In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
Background The Melnikov method is used in many cases to predict the occurrence of chaotic orbits in non-autonomous smooth nonlinear systems under periodic perturbation. According to the method, it is possible to construct a function called the "Melnikov function" which can be used to predict either regular or chaotic behavior of a dynamical system. Thus, the Melnikov function will be used to determine a measure of distance between stable and unstable manifolds in the Poincaré map. Moreover, when this measure is equal to zero, by the method, those manifolds crossed each other transversally and from that crossing the system will become chaotic. This method appeared in 1890 by H. Poincaré and by V. Melnikov in 1963 and could be called the "Poincaré-Melnikov Method". Moreover, it was described by several textbooks as Guckenheimer & Holmes, Kuznetsov, S. Wiggins, Awrejcewicz & Holicke and others. There are many applications for Melnikov distance as it can be used to predict chaotic vibrations. In this method, critical amplitude is found by setting the distance between homoclinic orbits and stable manifolds equal to zero. Just like in Guckenheimer & Holmes where they were the first who based on the KAM theorem, determined a set of parameters of relatively weak perturbed Hamiltonian systems of two-degrees-of-freedom, at which homoclinic bifurcation occurred.
The Melnikov distance Consider the following class of systems given by
x ˙ = ∂ H ∂ y ( x , y ) + ϵ g 1 ( x , y , t , ϵ ) y ˙ = − ∂ H ∂ x ( x , y ) + ϵ g 2 ( x , y , t , ϵ ) , ( 1 ) {\displaystyle {{\begin{array}{lcl}{\dot {x}}&=&{\frac {\partial H}{\partial y}}(x,y)+\epsilon g_{1}(x,y,t,\epsilon )\\{\dot {y}}&=&-{\frac {\partial H}{\partial x}}(x,y)+\epsilon g_{2}(x,y,t,\epsilon ),\end{array}}{(1)}}} or in vector form q ˙ = J D H ( q ) + ϵ g ( q , t , ϵ ) ( 2 ) {\displaystyle {{\dot {q}}=JDH(q)+\epsilon g(q,t,\epsilon )~\ ~\ {(2)}}}
where q = ( x , y ) {\displaystyle q=(x,y)} , D H = ( ∂ H ∂ x , ∂ H ∂ y ) {\displaystyle DH=\left({\frac {\partial H}{\partial x}},{\frac {\partial H}{\partial y}}\right)} , g = ( g 1 , g 2 ) {\displaystyle g=(g_{1},g_{2})} and
J = ( 0 1 − 1 0 ) . {\displaystyle J=\left({\begin{array}{cc}0&1\\-1&0\\\end{array}}\right).}
Assume that system (1) is smooth on the region of interest, ϵ {\displaystyle \epsilon } is a small perturbation parameter and g {\displaystyle g} is a periodic vector function in t {\displaystyle t} with the period T = 2 π ω {\displaystyle T={\dfrac {2\pi }{\omega }}} . If ϵ = 0 {\displaystyle \epsilon =0} , then there is an unperturbed system
q ˙ = J D H ( q ) . ( 3 ) {\displaystyle {{\dot {q}}=JDH(q).~\ ~\ {(3)}}}
From this system (3), looking at the phase space in Figure 1, consider the following assumptions
A1 - The system has a hyperbolic fixed point p 0 {\displaystyle p_{0}} , connected to itself by a homoclinic orbit q 0 ( t ) = ( x 0 ( t ) , y 0 ( t ) ) ; {\displaystyle q_{0}(t)=(x_{0}(t),y_{0}(t));}
A2 - The system is filled inside Γ p 0 {\displaystyle \Gamma _{p_{0}}} by a continuous family of periodic orbits q α ( t ) {\displaystyle q^{\alpha }(t)} of period T α {\displaystyle T^{\alpha }} with α ∈ ( − 1 , 0 ) , {\displaystyle \alpha \in (-1,0),} where Γ p 0 = { q ∈ R 2 | q = q 0 ( t ) , t ∈ R } = W s ( p 0 ) ∩ W u ( p 0 ) ∪ { p 0 } . {\displaystyle \Gamma _{p_{0}}=\{q\in \mathbb {R} ^{2}|q=q_{0}(t),t\in \mathbb {R} \}=W^{s}(p_{0})\cap W^{u}(p_{0})\cup \{p_{0}\}.}
To obtain the Melnikov function, some tricks have to be used, for example, to get rid of the time dependence and to gain geometrical advantages new coordinate has to be used ϕ {\displaystyle \phi } that is cyclic type given by ϕ = ω t + ϕ 0 . {\displaystyle \phi =\omega t+\phi _{0}.} Then, the system (1) could be rewritten in vector form as follows
q ˙ = J D H ( q ) + ϵ g ( q , ϕ , ϵ ) ϕ ˙ = ω . ( 4 ) {\textstyle {{\begin{array}{lcl}{\dot {q}}&=&JDH(q)+\epsilon g(q,\phi ,\epsilon )\\{\dot {\phi }}&=&\omega .\end{array}}~\ ~\ {(4)}}}
Hence, looking at Figure 2, the three-dimensional phase space R 2 × S 1 , {\displaystyle \mathbb {R} ^{2}\times \mathbb {S} ^{1},} where q ∈ R 2 {\displaystyle q\in \mathbb {R} ^{2}} and ϕ ∈ S 1 {\displaystyle \phi \in \mathbb {S} ^{1}} has the hyperbolic fixed point p 0 {\displaystyle p_{0}} of the unperturbed system becoming a periodic orbit γ ( t ) = ( p 0 , ϕ ( t ) ) . {\displaystyle \gamma (t)=(p_{0},\phi (t)).} The two-dimensional stable and unstable manifolds of γ ( t ) {\displaystyle \gamma (t)} by W s ( γ ( t ) ) {\displaystyle W^{s}(\gamma (t))} and W u ( γ ( t ) ) {\displaystyle W^{u}(\gamma (t))} are denoted, respectively. By the assumption A 1 , {\displaystyle A1,} W s ( γ ( t ) ) {\displaystyle W^{s}(\gamma (t))} and W u ( γ ( t ) ) {\displaystyle W^{u}(\gamma (t))} coincide along a two-dimensional homoclinic manifold. This is denoted by Γ γ = { ( q , ϕ ) ∈ R 2 × S 1 | q = q 0 ( − t 0 ) , t 0 ∈ R ; ϕ = ϕ 0 ∈ ( 0 , 2 π ] } , {\displaystyle \Gamma _{\gamma }=\{(q,\phi )\in \mathbb {R} ^{2}\times \mathbb {S} ^{1}|q=q_{0}(-t_{0}),t_{0}\in \mathbb {R} ;\phi =\phi _{0}\in (0,2\pi ]\},} where t 0 {\displaystyle t_{0}} is the time of flight from a point q 0 ( − t 0 ) {\displaystyle q_{0}(-t_{0})} to the point q 0 ( 0 ) {\displaystyle q_{0}(0)} on the homoclinic connection. In the Figure 3, for any point p ≡ ( q 0 ( − t 0 ) , ϕ 0 ) , {\displaystyle p\equiv (q_{0}(-t_{0}),\phi _{0}),} a vector is constructed π p {\displaystyle \pi _{p}} , normal to the Γ γ {\displaystyle \Gamma _{\gamma }} as follows π p ≡ ( D H ( q 0 ( − t 0 ) , 0 ) . {\displaystyle \pi _{p}\equiv (DH(q_{0}(-t_{0}),0).} Thus varying t 0 {\displaystyle t_{0}} and ϕ 0 {\displaystyle \phi _{0}} serve to move π p {\displaystyle \pi _{p}} to every point on Γ γ . {\displaystyle \Gamma _{\gamma }.}
Splitting of stable and unstable manifolds If ϵ ≠ 0 {\displaystyle \epsilon \neq 0} is sufficiently small, which is the system (2), then γ ( t ) {\displaystyle \gamma (t)} becomes γ ϵ ( t ) , {\displaystyle \gamma _{\epsilon }(t),} Γ γ {\displaystyle \Gamma _{\gamma }} becomes Γ γ ϵ , {\displaystyle \Gamma _{\gamma _{\epsilon }},} and the stable and unstable manifolds become different from each other. Furthermore, for this sufficiently small ϵ {\displaystyle \epsilon } in a neighborhood N ( ϵ 0 ) , {\displaystyle {\mathcal {N}}(\epsilon _{0}),} the periodic orbit γ ( t ) {\displaystyle \gamma (t)} of the unperturbed vector field (3) persists as a periodic orbit, γ ϵ ( t ) = γ ( t ) + O ( ϵ ) . {\displaystyle \gamma _{\epsilon }(t)=\gamma (t)+{\mathcal {O}}(\epsilon ).} Moreover, W l o c s ( γ ϵ ( t ) ) {\displaystyle W_{loc}^{s}(\gamma _{\epsilon }(t))} and W l o c u ( γ ϵ ( t ) ) {\displaystyle W_{loc}^{u}(\gamma _{\epsilon }(t))} are C r {\displaystyle C^{r}} ϵ {\displaystyle \epsilon } -close to W l o c s ( γ ( t ) ) {\displaystyle W_{loc}^{s}(\gamma (t))} and W l o c u ( γ ( t ) ) {\displaystyle W_{loc}^{u}(\gamma (t))} respectively.
Consider the following cross-section of the phase space Σ ϕ 0 = { ( q , ϕ ) ∈ R 2 | ϕ = ϕ 0 } , {\displaystyle \Sigma ^{\phi _{0}}=\{(q,\phi )\in \mathbb {R} ^{2}|\phi =\phi _{0}\},} then ( q ( t ) , ϕ ( t ) ) {\displaystyle (q(t),\phi (t))} and ( q ϵ ( t ) , ϕ ( t ) ) {\displaystyle (q_{\epsilon }(t),\phi (t))} are the trajectories of the unperturbed and perturbed vector fields, respectively. The projections of these trajectories onto Σ ϕ 0 {\displaystyle \Sigma ^{\phi _{0}}} are given by ( q ( t ) , ϕ 0 ( t ) ) {\displaystyle (q(t),\phi _{0}(t))} and ( q ϵ ( t ) , ϕ 0 ( t ) ) . {\displaystyle (q_{\epsilon }(t),\phi _{0}(t)).} Looking at the Figure 4, splitting of W s ( γ ϵ ( t ) ) {\displaystyle W^{s}(\gamma _{\epsilon }(t))} and W u ( γ ϵ ( t ) ) , {\displaystyle W^{u}(\gamma _{\epsilon }(t)),} is defined hence, consider the points that intersect π p {\displaystyle \pi _{p}} transversely as p ϵ s {\displaystyle p_{\epsilon }^{s}} and p ϵ u {\displaystyle p_{\epsilon }^{u}} , respectively. Therefore, it is natural to define the distance between W s ( γ ϵ ( t ) ) {\displaystyle W^{s}(\gamma _{\epsilon }(t))} and W u ( γ ϵ ( t ) ) {\displaystyle W^{u}(\gamma _{\epsilon }(t))} at the point p , {\displaystyle p,} denoted by d ( p , ϵ ) ≡ | p ϵ s − p ϵ u | {\displaystyle d(p,\epsilon )\equiv |p_{\epsilon }^{s}-p_{\epsilon }^{u}|} and it can be rewritten as d ( p , ϵ ) = ( p ϵ s − p ϵ u ) ⋅ ( D H ( q 0 ( − t 0 ) , 0 ) ∥ ( D H ( q 0 ( − t 0 ) , 0 ) ∥ . {\displaystyle d(p,\epsilon )={\dfrac {(p_{\epsilon }^{s}-p_{\epsilon }^{u})\cdot (DH(q_{0}(-t_{0}),0)}{\parallel (DH(q_{0}(-t_{0}),0)\parallel }}.} Since p ϵ s {\displaystyle p_{\epsilon }^{s}} and p ϵ u {\displaystyle p_{\epsilon }^{u}} lie on π p , p ϵ s = ( q ϵ s , ϕ 0 ) {\displaystyle \pi _{p},p_{\epsilon }^{s}=(q_{\epsilon }^{s},\phi _{0})} and p ϵ u = ( q ϵ u , ϕ 0 ) , {\displaystyle p_{\epsilon }^{u}=(q_{\epsilon }^{u},\phi _{0}),} and then d ( p , ϵ ) {\displaystyle d(p,\epsilon )} can be rewritten by
d ( t 0 , ϕ 0 , ϵ ) = D H ( q 0 ( − t 0 ) ) ⋅ ( q ϵ u − q ϵ s ) ∥ ( D H ( q 0 ( − t 0 ) ) ∥ . ( 5 ) {\textstyle {d(t_{0},\phi _{0},\epsilon )={\dfrac {DH(q_{0}(-t_{0}))\cdot (q_{\epsilon }^{u}-q_{\epsilon }^{s})}{\parallel (DH(q_{0}(-t_{0}))\parallel }}.~\ ~\ {(5)}}}
The manifolds W s ( γ ϵ ( t ) ) {\displaystyle W^{s}(\gamma _{\epsilon }(t))} and W u ( γ ϵ ( t ) ) {\displaystyle W^{u}(\gamma _{\epsilon }(t))} may intersect π p {\displaystyle \pi _{p}} in more than one point as shown in Figure 5. For it to be possible, after every intersection, for ϵ {\displaystyle \epsilon } sufficiently small, the trajectory must pass through N ( ϵ 0 ) {\displaystyle {\mathcal {N}}(\epsilon _{0})} again.
Deduction of the Melnikov function Expanding in Taylor series the eq. (5) about ϵ = 0 , {\displaystyle \epsilon =0,} gives us d ( t 0 , ϕ 0 , ϵ ) = d ( t 0 , ϕ 0 , 0 ) + ϵ ∂ d ∂ ϵ ( t 0 , ϕ 0 , 0 ) + O ( ϵ 2 ) , {\displaystyle d(t_{0},\phi _{0},\epsilon )=d(t_{0},\phi _{0},0)+\epsilon {\frac {\partial d}{\partial \epsilon }}(t_{0},\phi _{0},0)+{\mathcal {O}}(\epsilon ^{2}),} where d ( t 0 , ϕ 0 , 0 ) = 0 {\displaystyle d(t_{0},\phi _{0},0)=0} and ∂ d ∂ ϵ ( t 0 , ϕ 0 , 0 ) = D H ( q 0 ( − t 0 ) ) ⋅ ( ∂ q ϵ u ∂ ϵ | ϵ = 0 − ∂ q ϵ s ∂ ϵ | ϵ = 0 ) ∥ ( D H ( q 0 ( − t 0 ) ) ∥ . {\displaystyle {\frac {\partial d}{\partial \epsilon }}(t_{0},\phi _{0},0)={\dfrac {DH(q_{0}(-t_{0}))\cdot \left({\frac {\partial q_{\epsilon }^{u}}{\partial \epsilon }}{\Big |}_{\epsilon =0}-{\frac {\partial q_{\epsilon }^{s}}{\partial \epsilon }}{\Big |}_{\epsilon =0}\right)}{\parallel (DH(q_{0}(-t_{0}))\parallel }}.}
When d ( t 0 , ϕ 0 , ϵ ) = 0 , {\displaystyle d(t_{0},\phi _{0},\epsilon )=0,} then the Melnikov function is defined to be
M ( t 0 , ϕ 0 ) ≡ D H ( q 0 ( − t 0 ) ) ⋅ ( ∂ q ϵ u ∂ ϵ | ϵ = 0 − ∂ q ϵ s ∂ ϵ | ϵ = 0 ) , ( 6 ) {\displaystyle {M(t_{0},\phi _{0})\equiv DH(q_{0}(-t_{0}))\cdot \left({\frac {\partial q_{\epsilon }^{u}}{\partial \epsilon }}{\Big |}_{\epsilon =0}-{\frac {\partial q_{\epsilon }^{s}}{\partial \epsilon }}{\Big |}_{\epsilon =0}\right),~\ ~\ {(6)}}}
since D H ( q 0 ( − t 0 ) ) = ( ∂ H ∂ x ( q 0 ( − t 0 ) ) , ∂ H