)as τ → ∞ . Therefore, if σ is the simple root of (4) such that P ′ ( σ ; δ, ν, κ ) <
0, then z ± ( τ ) are real ofdifferent signs: z ± ( τ ) = ± τ (4 λ ) p −P ′ ( σ ; δ, ν, κ ) + O (1) , τ → ∞ . This implies that the fixed point (0 ,
0) of (8) is a saddle in the asymptotic limit, and the correspondingsolutions ρ ∗ ( τ ), ψ ∗ ( τ ) to system (1) are unstable (see, for example, [34]).Similarly, if σ is the root of multiplicity 2 such that ∓P ′′ ( σ ; δ, ν, κ ) > ψ ( τ ) = ± φτ − / + O ( τ − ), then z ± ( τ ) = ± τ (4 λ ) p | φ P ′′ ( σ ; δ, ν, κ ) | + O (1) , τ → ∞ . In this case, the fixed point (0 ,
0) of (8) and the corresponding particular solution to system (1) areboth unstable.In the same way, the fixed point of the linearized system is unstable when σ is the root of multiplicity3 and P ′′′ ( σ ; δ, ν ) <
0. In this case, the eigenvalues have the following asymptotics: z ± ( τ ) = ± τ λ p − χ P ′′′ ( σ ; δ, ν, κ ) + O (1) , τ → ∞ . If σ is the root of multiplicity 4 and ˆ ψ ( τ ) = − ξτ − / + O ( τ − / ), then z ± ( τ ) = ± τ λ ξ + O (1) , τ → ∞ , and the corresponding particular solution to system (1) is unstable.Thus we have Theorem 3.
Let σ be a root of equation (4) . • If P ′ ( σ ; δ, ν, κ ) < , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotics (3) is unstable. • If P ′ ( σ ; δ, ν, κ ) = 0 and ∓P ′′ ( σ ; δ, ν, κ ) > , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotics (3) , ψ = ± φ is unstable. • If P ′ ( σ ; δ, ν, κ ) = 0 , P ′′ ( σ ; δ, ν, κ ) = 0 , and P ′′′ ( σ ; δ, ν, κ ) < , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) withasymptotics (6) is unstable. • If P ′ ( σ ; δ, ν, κ ) = 0 , P ′′ ( σ ; δ, ν, κ ) = 0 , and P ′′′ ( σ ; δ, ν, κ ) = 0 , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) withasymptotics (7) , ψ = − ξ is unstable. Let us consider the following cases that are not covered by Theorem 3:
Case I: P ′ ( σ ; δ, ν, κ ) > Case II: P ′ ( σ ; δ, ν, κ ) = 0, ±P ′′ ( σ ; δ, ν ) > ψ ( τ ) = ± φτ − + O ( τ − ) as τ → ∞ . Case III: P ′ ( σ ; δ, ν, κ ) = 0, P ′′ ( σ ; δ, ν, κ ) = 0, and P ′′′ ( σ ; δ, ν, κ ) > Case IV: P ′ ( σ ; δ, ν, κ ) = 0, P ′′ ( σ ; δ, ν, κ ) = 0, P ′′′ ( σ ; δ, ν, κ ) = 0 and ˆ ψ ( τ ) = ξτ − + O ( τ − ) as τ → ∞ . In these cases, the roots of the characteristic equation are complex. In particular, z ± ( τ ) = ± iτ (4 λ ) p P ′ ( σ ; δ, ν, κ ) + O (1) , in Case I ,z ± ( τ ) = ± iτ (4 λ ) p | φ P ′′ ( σ ; δ, ν, κ ) | + O (1) , in Case II ,z ± ( τ ) = ± iτ λ p χ P ′′′ ( σ ; δ, ν, κ ) + O (1) , in Case III ,z ± ( τ ) = ± iτ λ ξ + O (1) , in Case IV , and ℜ z ± ( τ ) = O (1) as τ → ∞ . In such cases, the linear stability analysis fails (see, for example, [32,33]), and the nonlinear terms of the equations must be taken into account.3.2. Lyapunov functions.
Let us specify the definition of stability that will be used in this section.
Definition 1.
The solution ρ ∗ ( τ ) , ψ ∗ ( τ ) to system (1) is stable with the weights τ w and τ w as τ ≥ τ if for all ǫ > there exists δ ǫ > such that for all ( ρ , ψ ) : ( ρ − ρ ∗ ( τ )) + ( ψ − ψ ∗ ( τ )) < δ ǫ the solution ρ ( τ ) , ψ ( τ ) to system (1) with initial data ρ ( τ ) = ρ , ψ ( τ ) = ψ satisfies the followinginequality: ( ρ ( τ ) − ρ ∗ ( τ )) τ w + ( ψ ( τ ) − ψ ∗ ( τ )) τ w < ǫ (9) for all τ > τ . This definition modifies classical concept of stability with w = w = 0. Inequality (9) can beconsidered as the estimate for the norm in the space of continuous functions with the weights τ w and τ w . Thus, if the solution is stable with the weights τ w , τ w and w , w ≥
0, then the perturbedsolutions with initial data sufficiently close to ρ ∗ ( τ ), ψ ∗ ( τ ) have the following asymptotics: ρ ( τ ) = ρ ∗ ( τ ) + O ( τ − w ), ψ ( τ ) = ψ ∗ ( τ ) + O ( τ − w ) as τ → ∞ . Theorem 4.
Let σ be a root of equation (4) . • In Case I , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotics (3) is stable. • In Case II , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotics (3) is stable with the weights τ / and τ / . • In Case III , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotics (6) is stable with the weights τ / and τ / . • In Case IV , the solution ρ ∗ ( τ ) , ψ ∗ ( τ ) with asymptotics (7) is stable with the weights τ / and τ / .Proof. Note that system (8) can be rewritten in a near-Hamiltonian form: dRdτ = − ∂ Ψ H ( R, Ψ , τ ) , d Ψ dτ = ∂ R H ( R, Ψ , τ ) + F ( R, Ψ , τ ) , (10)where H ( R, Ψ , τ ) := ρ ∗ R + α (cid:0) cos( ψ ∗ + Ψ) − cos ψ ∗ + Ψ sin ψ ∗ (cid:1) − βρ ∗ (cid:16) cos(2 ψ ∗ + 2Ψ + ν ) − cos(2 ψ ∗ + ν ) + 2Ψ sin(2 ψ ∗ + ν ) (cid:17) + γR Ψ + R − βR (cid:16) cos(2 ψ ∗ + 2Ψ + ν ) − cos(2 ψ ∗ + ν ) (cid:17) and F ( R, Ψ , τ ) := α (cid:18) cos( ψ ∗ + Ψ) ρ ∗ + R − cos ψ ∗ ρ ∗ (cid:19) − β (cid:16) cos(2 ψ ∗ + 2Ψ + ν ) − cos(2 ψ ∗ + ν ) (cid:17) − γ Ψ . UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 11
Taking into account the asymptotic formulas for the particular solutions ρ ∗ ( τ ) = √ λτ (1 + ˆ ρ ( τ )) and ψ ∗ ( τ ) = σ + ˆ ψ ( τ ), we obtain the following asymptotic estimates: H = τ (cid:16) √ λR + Ψ Z P ( σ + ζ ; δ, ν, κ ) dζ (cid:17) + τ ˆ ψ (cid:16) P ( σ + Ψ; δ, ν, κ ) − P ′ ( σ ; δ, ν, κ )Ψ (cid:17) + τ ˆ ψ (cid:16) P ′ ( σ + Ψ; δ, ν, κ ) − P ′′ ( σ ; δ, ν, κ )Ψ − P ′ ( σ ; δ, ν, κ ) (cid:17) + τ ˆ ψ (cid:16) P ′′ ( σ + Ψ; δ, ν, κ ) − P ′′′ ( σ ; δ, ν, κ )Ψ − P ′′ ( σ ; δ, ν, κ ) (cid:17) + γ R Ψ + R − β R (cid:16) cos(2 σ + 2Ψ + ν ) − cos(2 σ + ν ) (cid:17) ++ β R ˆ ψ (cid:16) sin(2 σ + 2Ψ + ν ) − sin(2 σ + ν ) (cid:17) + O ( ˆ ψ ) + O ( τ ˆ ψ ) + O ( τ − )and F = 1 √ λ Ψ Z (cid:16) P ( σ + ζ ; δ, ν, κ ) − κ (cid:17) dζ + O ( ˆ ψ ) + O ( τ − )as τ → ∞ , for all ( R, Ψ) ∈ B ( d ∗ ), where B ( d ∗ ) := { ( R, Ψ) ∈ R : d = p R + Ψ ≤ d ∗ } , d ∗ = const > . Consider
Case I , when σ is the simple root to equation (4). In this case, ˆ ψ = θτ − + O ( τ − / ),and we get H = τ (cid:16) √ λR + ω Ψ O ( d ) (cid:17) + O ( τ − d ) , F = − γ Ψ + O ( d ) + O ( τ − d )as d → τ → ∞ , where ω = P ′ ( σ ; δ, ν, κ ) >
0. Note that the asymptotic estimates are uniformwith respect to ( R, Ψ , τ ) in the domain { ( R, Ψ , τ ) ∈ R : d ≤ d ∗ , τ ≥ τ ∗ } , where d ∗ , τ ∗ = const > V ( R, Ψ , τ ) := τ − (cid:16) H ( R, Ψ , τ ) − γ R Ψ (cid:17) as a Lyapunov function candidate for system (10). The function V ( R, Ψ , τ ) has the following asymp-totics: V = √ λR + ω Ψ O ( d ) + O ( τ − d )as d → τ → ∞ . It can easily be checked that for all 0 < κ < d > τ > − κ ) W ( R, Ψ) ≤ V ( R, Ψ , τ ) ≤ (1 + κ ) W ( R, Ψ)for all ( R, Ψ , τ ) ∈ D W ( d , τ ), where D W ( d , τ ) := { ( R, Ψ , τ ) ∈ R : W ( R, Ψ) ≤ d , τ ≥ τ } , W ( R, Ψ) := √ λR + ω Ψ . The total derivative of the function V ( R, Ψ , τ ) with respect to τ along the trajectories of system (10)has the form: dV dτ (cid:12)(cid:12)(cid:12) (10) = ∂V ∂τ − ∂V ∂R ∂ Ψ H + ∂V ∂ Ψ (cid:16) ∂ R H + F (cid:17) = − γ (cid:16) √ λR + ω Ψ O ( d ) (cid:17) + O ( τ − d )as d → τ → ∞ . Hence, for all 0 < κ < d > τ > dV dτ (cid:12)(cid:12)(cid:12) (10) ≤ − γ κ V , γ κ := γ (cid:16) − κ κ (cid:17) > , (11) for all ( R, Ψ , τ ) ∈ D W ( d , τ ). Thus, for all 0 < ε < d there exist δ ε := ε p (1 − κ ) / (1 + κ ) / W ≤ δ ε V ( R, Ψ , τ ) ≤ (1 + κ ) δ ε < (1 − κ ) ε ≤ inf W = ε V ( R, Ψ , τ )for all τ > τ , where d = min { d , d } and τ = max { τ , τ } . The last estimates and the negativityof the total derivative of the function V ( R, Ψ , τ ) ensure that any solution of system (10) with initialdata W ( R ( τ ) , Ψ( τ )) ≤ δ ε cannot leave the domain { ( R, Ψ) ∈ R : W ( R, Ψ) ≤ ε } as τ > τ . Hence,the fixed point (0 ,
0) is stable as τ > τ . The stability on the finite time interval (0 , τ ] follows fromthe theorem on the continuity of the solutions to the Cauchy problem with respect to the initial data.Consider Case II . Let σ be a root of multiplicity 2 to equation (4) such that P ′ ( σ ; δ, ν, κ ) = 0, ±P ′′ ( σ ; δ, ν, κ ) ˆ ψ = ω τ − / + O ( τ − ), where ω = φ P ′′ ( σ ; δ, ν, κ ) >
0. Hence, H = τ (cid:16) √ λR + P ′′ ( σ ; δ, ν, κ ) Ψ O ( d ) (cid:17) + ω Ψ O ( d ) + O ( τ − d )as d → τ → ∞ . In this case, H ( R, Ψ , τ ) is sign indefinite and the function V ( R, Ψ , τ ) can notbe used as a Lyapunov function. Consider the change of variables R ( τ ) = τ − r ( τ ) , Ψ( τ ) = τ − ϕ ( τ )(12)in system (10). The transformed system is drdτ = − ∂ ϕ H ( r, ϕ, τ ) , dϕdτ = ∂ r H ( r, ϕ, τ ) + F ( r, ϕ, τ ) , (13)where H ( r, ϕ, τ ) := τ H ( τ − r, τ − ϕ, τ ) − τ − rϕ , F ( r, ϕ, τ ) := τ F ( τ − r, τ − ϕ, τ ) + τ − ϕ . Taking into account (3), we see that H = τ (cid:16) √ λR + ω ϕ P ′′ ( σ ; δ, ν, κ ) ϕ (cid:17) + O (∆ ) , F = − γ ϕ + O (∆ τ − )as t → ∞ and ∆ = p r + ϕ →
0. Note that the function H ( r, ϕ, τ ) is suitable for the basis of aLyapunov function candidate: V ( r, ϕ, τ ) = τ − (cid:16) H ( r, ϕ, τ ) − γ rϕ (cid:17) It follows easily that for all 0 < κ < > τ > − κ ) W ( r, ϕ ) ≤ V ( r, ϕ, τ ) ≤ (1 + κ ) W ( r, ϕ )for all ( r, ϕ, τ ) ∈ D W (∆ , τ ), where W ( r, ϕ ) := √ λr + ω ϕ /
2. The total derivative of the function V ( r, ϕ, τ ) has a sign definite leading term of the asymptotics: dV dτ (cid:12)(cid:12)(cid:12) (13) = ∂V ∂τ + ∂V ∂ϕ F + γ τ − (cid:16) ϕ ∂H ∂ϕ − r ∂H ∂r (cid:17) = − γ (cid:16) √ λr + ω ϕ O (∆ ) (cid:17) + O ( τ − ∆ )as ∆ → τ → ∞ . Hence, for all 0 < κ < > τ > dV dτ (cid:12)(cid:12)(cid:12) (13) ≤ − γ κ V ≤ r, ϕ, τ ) ∈ D W (∆ , τ ). As above, the last inequality implies the stability of the equilibrium(0 ,
0) to system (13). Returning to the original variables, we obtain the result of the Theorem.In
Case III , σ is a root of multiplicity 3 to equation (4) such that P ′ ( σ ; δ, ν, κ ) = P ′′ ( σ ; δ, ν, κ ) = 0and P ′′′ ( σ ; δ, ν, κ ) >
0. The solution ρ ∗ ( τ ), ψ ∗ ( τ ) has the asymptotics (6), and ˆ ψ = χτ − / + O ( τ − / ).In this case, H ( R, Ψ , τ ) = τ (cid:16) √ λR + P ′′′ ( σ ; δ, ν, κ ) Ψ
24 + O ( d ) (cid:17) + O ( τ d ) + O ( d ) UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 13 as d → τ → ∞ . Note that this function is sign indefinite in a neighborhood of the equilibriumand can not be used in the construction of a Lyapunov function. Indeed, if Ψ ∼ ǫ / and τ ∼ ǫ − as ǫ →
0, the leading and the remainder terms in the last expression can be of the same order. Considerthe change of variables: R ( τ ) = τ − r ( τ ) , Ψ( τ ) = τ − ϕ ( τ )in system (10). It can easily be checked that the transformed system has the form drdτ = − ∂ ϕ H ( r, ϕ, τ ) , dϕdτ = ∂ r H ( r, ϕ, τ ) + F ( r, ϕ, τ ) , (15)where H ( r, ϕ, τ ) := τ H ( τ − r, τ − ϕ, τ ) − τ − rϕ , F ( r, ϕ, τ ) := τ F ( τ − r, τ − ϕ, τ ) + τ − ϕ. Using asymptotic formulas for the particular solution, we obtain H = τ (cid:16) √ λr + ω ϕ − P ′′′ ( σ ; δ, ν, κ ) ϕ (cid:0) χ − ϕ (cid:1)(cid:17) + O (∆ ) , F = − γ ϕ + O (∆ τ − )as ∆ → τ → ∞ , where ω := χ P ′′′ ( σ ; δ, ν ) / >
0. Consider the combination V ( r, ϕ, τ ) = τ − (cid:16) H ( r, ϕ, τ ) − γ rϕ (cid:17) as a Lyapunov function candidate for system (15). It follows easily that for all 0 < κ < > τ > − κ ) W ( r, ϕ ) ≤ V ( r, ϕ, τ ) ≤ (1 + κ ) W ( r, ϕ )for all ( r, ϕ, τ ) ∈ D W (∆ , τ ), where W ( r, ϕ ) := √ λr + ω ϕ /
2. The derivative of this function withrespect to τ along the trajectories of system (15) satisfies: dV dτ (cid:12)(cid:12)(cid:12) (15) = − γ (cid:16) √ λr + ω ϕ O (∆ ) (cid:17) + O ( τ − ∆ )as ∆ → τ → ∞ . Hence, for all 0 < κ < > τ > dV dτ (cid:12)(cid:12)(cid:12) (15) ≤ − γ κ V ≤ r, ϕ, τ ) ∈ D W (∆ , τ ). The last inequality implies that the fixed point (0 ,
0) of system (15) isstable and the solution ρ ∗ ( τ ), ψ ∗ ( τ ) is stable with the weights τ / and τ / .Finally, consider Case IV . The change of variables R ( τ ) = τ − r ( τ ) , Ψ( τ ) = τ − ϕ ( τ )transforms system (10) into drdτ = − ∂ ϕ H ( r, ϕ, τ ) , dϕdτ = ∂ r H ( r, ϕ, τ ) + F ( r, ϕ, τ ) , (17)where H ( r, ϕ, τ ) := τ H ( τ − r, τ − ϕ, τ ) − τ − rϕ , F ( r, ϕ, τ ) := τ − ϕ τ F ( τ − r, τ − ϕ, τ ) . Taking into account (7), it can easily be checked that H ( r, ϕ, τ ) = τ (cid:16) √ λR + ω ϕ ξ ϕ
12 + ξϕ ϕ (cid:17) + O (∆ ) , F ( r, ϕ, τ ) = − γ ϕ + O (∆ τ − )as ∆ → τ → ∞ , where ω := ξ / >
0. Consider V ( r, ϕ, τ ) = τ − (cid:16) H ( r, ϕ, τ ) − γ rϕ (cid:17) as a Lyapunov function candidate to system (17). We see that for all 0 < κ < > τ > − κ ) W ( r, ϕ ) ≤ V ( r, ϕ, τ ) ≤ (1 + κ ) W ( r, ϕ ) for all ( r, ϕ, τ ) ∈ D W (∆ , τ ), where W ( r, ϕ ) := √ λr + ω ϕ /
2. The total derivative of V withrespect to τ satisfies: dV dτ (cid:12)(cid:12)(cid:12) (17) = − γ (cid:16) √ λr + ω ϕ O (∆ ) (cid:17) + O ( τ − ∆ )as ∆ → τ → ∞ . Hence, for all 0 < κ < > τ > dV dτ (cid:12)(cid:12)(cid:12) (17) ≤ − γ κ V ≤ r, ϕ, τ ) ∈ D W (∆ , τ ). The last inequality implies that the fixed point (0 ,
0) of system (15) isstable. Returning to the variables ( ρ, ψ ) we obtain the result of the theorem. (cid:3)
Thus, the particular autoresonant solutions ρ ∗ ( τ ), ψ ∗ ( τ ) with power-law asymptotics are stable inall four cases. The stability ensures the existence of a family of solutions with a similar behaviour. Inparticular, we have the following. Corollary 1.
There exist ∆ ∗ > and T ∗ > such that for all ( ρ , ψ ) : ( ρ − ρ ∗ ( T ∗ )) +( ψ − ψ ∗ ( T ∗ )) < ∆ ∗ the solution ρ ( τ ) , ψ ( τ ) to system (1) with initial data ρ ( T ∗ ) = ρ , ψ ( T ∗ ) = ψ has the followingestimates as τ → ∞ : ρ = √ λτ + O ( τ − ) , ψ = σ + O ( τ − ) in Case I ;(18) ρ = √ λτ + O ( τ − ) , ψ = σ + O ( τ − ) in Case II ;(19) ρ = √ λτ + O ( τ − ) , ψ = σ + O ( τ − ) in Case III ;(20) ρ = √ λτ + O ( τ − ) , ψ = π O ( τ − ) in Case IV . (21) Proof.
Consider
Case I . Let R ( τ ), Ψ( τ ) be a solution to system (10) starting from the ball B (∆ ∗ )at τ = T ∗ , where ∆ ∗ = d and T ∗ = τ (see Theorem 4). Then it follows from (11) that the function v ( τ ) = V ( R ( τ ) , Ψ( τ ) , τ ) satisfies the inequality: dvdτ ≤ − γ κ v (22)as τ ≥ T ∗ , where γ κ = γ (1 − κ ) / (1 + κ ), κ ∈ (0 , τ , we obtain 0 ≤ v ( τ ) ≤ v ( T ∗ ) exp (cid:0) − γ κ ( τ − T ∗ ) (cid:1) , where 0 ≤ v ( T ∗ ) ≤ C ∗ ∆ ∗ , C ∗ = const. Thus wehave R ( τ ) = O ( e − γ κ τ ), Ψ( τ ) = O ( e − γ κ τ ) as τ → ∞ . Since ρ ( τ ) = ρ ∗ ( τ ) + R ( τ ), ψ ( τ ) = ψ ∗ ( τ ) + Ψ( τ )and ρ ∗ ( τ ) = √ λτ + O ( τ − / ), ψ ∗ ( τ ) = σ + O ( τ − ) as τ → ∞ , we have (18). Case II . Let r ( τ ), ϕ ( τ ) be a solution to system (13) starting from { ( r , ϕ ) : W ( r , ϕ ) ≤ ∆ ∗ } at τ = T ∗ , where ∆ ∗ = min { ∆ , ∆ } , T ∗ = max { τ , τ } . From (14) it follows that the derivative of thefunction v ( τ ) = V (cid:0) r ( τ ) , ϕ ( τ ) , τ (cid:1) satisfies (22) as τ ≥ T ∗ . By integrating this estimate with respectto τ and taking into account (12), we get R ( τ ) = O ( τ − / e − γ κ τ ), Ψ( τ ) = O ( τ − / e − γ κ τ ) as τ → ∞ .Returning to the variables ( ρ, ψ ) and using (3), we derive (19). Case III . Let r ( τ ), ϕ ( τ ) be a solution to system (15) with initial data from { ( r , ϕ ) : W ( r , ϕ ) ≤ ∆ ∗ } at τ = T ∗ , where ∆ ∗ = min { ∆ , ∆ } , T ∗ = max { τ , τ } . From (16) it follows that the func-tion v ( τ ) = V (cid:0) r ( τ ) , ϕ ( τ ) , τ (cid:1) satisfies (22) as τ ≥ T ∗ . Hence, R ( τ ) = O ( τ − / e − γ κ τ ), Ψ( τ ) = O ( τ − / e − γ κ τ ) as τ → ∞ . Combining this with (6), we get (20).Finally, consider Case IV . Let r ( τ ), ϕ ( τ ) be a solution to system (17) with initial data from { ( r , ϕ ) : W ( r , ϕ ) ≤ ∆ ∗ } at τ = T ∗ , where ∆ ∗ = min { ∆ , ∆ } , T ∗ = max { τ , τ } . It follows easilythat the function v ( τ ) = V (cid:0) r ( τ ) , ϕ ( τ ) , τ (cid:1) satisfies (22) as τ ≥ T ∗ . Hence, R ( τ ) = O ( τ − / e − γ κ τ ),Ψ( τ ) = O ( τ − / e − γ κ τ ) as τ → ∞ . Combining this with (7), we get (21). (cid:3) Conclusion
Thus, we have described possible autoresonant modes in oscillating systems with a combined excita-tion and a weak dissipation. The presence of dissipation in the model leads to exponential stability of apart of autoresonant modes in comparison with a systems without dissipation, where only polynomial
UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 15 - - ∆ Σ s - s + (a) κ = 0 . ν = π - - - ∆ Σ s + s - s (b) κ = 0 . ν = π - - - ∆ Σ s + s - s s (c) κ = 1 . ν = π Figure 5.
The roots to equation (4) as functions of the parameter δ (black solid lines).The vertical dotted lines correspond to s − , s and s + . The shaded areas correspond to P ′ ( σ ; δ, ν, κ ) >
0, where the particular solutions to system (1) with asymptotics (3) arestable. - - ∆ Σ s - s + (a) κ = 0 . ν = π - - - ∆ Σ s + s - s (b) κ = 0 . ν = π - - - ∆ Σ s + s - s s (c) κ = 1 . ν = π Figure 6.
The roots to equation (4) as functions of the parameter δ (dashed lines).The vertical dotted lines correspond to s − , s and s + . The black points correspond tothe roots of multiplicity 2. The shaded areas correspond to P ′′ ( σ ; δ, ν, κ ) >
0, wherethe particular solution to system (1) with asymptotics (3), ψ = σ , ψ = φ is stable.stability takes place [27]. Depending on the values of the excitation parameters and the dissipation co-efficient, the system can have different number of autoresonant modes with different phase mismatch: ψ ( τ ) ∼ σ as τ → ∞ , where σ is the root of equation (4). For every κ = γ √ λ > s ± and s consist of bifurcations points on the plane ( δ, ν ), where δ = β √ λ . Outside of this curves the rootsof equation (4) are simple and system (1) can have two or four different autoresonant modes. Theirstability depends on the sign of the value P ′ ( σ ; δ, ν, κ ) (see the shaded areas in Fig. 5).Some of the autoresonant modes coalesce, when the parameters ( δ, ν ) pass through the bifurcationcurves. The following cases are possible. (I) Equation (4) has three different roots: two simple rootsand one root of multiplicity 2 (see Fig. 6, a). (II) Equation (4) has two different roots: one simpleroot and one root of multiplicity 3 (see Fig. 7, a), or two roots of multiplicity 2 (see Fig. 6, b). (III)Equation (4) has only one root: a root of multiplicity 2 (see Fig. 6, c), or a root of multiplicity 4 (seeFig. 7, b).If σ is the root of multiplicity 2, system (1) has two autoresonant modes associated with theparticular solutions having asymptotics (3), where ψ = σ and ψ = ± φ . In this case, the solutionwith ψ = ± φ is stable if ±P ′′ ( σ ; δ, ν, κ ) > σ is the root of multiplicity 3, there - - ∆ Σ s - s + (a) κ = 0 . ν ≈ . - - ∆ Σ s - s s + (b) κ = 0 . ν = π - - - ∆ Σ s - s + (c) κ = √ . ν ≈ . Figure 7.
The roots to equation (4) as functions of the parameter δ (dashed lines).The vertical dotted lines correspond to s − , s and s + . The black points correspond tothe roots of multiplicity 3 and 4. (a) The shaded areas correspond to P ′′′ ( σ ; δ, ν, κ ) > P (4) ( σ ; δ, ν, κ ) >
0. (c) The shaded areas correspond to P ′ ( σ ; δ, ν, κ ) > Τ Ρ - Τ Ψ Figure 8.
The evolution of ρ ( τ ) and ψ ( τ ) for solutions of system (1) with λ = 1, α ( τ ) ≡ √ τ , β ( τ ) ≡ δ , γ ( τ ) ≡ κ . The black curves correspond to δ = − κ = 1, ν = 5 π/
6. The gray curves correspond to δ = − / √ κ = 1, ν = 2 π/
3. The bluecurves correspond to δ = − / κ = 1 / ν = π/ ψ = σ , ψ = 0 and ψ = χ . This mode is stable if P ′′′ ( σ ; δ, ν, κ ) > σ is the root of multiplicity4, there are two autoresonant modes with asymptotics (7), where ψ = σ , ψ = ± ξ . In this case, themode with ψ = ξ is exponentially stable, while the mode with ψ = − ξ is unstable (see Fig. 7, b).Note that the combined excitation allows to expand the use of autoresonant method for controlthe dynamics of nonlinear systems. In particular, unstable autoresonant modes in systems with pureexternal excitation ( δ = 0) can be stabilized by switching on parametric pumping (see, for example,Fig. 7, c, where the mode with σ = arcsin(4 κ/
3) becomes stable as δ < −√ . γ >
0, the parameters λ > β = 0 and ν ∈ [0 , π ) of the combined excitation can be chosenin such a way to guarantee the existence and stability of autoresonant mode with any prescribedphase shift ψ ( τ ) ≈ σ , σ ∈ [0 , π ). For example, for σ = 0, we should take ν = π − arcsin( − κδ − )and δ < − p κ + 1 /
4. In this case, P ( σ ; δ, ν, κ ) = 0, P ′ ( σ ; δ, ν, κ ) > σ = π , we should take ν = π − arcsin( − κδ − ) and δ < − κ . For σ = π/
2, one can take( δ, ν, κ ) such that 0 < ( κ − δ − ≤ ν = arcsin(( κ − δ − ) and δ cos ν < UTORESONANCE IN SYSTEMS WITH COMBINED EXCITATION AND WEAK DISSIPATION 17
Acknowledgements
The research presented in Section 3 is funded in the framework of executing the development pro-gram of Scientific Educational Mathematical Center of Privolzhsky Federal Area, additional agreementno. 075-02-2020-1421/1 to agreement no. 075-02-2020-1421.