In this paper, some stability results were reviewed. A suitable and complete Lyapunov function for the hard spring model was constructed using the Cartwright method. This approach was compared with the existing results which confirmed a superior global stability result. Our contribution relies on its application to high damping door constructions. (2010 Mathematics Subject Classification: 34B15, 34C15, 34C25, 34K13.)
In real life, most problems that occur are non-linear in nature and may not have analytic solutions except by approximations or simulations and so trying to find an explicit solution may in general be complicated and sometimes impossible. Duffing’s equation is a second order non-linear differential equation used to model such problems of non-linear nature [
x ¨ + c x ˙ + g ( t , x ) = p ( t ) (1.1)
where p ( t ) is continuous and 2π-periodic in t ∈ ℝ .
The study of periodic solution of Duffing’s equation is like that of the study of classical Hamiltonian equation of motion which is characterized by multiplicity of periodic solution. Various techniques for investigating the stability of periodic solutions of Duffing’s equation have been reported, (see [
Our construction of suitable Lyapunov function rests squarely on the approach of [
u ¨ + δ u ˙ + α u + β u 3 + μ u 5 = 0 (1.2)
was carried out. Using Equation (1.2) the existence of a three term valid solution was obtained using derivative expansion method. The derivative expansion method is one of the perturbative methods which require the existence of a small parameter and is therefore not valid in principle for the Duffing equation in which the nonlinearity is large. Its stability analysis of the equilibrium point was carried out using the eigenvalue approach. Secondly, our motivation was augmented by reading the works of [
x ¨ + c x ˙ + g ( t , x ) = 0 (1.3)
were derived. This approach exposed by [
x ¨ + c x ˙ + a x + b x 2 + 2 x 3 = h ( t ) (1.4)
with boundary conditions as:
x ( 0 ) = x ( 2π ) (1.5)
x ˙ ( 0 ) = x ˙ (2π)
where a , b , c are real constants and h : [ 0 , 2 π ] → ℝ N is continuous. In Equation (1.4), a is the stiffness constant, c is the coefficient of viscous damping and b x 2 + 2 x 3 represents the nonlinearity in the restoring force acting like a hard spring. Equation (1.4) has received wide interest in neurology, modelling of mechanical systems such as shock absorbers in most vehicles. It can be used to model plant stems to better understand the effect of nonlinear stiffness or resonant behavior of plants. It is a model arising in many branches of Physics and Engineering such as oscillation of rigid pendulum using moderately large amplitude motion [
Definition 2.1. (Properties of Lyapunov Function)
The Lyapunov function has the following properties:
a) Continuity: V ( t , X ) is continuous and single valued under the condition t ≥ T and | x i | < H and V ( t , 0 ) ≡ 0 .
b) V ( t , X ) is positive definite; and
c) V ˙ = ∂ V ∂ x 1 x ˙ 1 + ∂ V ∂ x 2 x ˙ 2 + ⋯ + ∂ V ∂ x n x ˙ n + ∂ V ∂ t , representing the total derivative with respect to t is negative definite.
Definition 2.2. (Complete Lyapunov function)
A Lyapunov function V defined as V : I × ℝ n → ℝ is said to be COMPLETE if for X ∈ ℝ n ,
1) V ( t , X ) ≥ 0 ；
2) V ( t , X ) = 0 if and only if X = 0 and
3) V ˙ ( t , X ) ≤ − c | X | where c is any positive constant and | X | given by
| X | = ( ∑ i = 1 n ( x i 2 ) ) 1 2 → ∞
It is INCOMPLETE if 3) is not satisfied.
Theorem 2.3. Consider a system of differential equations
X ˙ = f ( x ) (1.6)
f : D → ℝ 2 . Let X = 0 be an equilibrium point of Equation (1.6).
Let V : D → ℝ be a continuously differentiable function; such that:
1) V ( 0 ) = 0 ；
2) V ( x ) > 0 in D − { 0 } ；
3) V ˙ ( x ) ≤ 0 in D − { 0 } .
Then X = 0 is stable.
Theorem 2.4. Let V : D → ℝ be a continuously differentiable function such that:
1) V ( 0 ) = 0 ;
2) V ( x ) > 0 in D − { 0 } ;
3) V ˙ ( x ) < 0 in D − { 0 } .
Then X = 0 is “asymptotically stable”.
Theorem 2.5. Let V : D → ℝ be a continuously differentiable function such that:
1) V ( 0 ) = 0 ;
2) V ( x ) > 0 in D − { 0 } ;
3) V ( x ) is “radially bounded”;
4) V ˙ ( x ) < 0 in D − { 0 } .
Then X = 0 is “globally asymptotically stable”.
Theorem 2.6. Suppose all conditions for asymptotic stability are satisfied. In addition to it, suppose there exists constants K 1 , K 2 , K 3 , P
1) K 1 ‖ X ‖ P ≤ V ( X ) ≤ K 2 ‖ X ‖ P ;
2) V ˙ ( X ) ≤ − K 3 ‖ X ‖ P .
Then the origin X = 0 is “exponentially stable”.
Moreover, if this conditions hold globally, then the origin X = 0 is globally exponentially stable.
We adapt the method of construction of Lyapunov function used in [
x ¨ + c x ˙ + a x + b x 2 + 2 x 3 = p ( t ) (1.7)
The equivalent systems of Equation (1.7) is
x ˙ 1 = x 2 x ˙ 2 = − c x 2 − a x 1 − h ( x 1 ) (1.8)
where h ( x 1 ) = b x 1 2 + 2 x 1 3 and p ( t ) = 0 .
Writing the equivalent systems of Equation (1.7) in compact form, we have
X ˙ = A X (1.9)
where A = [ 0 1 − a − h ( x ) − c ] , X = [ x 1 x 2 ] .
The method discussed here is based on the fact that the matrix A has all its eigenvalues as negative real parts. Then from the general theory which corresponds to any positive quadratic form U ( x ) , there exists another positive definite quadratic form V ( x ) such that
V ˙ = − U (1.10)
Choosing the most general quadratic form of order two and picking the coefficient in the quadratic form to satisfy Equation (1.10) along the solution paths of Equation (1.10) we assume V to be defined by
2 V = A x 2 + B y 2 + 2 K x y (1.11)
V ˙ = A x x ˙ + B y y ˙ + K ( x ˙ y + y ˙ x ) = A x y + B ( − c y − a x − h ( x ) ) + K y 2 + K x ( − c y − a x − h ( x ) ) = A x y − B c y 2 − B y a x − B y h ( x ) + K y 2 − K c x y − K a x 2 − K x h (x)
Simplifying the coefficients we have
V ˙ = ( A − B a − K c ) x y + ( K − B c ) y 2 − ( K a ) x 2 − ( K x + B y ) h ( x ) (1.12)
To make V ˙ negative definite, we equate the coefficient of mixed variable to zero and the coefficients of x 2 and y 2 to any positive constant (say δ ) as outlined in
A − B a − K c = 0 (1)
K − B c = δ (2)
− K a = δ (3)
− ( K x + B y ) = 0 (4)
From Equation (3) above
− K a = δ
K = − δ a (1.13)
Then substituting the value of K into Equation (2) we obtain
− δ a − B c = δ
− B c = δ + δ a
B = − δ ( a + 1 ) c a (1.14)
Substituting for K and B in (1) we have
A = B a + K c = − δ ( a + 1 ) c − δ c a (1.15)
which by further simplification gives that
A = − δ c a [ ( a + 1 ) a + c 2 ] (1.16)
Substituting for the values of the constant A, B, K in Equation (1.11) gives
2 V = − δ c a [ ( a + 1 ) a + c 2 ] x 2 − δ c a [ a + 1 ] y 2 − 2 δ a x y
| Terms | Coefficient |
|---|---|
| x y | A − B a − K c |
| y 2 | K − B c |
| x 2 | − K a |
| h ( x ) | − ( K x + B y ) |
V = − δ 2 c a [ ( ( a + 1 ) a + c 2 ) x 2 + ( a + 1 ) y 2 + 2 c x y ] (1.17)
By choosing δ c a = − 1
V ( x ) = 1 2 [ ( ( a + 1 ) a + c 2 ) x 2 + ( a + 1 ) y 2 + 2 c x y ] > 0 (1.18)
Using Equation (1.19) and the fact that V ˙ < 0 , the equilibrium point is asymptotically stable.
Considering the Duffing equation of the form (1.7), the first equivalent systems of (1.7) is given by
x ˙ = y y ˙ = − c y − a x − b x 2 − 2 x 3 + h ( t ) (1.19)
For the unforced case, Equation (1.19) is reduced to
x ˙ = y y ˙ = − c y − a x − b x 2 − 2 x 3 (1.20)
At fixed points, x ˙ = y = 0.
So that y = 0 and y ˙ = − a x − b x 2 − 2 x 3 = x ( − a x − b x − 2 x 2 ) = 0.
Giving us x = 0 , x 1 = − b + b 2 − 8 a 4 and x 2 = − b − b 2 − 8 a 4 which correspond to ( 0 , 0 ) , ( x 1 , 0 ) and ( x 2 , 0 ) at fixed point. Analysis of the stability of the fixed points can be done by linearizing Equation (1.20) which gives
x ¨ = y ˙ y ¨ = − c y ˙ − ( a + 2 b x + 6 x 2 ) x ˙ (1.21)
The matrix equation for (1.21) can be written as
[ x ¨ y ¨ ] = [ 0 1 − ( a + 2 b x + 6 x 2 ) − c ] [ x ˙ y ˙ ] (1.22)
Examining the stability at the point (0,0) gives
[ 0 − λ 1 − ( a + 2 b x + 6 x 2 ) − λ − c − λ ]
λ c + λ 2 + a + 2 b x + 6 x 2 + λ = 0
λ 2 + λ ( 1 + c ) + g = 0 (1.23)
λ = − ( 1 + c ) ± ( 1 + c ) 2 − 4 g 2 (1.24)
where λ 1 = 1 2 ( − ( 1 + c ) + ( 1 + c ) 2 − 4 g ) and λ 2 = 1 2 ( − ( 1 + c ) − ( 1 + c ) 2 − 4 g ) are the roots of Equation (1.23). λ 1 and λ 2 can be written as
λ 1 = 1 2 ( − δ + δ 2 − 4 g ) and λ 2 = 1 2 ( − δ − δ 2 − 4 g ) where δ = 1 + c and
g = a + 2 b x + 6 x 2
The coefficient of β shows that the Duffing equation is highly damped representing a hard spring. This hard spring is represented in Equation (1.21) where the coefficient of x 2 is 6.
At the origin, λ ± ( 0 , 0 ) = 1 2 ( − δ ± δ 2 − 4 g ) .
With δ = 0 , λ 1 , 2 = 1 2 ( ± − 4 g ) .
For this, we consider the following cases:
1) When g = 0 , λ 1 , 2 = 0 and this implies that b , x , a are all zero;
2) When g > 0 , λ 1 , 2 = ± i g which corresponds to critical points that are centres for which stability is ensured;
3) When g < 0 , λ 1 , 2 = ± g which corresponds to saddles giving rise to instability [
With δ > 0 , λ 1 , 2 = 1 2 ( − δ ± δ 2 − 4 g ) .
For this, we consider the following cases:
1) When g = 0 , λ 1 , 2 = 0 , − δ ;
2) When g > 0 , λ 1 , 2 = 1 2 ( − δ ± δ 2 − 4 g ) .
For the discriminate, we have the following cases:
When δ 2 < 4 g , λ 1 , 2 = 1 2 ( − δ ± i δ 2 − 4 g ) shows that the roots are imaginary which to lead to spiral and asymptotic stability.
When δ 2 > 4 g , λ 1 , 2 = 1 2 ( − δ ± δ 2 − 4 g ) shows that the roots are real which leads to saddles and instability.
When δ 2 = 4 g , λ 1 , 2 = ± g shows that the roots are real corresponding to saddles and instability.
When g < 0 , λ 1 , 2 = 1 2 ( δ ± δ 2 + 4 g ) .
For the discriminate, we consider the following cases:
1) When δ 2 + 4 g < 0 , λ 1 , 2 = 1 2 ( δ ± i δ 2 + 4 g ) which corresponds to spirals and asymptotic stability.
2) When δ 2 + 4 g > 0 , λ 1 , 2 = 1 2 ( δ ± i δ 2 + 4 g ) which leads to instability.
3) When δ 2 + 4 g = 0 , λ 1 , 2 = ± ( i g ) which leads to centres and instability.
Interestingly, for special case when c = 0 with no forcing term equation (1.20) becomes
x ˙ = y (1.25)
y ˙ = − a x − b x 2 − 2 x 3 (1.26)
The above can be integrated by quadrature, differentiating (1.25) and plugging in (1.26) gives
x ˙ = y ˙ = − a x − b x 2 − 2 x 3 (1.27)
Multiplying both sides by x ˙ gives
x ˙ x ˙ − a x x ˙ − b x ˙ x 2 − 2 x ˙ x 3 = 0 (1.28)
Equation (1.28) can be written as
d d t [ 1 2 x ˙ 2 − 1 2 a x 2 − 1 3 b x 3 − 1 2 x 4 ] = 0 (1.29)
So we have a variant of motion
h = 1 2 x ˙ 2 − 1 2 a x 2 − 1 3 b x 3 − 1 2 x 4 (1.30)
solving for x ˙ 2 gives x ˙ 2 = ( d x d t ) 2 = 2 h + a x 2 + 2 3 b x 3 + x 4
d x d t = 2 h + a x 2 + 2 3 b x 3 + x 4
t = ∫ d t = ∫ d t 2 h + a x 2 + 2 3 b x 3 + x 4 ( [
Note that the invariant of motion h satisfies x ˙ = ∂ h ∂ x ˙ = ∂ h ∂ y where
∂ h ∂ x = a x + b x 2 + 2 x 3 = y ˙ (1.31)
So the equation of Duffing oscillator are given by the Hamiltonian system
x ˙ = ∂ h ∂ y and y ˙ = − ∂ h ∂ x ( [
We consider the stability analysis of Duffing’s equation for different values of a, b, c, as illustrated in
| a = 0.1 b = 0.2 c = 0.1 | x 12 = − 0.2 ± − 0.76 4 x 21 = − 0.2 ± − 0.76 4 | −0.05 −0.05 0.168 −1.07 | Spiral Spiral Saddle Spiral | Asymptotically stable Asymptotically stable Unstable Stable |
|---|---|---|---|---|
| a = 0.6 b = 0.3 c = 0.5 | x 12 = − 0.2 ± − 4.71 4 x 21 = − 0.2 ± 4.71 4 | −0.0075 −0.0075 0.468 −0.618 | Spiral Spiral Saddle Spiral | Asymptotically stable Asymptotically stable Unstable Stable |
| a = 0.01 b = 0.02 c = 0.03 | x 12 = − 0.2 ± − 0.0796 4 x 21 = − 0.2 ± 0.0796 4 | −0.02 −0.02 0.066 −0.076 | Spiral Spiral Saddle Spiral | Asymptotically stable Asymptotically stable Unstable Stable |
| a = 0.2 b = 0.2 c = 0.03 | x 12 = − 0.2 ± − 1.56 4 x 21 = − 0.2 ± 1.56 4 | −0.2 −0.2 0.262 −0.362 | Spiral Spiral Saddle Spiral | Asymptotically stable Asymptotically stable Unstable Stable |
Field work 2017.
a : = 0.1 b : = 0.2 c : = 0.1
t 0 : = 0 t 1 : = 150 Solution interval endpoints;
i c : = [ 0 1 ] Initial condition vector;
N : = 1500 Number of solution values on [ t 0 , t 1 ] .
D ( t , X ) : = [ X 1 − a ⋅ X 0 − b ⋅ ( X 0 ) 2 − 2 ⋅ ( X 0 ) 3 − c ⋅ X 1 ] Derivative function.
S : = r k f i x e d ( i c , t 0 , t 1 , N , D ) .
T : = S 〈 0 〉 Independent variable values.
X 1 : = S 〈 1 〉 Solution function values.
X 2 : = S 〈 2 〉 Derivative function values.
The unforced and damped cubic-quintic Duffing oscillator is given by
u ¨ + δ u ˙ + α u + β u 3 + μ u 5 = 0 (1.32)
From Equation (1.32) we obtain the autonomous dynamical system
y = u ˙ y ˙ = − δ u ˙ − α u − β u 3 − μ u 5 } (1.33)
From the condition u ˙ = 0 we obtain y = 0 which gives us
α u + β u 3 + μ u 5 = 0 (1.34)
Equation (1.34) can be written as
u [ α + β u 2 + μ u 4 ] = 0 (1.35)
We obtain the roots of (1.35) as
u 0 = 0 , u 1 , 2 , 3 , 4 = ± 1 2 μ [ − β ± β 2 − 4 μ α ]
Then the equation satisfied by the eigenvalues of our systems stability matrix is
λ 2 + δ λ = − α − 3 β u 2 − 5 μ u 4 (1.36)
where u _ denotes u 1 u 2 u 3 and u 4 or the u-coordinate of an equilibrium point. Whether our eigenvalues will be complex, real or imaginary will be determined by the values of δ and g = 5 μ u 4 + 3 β u 2 + α with = 0, λ 1 , 2 = 1 2 [ ± − ( 4 g ) ] . For this we consider the following cases:
1) g = 0 : for this case λ 1 , 2 = 0.
This contradicts our assumption that det J ≠ 0 and it also implies that α , β , μ are all zero.
2) g > 0 : for this case λ 1 , 2 = ± i g .
This corresponds to critical points that are centers for which stability is ensured.
3) g < 0 : for this case λ 1 , 2 = ± g which corresponds to saddles giving rise to instability in (1.35) with δ > 0 ,
λ 1 , 2 = 1 2 [ − δ ± δ 2 − 4 g ]
Then considering the case below we have:
1) g = 0 : giving the values λ 1 , 2 = 0 , − δ which goes contrary to our aassumption that det j ≠ 0 . It also implies that α , β , μ are all zero.
2) g > 0 : for this case
λ 1 , 2 = 1 2 [ − δ ± δ 2 − 4 g ]
Considering the discriminant fot this case, we have three cases.
a) δ 2 < 4 g , λ 1 , 2 = 1 2 [ − δ ± δ 2 − 4 g ] which corresponds to spirals and asymptotic stability.
b) The case δ 2 > 4 g gives the values of the form λ 1 , 2 = 1 2 [ − δ ± p ] , p > 0 which corresponds to nodes resulting in asymptotic stability if δ > p and to saddles and consequently instability if δ < p .
c) δ 2 = 4 g gives λ 1 , 2 = ± g . For this case we have saddles and hence instability.
3) g < 0 leads us to have λ 1 , 2 = 1 2 [ δ ± δ 2 − 4 g ] .
Considering the discriminate we consider the three cases as follows:
a) δ 2 + 4 g < 0 : this case gives λ 1 , 2 = 1 2 [ δ ± i δ 2 − 4 g ] and we have spirals and consequently asymptotic stability.
b) δ 2 + 4 g > 0 gives λ 1 , 2 = 1 2 [ δ ± i Q ] , Q > 0 and this leads us to the existence of node and instability if δ > Q or saddles and instability if δ < Q .
c) δ 2 + 4 g = 0 , λ 1 , 2 = ± [ i g ] which yield centers and stability.
We now consider the stability analysis of the dynamics for a few choices of α , β , μ and δ by using employing Equations (1.35) and (1.36). These are illustrated in
We consider the Duffing’s equation of the form
x ¨ + c x ˙ + g ( t , x ) = 0 (1.37)
with c > 0 and g : [ 0 , 2 π ] × R → R is a L 1 -caratheodory function such that the partial derivative g x exists and it is:
L 1 -caratheodory. For a given 1 ≤ p ≤ + ∞ , let us define the set
Ω p , c = { a ∈ L p ( 0 , 2 π ) : a > 0 , ‖ a ‖ p < ( 1 + c 2 4 ) k ( 2 p * ) }
with p * , k defined. Let φ be a 2π periodic solution of Equation (1.37), we
| α = β = μ = 1 | U 1 , 2 = ± − 1 + i 3 2 U 3 , 4 = ± − 1 − i 3 2 | −1 −1 1 −1 | Spirals Spirals Spirals | Asymptotically stable Unstable Unstable |
|---|---|---|---|---|
| α = 1 , β = μ = − 1 | U 1 , 2 = ± − 1 − i 5 2 U 3 , 4 = ± − 1 + i 5 2 | −1 −1 1 −1 | Saddles Saddles Saddles Saddles | Unstable Unstable Unstable Unstable |
| α = 1 , β = − 1 , μ = 0.01 | U 1 , 2 = ± − 1 + 0.96 0.02 U 3 , 4 = ± 1 − 0.96 0.02 | −1 −1 1 −1 | Spirals Spirals Saddle | Asymptotically stable Unstable Unstable Unstable Unstable |
| α = − 1 , β = μ = 1 | U 1 , 2 = ± − 1 + 5 2 U 3 , 4 = ± − 1 − 5 2 | −1 −1 1 −1 | Spirals Spirals Spirals Spirals | Asymptotically stable Unstable Asymptotically stable Unstable |
| α = − 1 , β = 1 , μ = − 0.02 | U 1 , 2 = ± 1 − 0.92 0.04 | −1 −1 1 −1 | Spirals Saddles Saddles Saddles | Asymptotically stable Unstable Unstable Unstable |
assume that a ∈ Ω p , c exists such that
g x ( t , φ ( x ) ) < a ( t ) for a.e t ∈ [ 0 , 2 π ] (1.38)
For such an a, the operator L as defined in maximum principle is inversely positive. By defining the operator.
F : C [ 0 , 2 π ] → L 1 ( 0 , 2 π ) , H : L 1 ( 0 , 2 π ) → W
G x = a ( x ) → g ( x ) H = L − 1 F .
The solution φ can be seen as a fixed point of the operator H. Next, we recall that an isolated 2π-periodic solution has a well-defined index γ ( φ ) and | γ ( φ ) ≤ 1 | .
The study of the stability analysis of periodic solution using Lyapunov direct
method and Matcad has yielded successful results. We discovered that global stability of periodic solution was achieved through the construction of a suitable and complete Lyapunov function for the hard spring system using Cartwright method. From the above tables discussed and outlined, we could see that the stability behaviours of the solutions are similar despite the different methods. Our technique showed superiority above others because stability is a property of the equilibrium point and of the system.
The authors declare no conflicts of interest regarding the publication of this paper.
Eze, E.O., Obasi, U.E. and Agwu, E.U. (2019) Stability Analysis of Periodic Solutions of Some Duffing’s Equations. Open Journal of Applied Sciences, 9, 198-214. https://doi.org/10.4236/ojapps.2019.94017