Lin's method
| Xiao-Biao Lin (2008), Scholarpedia, 3(9):6972. | doi:10.4249/scholarpedia.6972 | revision #185179 [link to/cite this article] |
Lin's method refers to an implementation of the Lyapunov-Schmidt method to construct solutions that stay near a finite or infinite chain of heteroclinic solutions (LSH).
Consider the system of \(n\) equations \[\tag{1} \dot x(t) = f(x(t),\mu), \]
where \(f:\mathbb{R}^n \times M \to \mathbb{R}^n \) and \(\mu\) is a parameter in a linear space \( M \ .\)
An infinite sequence of heteroclinic solutions \(\{q_i(t)\}_{i\in\mathbb{Z}}\) is called a chain if there is a sequence of equilibria \(\{p_i\}_{i\in\mathbb{Z}}\) such that \[\lim_{t\to -\infty} q_i(t) = p_i,\quad \lim_{t\to\infty} q_i(t)=p_{i+1}. \] An infinite chain may come from a finite sequence of heteroclinic orbits by repeating its entries. Assume that Eqn. (#dotxeqf) has a chain when \(\mu=0\ .\) Let \(\Sigma_i\) be a codimension one plane through \(q_i(0)\) that is orthogonal to \(\dot q_i(0)\ .\) \[ \Sigma_i:=\left\{x|\langle \dot q_i(0), x-q_i(0)\rangle=0\right\}. \] We look for conditions on \(\{\omega_i\}_{i\in\mathbb{Z}}\) and \(\mu\) such that there exists a solution \(x(t)\) that lies near the chain and the time spent by \(x(t)\) from \(\Sigma_{i-1}\) to \(\Sigma_i\) is \(2\omega_i\ .\) For a finite chain, \(x(t)\) starts and ends at equilibrium points and \(\{\omega_i\}\) starts and ends at the symbol \(\infty\ .\)
Let the orbit of \(x(t)\) be the union of those of \(x_i(t)\ ,\) defined on \(t\in [-\omega_i,\omega_{i+1}]\) and subject to the phase condition \(x_i(0)\in\Sigma_i\ .\) If \(\{t_i\}\) is a sequence such that \(x(t_i) \in \Sigma_i\ ,\) then \(x_i(t)=x(t+t_i)\ .\) Each segment \(\{x_i(t), -\omega_i\leq t \leq \omega_{i+1}\}\) is a perturbation of the heteroclinic segment \(\Gamma_i=\{q_i(t), -\omega_i \leq t \leq \omega_{i+1}\}\ .\) Assume that the equilibrium points \(p_i\) are hyperbolic. Then the linearized system \(\dot x = D_x f(q_i(t),0)x\) has exponential dichotomies on \([-\omega_i,0]\) and \([0,\omega_{i+1}]\ .\) A modified shadowing lemma for continuous systems (Lin 1989) can be used to glue the end points of \(\Gamma_i\) and \(\Gamma_{i+1}\) together. To compensate for the non-transversal intersection of the unstable subspace at \(t=0-\) and the stable subspace at \(t=0+\) of the dichotomies, \(x_i(t)\) is allowed to have a gap at \(t=0\) along a specified direction \(\Delta_i\) that is in a linear space complementary to \(TW^u(p_i) + TW^s(p_{i+1})\) at \(q(0)\ .\) The result is a unique piecewise-smooth solution \(x(t)\) with jumps \(x(t_i -)-x_i(t_i +) = \xi_i \Delta_i,\; \xi_i\in\mathbb{R}\ .\) Let \(G_i(\{\omega_i\}, \mu):= \xi_i\ .\) The existence of a smooth solution \(x(t)\) is reduced to a system of bifurcation equations \(G_i(\{\omega_i\},\mu)=0\ ,\) each associated to a heteroclinic orbit \(q_i\) in the chain.
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Intuitive ideas
Assume that for \(\mu=0\ ,\) a system in \(\mathbb{R}^2\) has a homoclinic orbit \(q(t)=(q_1(t),q_2(t))\) asymptotic to the hyperbolic equilibrium \((0,0)\ .\) For simplicity, assume that in a neighborhood \(\mathcal{O}\) of \((0,0)\) the system is linear: \[ \dot x_1 = \lambda_1 x_1,\quad \dot x_2 = -\lambda_2 x_2, \quad \lambda_1,\lambda_2 >0. \]
Let \(\Sigma\) be a cross-section passing through \(q(0)\) and orthogonal to \(\dot q(0)\ .\) For small \(\mu\ ,\) we look for a periodic orbit \(x(t)\) of period \(2\omega\) where \(\omega\) is sufficiently large so that the \(x(t)\) spends most of the time inside \(\mathcal{O}\ .\)
As a first approximation, let \(x(t) = q(t), -\omega \leq t \leq \omega\ .\) The approximation has a small jump error \(E:=x(\omega)-x(-\omega)=q(\omega)-q(-\omega)=O(e^{-\alpha \omega})\) where \(\alpha = \min\{\lambda_1,\lambda_2\}\ .\) We use an iteration method to glue the two ends of the homoclinic segment \(q(t), -\omega \leq t \leq \omega \ ,\) together at the cost of allowing \(x(t)\) to have a gap at \(t=0\ .\) Each iteration consists of two steps:
(i) Project \(E\) into the \(x_1\) and \(x_2\) axes and let \(H=(q_1(-\omega),q_2(\omega))\ .\) Let \(x^-(t)=\Phi_{t+\omega} H,\; -\omega \leq t \leq 0\ ,\) and let \(x^+(t)=\Phi_{t-\omega} H,\; 0\leq t \leq \omega\) where \(\Phi_t\) is the flow. Then \(x^-(-\omega)=x^+(\omega)\) but \(x^-(0) \neq x^+(0)\ .\) Since in the open set \(\mathcal{O}\ ,\) the system is stable in \(x_2\) and is backward stable in \(x_1\ ,\) the gap \(x^+(0) - x^-(0)\) is exponentially smaller than \(E\ ,\) see Figure 1.
(ii) Since \(\Sigma\) is a cross section to the local flow, replacing \(x^\pm(t)\) by \(x^\pm(t+\tau^\pm)\) where \(\tau^\pm\) is a proper time shift, we make \(x^\pm(0)\in\Sigma\ ,\) see Figure 3.
After steps (i) and (ii), there is a jump \(\tilde E= x^+(\omega) - x^-(-\omega)\ ,\) but \(\tilde E\) is smaller than \(E\ .\) We now project the jump \(\tilde E\) into \(x_1\) and \(x_2\) axes and repeat steps (i) and (ii) to further reduce the jump. The iteration yields a sequence \(\{x_j^-(t)\},-\omega \leq t \leq 0\) and \(\{x_j^+(t)\}, 0\leq t\leq \omega\) of which the limit is a piecewise smooth function \(x(t), -\omega \leq t \leq \omega\) with \(x(-\omega)=x(\omega)\) and \(x(0\pm)\in\Sigma\ ,\) see Figure 4.
The first guess of the solution in the iteration provides an estimate of the solution and it is standard to find its error bound from \(|\tilde E|/|E|\ .\) If \((P_s, P_u)\) are the spectral projections of \(D_x f(0,0)\ ,\) then the fact \(|\tilde E| < < |E|\) suggests \[ P_s x(-\omega) \approx P_s q(\omega), \quad P_u x(\omega) \approx P_u q(-\omega). \]
To have a smooth solution, we need to solve \(G(\omega,\mu)=0\ .\) Since the orbit \(x(t)\) is close to \(q(t)\) and \(\omega\) is large, \(D_\mu G(\omega,\mu)\) can be approximated by the Melnikov integral: \[D_\mu G(\omega,\mu) \approx \int_{-\infty}^\infty e^{-\int_0^t \nabla\cdot f(q(s),0)ds} f(q(t),0)\wedge \partial_\mu f(q(t),0) dt. \] If we assume that the Melnikov integral is non-zero, the bifurcation equation \(G(\omega,\mu)=0\) has a local solution \(\mu = \mu^*(\omega)\ .\) For a fixed \(\mu\ ,\) whether the system has a unique periodic solution or many such solutions is determined by whether \(\mu^*(\omega)\) is a monotone or oscillatory function of \(\omega\ .\)
For systems in \(\mathbb{R}^n\ ,\) generically the intersection of the unstable manifold \(W^u(0)\) and the stable manifold \(W^s(0)\) is one dimensional, and \(T W^u(0)+ TW^s(0)\) is \(n-1\) dimensional. Let \(\Delta\in\Sigma\) be a unit vector orthogonal to \(T W^u(0)+ TW^s(0)\) at \(q(0)\ .\) The iteration process should be modified as follows: after step (ii) of the iteration, we project the jump \(x^+(0)-x^-(0)\) on \(\Sigma\) according to the splitting \[(T\Sigma \cap TW^s(0)) \oplus (T\Sigma \cap TW^u(0)) \oplus \text{span}\{\Delta\}. \] Then using the forward and backward flows respectively, the error components on \(T\Sigma \cap TW^s(0)\) and \(T\Sigma \cap TW^u(0)\) can be eliminated. The error component along the direction of \(\Delta\) will remain. The limit \(x(t)\) of the iteration has a gap at \(t=0\) along the direction of \(\Delta\ .\)
Important results on bifurcation of periodic orbits from a homoclinic orbit were obtained by Sil'nikov (1968, 1970), see (Guckenheimer et al. 1983, Deng 1989). To obtain those results, careful analysis is done to overcome the lack of \(C^1\) linearization near a saddle point. On the other hand, \(C^1\) linearizations on the unstable or stable manifold separately (Hartman 1960) can be used to show how \(q(\omega)\) and \(q(-\omega)\) approach the origin, which is sufficient for the LSH.
Precise results on a heteroclinic chain
Assume that the heteroclinic chain \(\{q_i(t)\}\) of the system \(\dot x=f(x,\mu)\) satisfies the following:
- (B1) The equilibrium points \(\{p_i\}\) are hyperbolic.
- (B2) The dimensions of the unstable eigenspaces of \(D_x f(p_i,0)\) are independent of \(i\ .\)
- (B3) \(\dot q_i(t)\) is the only bounded solution (up to scalar multiple) for the linear system \( \dot z_i(t) -A_i(t) z_i(t) =0,\quad A_i(t) = D_x f(q_i(t),0)\ .\)
- (B4) The chain consists of permutations of finitely many heteroclinic orbits .
From (B3), the sum of tangent spaces \(\mathcal{M}_i:=T W^u(p_i)+ TW^s(p_{i+1})\) is \((n-1)\)-dimensional. Let \(\Delta_i\) be a unit vector orthogonal to \(\mathcal{M}_i\) at \(q_i(0)\ .\) In particular, \(\Delta_i \perp \dot q_i(0)\ .\) Assume that \(x_i(t)\) is \(C^1\) on \([-\omega_i,0]\) and \([0,\omega_{i+1}]\) and may admit a jump at \(t=0\) along the direction of \(\Delta_i\ :\) \[ x_i(0\pm) \in \Sigma_i,\quad x_i(0-)-x_i(0+)= \xi_i \Delta_i,\quad \xi_i\in \mathbb{R}. \] Such functions form a Banach space \(E([-\omega_i, \omega_{i+1}], \Delta_i)\) with sup norms on \([-\omega_i,0]\) and \([0,\omega_{i+1}]\ .\)
Main Theorem: Under conditions (B1) - (B4), there exist positive constants \(\hat \omega, \hat \mu\) and \(\hat \epsilon\) such that for any sequence \(\{\omega_i\}_{i\in\mathbb{Z}}\) with \(\omega_i \geq \hat \omega\) and parameter \(|\mu| \leq \hat \mu\ ,\) there exists a unique piecewise-smooth solution \(x(t)\) for \(\dot x=f(x(t),\mu)\) that orbitally lies in a \(\hat \epsilon\) neighborhood of the chain \(\{q_i\}\) and satisfies the following: (1) The time spent by \(x(t)\) between \(\Sigma_{i-1}\) and \(\Sigma_i\) is \(2\omega_i\ .\) (2) \(x(t)\) has a jump of the form \(\xi_i \Delta_i,\; \xi_i\in\mathbb{R}\) each time it meets \(\Sigma_i\ ,\) i.e., \(x_i \in E([-\omega_i,\omega_{i+1}],\Delta_i)\ .\)
Denote the solution by \(x_i(t)=X_i(t;\{\omega_i\},\mu); \; \xi_i = G_i(\{\omega_i\},\mu)\ .\) Then for a fixed \(\{\omega_i\}\ ,\) \[ X_i(\cdot;\{\omega_i\},\mu): M \to E([-\omega_i,\omega_{i+1}],\Delta_i),\quad G_i(\{\omega_i\},\mu): M \to \mathbb{R}\] are \(C^k\) functions of \(\mu\ .\)
Let \(x_i(t) = q_i(t) + z_i(t),\; -\omega_i \leq t \leq \omega_{i+1}\ .\) Then \(z_i(t)\) satisfies a variational system with boundary conditions at \(-\omega_i\) and \(\omega_{i+1}\ .\) \[\tag{2} \dot z_i(t) -A_i(t) z_i(t) = g_i(z_i(t),\mu,t), \quad -\omega_i \leq t \leq \omega_{i+1},\]
\[z_{i-1}(\omega_i) - z_i(-\omega_i) = b_i, \] \[\langle\dot q_i(0), z_i(t)\rangle =0. \] Here \(A_i(t) = D_x f(q_i(t),0),\; g_i(z,\mu,t)=f(q_i(t)+z,\mu)- f(q_i(t),0) -A_i(t) z\) and \(b_i=q_i(-\omega_i)-q_{i-1}(\omega_i)\ .\)
By a result of Palmer, the adjoint system \(\dot y + A_i^*(t) y =0\) also has a unique (up to scalar multiple) bounded solution \(\psi_i(t), \; t\in\mathbb{R}\) and \(\psi_i(0) \perp \mathcal{M}_i\ .\) One can define \(\Delta_i =\psi_i(0)\) which makes it independent of any geometrical terms. Multiplying the \(i\)th equation of (2) by \(\psi_i(t)\) and integrating by parts for \(t < 0\) and \( t>0\) respectively, we can express the jump as \[\tag{3} G_i(\{\omega_i\},\mu)=\int_{-\omega_{i-1}}^{\omega_i}\psi_i(t) g_i(z_i(t),\mu,t)dt + \psi_i(-\omega_i)z_i(-\omega_i)-\psi_i(\omega_{i+1})z_{i}(\omega_{i+1}). \]
If \(\hat \omega\) is sufficiently large and \(\hat \mu\) is small, Eq. (2) becomes a weakly coupled system. Each \((z_i,G_i)\) is mainly determined by \(g_i(z_i,\mu,t)\) and the boundary conditions \((b_i, b_{i+1})\ .\) In particular, using Lemmas 3.2 and 3.3 in (Lin 1990) to (3), we have
\[\tag{4}
G_i(\{\omega_i\},\mu)=\psi_i(-\omega_i)(q_{i-1}(\omega_i)-p_i)-\psi_i(\omega_{i+1})(q_{i+1}(-\omega_{i+1})-p_{i+1})
+ \mu\int_{-\infty}^\infty \psi_i(t)D_\mu f(q_i(t),0) dt + \text{ small terms}.
\]
Estimates of the small terms depend on the rate \(q_i(t) \to p_i, p_{i+1}\ ,\) and the ratios \(\omega_{i\pm 1}\) to \(\omega_i\ .\) An example of such estimates is given in Section 4.1.
For many applications, more delicate jump estimates are necessary. Those can be gained by a strict separation of the influence of the splitting of the stable and unstable manifolds (of the involved equilibria) and the dependence on the transition times \(\{\omega_i\}\) (Sandstede 1993, Yew 2001, Knobloch 2004). First, bifurcation of the heteroclinic \(\{q_i(t)\}\) is considered with \(\{\omega_i\}=\{\infty\}\) and \(\mu\approx 0\ .\) The perturbed heteroclinic orbit \(q_i(t,\mu)\) usually breaks at \(t=0\ .\) Care must be taken in constructing bounded solutions \(\psi_i(t,\mu)\) to the adjoint equation of \(\dot x = D_x f(q_i(t,\mu),\mu) x \) which is discontinuous at \( t=0\ .\) Next, the sequence \(\{\infty\}\) is moved to \(\{\omega_i\}\) which further contributes to the jump at \(t=0\ .\) The idea is expressed as \(G_i(\{\omega_i\},\mu) = G_i(\{\infty\},\mu) + (G_i(\{\omega_i\},\mu)-G_i(\{\infty\},\mu) )\) and yields a better estimate: \[ G_i(\{\omega_i\},\mu)=\mu\int_{-\infty}^\infty \psi_i(t)D_\mu f(q_i(t),0) dt + \psi_i(-\omega_i,\mu)(q_{i-1}(\omega_i,\mu)-p_i)-\psi_i(\omega_{i+1},\mu)(q_{i+1}(-\omega_{i+1},\mu)-p_{i+1}) + \text{ small terms} . \]
Examples
Consider the bifurcation of periodic and aperiodic solutions near a homoclinic orbit \(q(t)\) that is asymptotic to the hyperbolic equilibrium \(p=0\ .\) Let \(p_i\equiv p\) and \(q_i(t) \equiv q(t)\) and \(\Sigma_i\equiv \Sigma\ .\)
Assume \(x=\dot q(t)\) is the only bounded solution of the linear equation \( \dot x - D_x f(q(t),0) x =0 \ .\) Denote the unique bounded solution of the adjoint system \(\dot y+ A^*(t) y =0\) by \(\psi(t)\ .\) For any large \(\{\omega_i\}\) and small \(\mu\) there exists a unique piecewise smooth solution \(x(t)\) such that the time between two consecutive intersections of \(x(t)\) with \(\Sigma\) is \(2\omega_i\ .\) Also \(x(t_i-)-x(t_i+)= \xi_i\Delta\) where \(x(t_i)\in\Sigma\ .\)
The sequence \(\{\omega_i\}\) can be chosen according to the type of bifurcating solutions that one seeks:
- For a \(2\omega\) periodic orbit that follows the orbit \(q(t)\) once, let \(\omega_i = \omega, i\in\mathbb{Z}\ .\)
- For a "multiple periodic orbit" that follows \(q(t)\) \(k\) times, let \(\{\omega_i\} = \{\dots, (\omega_1,\omega_2,\dots,\omega_k), \text{repeating},\dots \}. \)
- For an aperiodic orbit near \(q(t)\ ,\) let \(\{\omega_i\}\) be an aperiodic sequence.
- For a "multiple homoclinic orbit" that follows \(q(t)\) \(k\) times, let \(\{\omega_i\}=\{ \infty, \omega_1, \omega_2,\dots,\omega_{k-1}, \infty\}. \)
Bifurcation to a simple periodic orbit
We look for a \(2\omega\) periodic solution \(x(t)\) that stays near the orbit of \(q\ .\) If \(x=q+z\) and \(x(-\omega)=x(\omega)\ ,\) then \[ \dot z(t)= D_x f(q(t),0)z(t) + g(z(t),\mu,t),\quad z(\omega) - z(-\omega)= q(-\omega)-q(\omega). \]
Let \(\rho =\min\{\text{Re} \lambda| \lambda\in \sigma D_x f(p,0), \text{Re}\lambda>0\},\quad \nu =\min\{-\text{Re} \lambda| \lambda\in \sigma D_x f(p,0), \text{Re}\lambda<0\}\ .\) We assume \(0< \rho <\nu\) and we make the generic assumptions:
(H1) \(\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt \neq 0\ .\)
(H2) \(|q(t)| \sim C_1 e^{\rho t}, \; t\to-\infty\ ,\) and \(|\psi(t)| \sim C_2 e^{-\rho t}, \; t\to\infty\ .\)
There is one bifurcation equation \(G(\omega,\mu)=0\ .\) Since \(|q(t)| \sim C_3 e^{-\nu t}, \; t\to \infty\ ,\) and \(|\psi(t)| \sim C_4 e^{\nu t}, \; t\to -\infty\ ,\) we drop the smaller term \(\psi(-\omega)q(\omega)\) in the estimate: \[ G(\omega,\mu) =\psi(\omega)q(-\omega) + \mu \int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt + o(e^{-2\rho \omega}+|\mu|). \]
There are two important cases:
(I) The dominant eigenvalue is real and simple. In this case \(\psi(\omega)q(-\omega) \sim C_5 e^{-2\rho\omega}\ ,\) and \[\mu\sim C_5 e^{-2\rho\omega}\left(\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt\right)^{-1}. \] The bifurcation to periodic orbits can happen only on one side of \(\mu\ ,\) and \(\mu \to 0\) as \(\omega \to\infty\ .\)
(II) The dominant eigenvalues are a pair of simple complex eigenvalues \(\rho \pm i \theta\ .\) In this case \(\psi(\omega)q(-\omega)\sim C_6 e^{-2\rho\omega}\sin(2\theta\omega+\beta)\ ,\) and \[\mu\sim C_5 e^{-2\rho\omega}\sin(2\theta\omega + \beta)\left(\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt\right)^{-1}. \] There exist infinitely many periodic orbits when \(\mu=0\ .\) The values of \(\omega\) are almost equally spaced. The periodic orbits can occur on both sides of \(\mu=0\ .\) The number of periodic orbits decreases as \(|\mu|\) increases.
Bifurcation to an aperiodic orbit
Assume the dominant eigenvalues are complex and simple. For each \(i\in\mathbb{Z}\ ,\) we have to solve \[C_5 e^{-2\rho\omega_i}\sin(2\theta\omega_i + \beta)\left(\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt\right)^{-1}+ \text{ small terms } -\mu =0. \] If the small terms are ignored, then several solutions of \(\omega_i\) are possible for each small \(\mu\ .\) We can form many aperiodic sequences from these \(\omega_i\ .\) With the small terms added, we can still adjust \(\omega_i\) to make all \(G_i(\{\omega_i\},\mu)=0\ .\) Proofs can be given using degree theory (Lin 1990).
Twistedness of heteroclinic and homoclinic orbits
Suppose that there is a solution \(x(t) \) near a heteroclinic chain for \(\mu = 0\ .\) From the estimate of \(G_i\ ,\) for large \(\omega_i\ :\) \[\tag{5} \psi_i(-\omega_i)(q_{i-1}(\omega_i)-p_i)\sim \psi_i(\omega_{i+1})(q_{i+1}(-\omega_{i+1})-p_{i+1}). \]
Since \(\psi_i(t)\) continuously points to one side of the codimension-one surface \(TW^u(p_i)+ TW^s(p_{i+1})\ ,\) equation (5) implies that \(q_{i-1}(\omega)\) and \(q_{i+1}(-\omega_{i+1})\) are on the same side of that surface. This is a necessary condition for the existence of \(x(t)\) near the chain (Velummylum, NCSU, 1998).
The opposite case occurs when considering the bifurcation of a double homoclinic orbit. For the sequence \(\{\omega_i\}=\{\infty, \omega,\infty\},\) there are two bifurcation equations: \[ G_1(\omega,\mu) = -\psi(-\omega)q(\omega) + \int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt \cdot \mu + \text{ small terms},\] \[ G_2(\omega,\mu) =\psi(\omega)q(-\omega) + \int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt \cdot \mu+ \text{ small terms}. \] We look for a branch of solutions \((\mu_1(\omega),\mu_2(\omega))\) where the parameter \(\mu=(\mu_1,\mu_2)\ .\) Letting \(\omega\to\infty\) and \(\mu\to 0\ ,\) we have \( \psi(-\omega)q(\omega) + \psi(\omega)q(-\omega) \to 0,\, \omega \to \infty. \) Such \(q(t), -\omega \leq t \leq \omega\ ,\) is called a twisted homoclinic segment since \(q(\omega)\) and \(q(-\omega)\) are on different sides of the surface \(TW^u(p)+TW^s(p)\) (Lin 1990). A homoclinic segment can be twisted near the equilibrium by a pair of dominant complex eigenvalues as in Sil'nikov's systems, or on the global part of the orbit without involving complex eigenvalues (Yanagida 1987). For bifurcations to multiple homoclinic orbits and their stability, see (SJA 1997, Sandstede 1998).
Technical details
The heart of the LSH is the Fredholm property for the linearized system around a heteroclinic orbit.
Let \(T(t,s)\) be the principal matrix solution for \( \dot x = A(t) x\) and \(T^*(s,t) := (T(t,s)^{-1})^*\) be the principal matrix solution of the adjoint equation \( \frac{dy}{ds} + A^*(s) y = 0\ .\) Assume that \(T(t,s)\) has an exponential dichotomy on an interval \(J\) with projections to the stable and unstable subspaces \(P_s(t)\) and \(P_u(t) = I-P_s(t)\ .\) Then \(T^*(s,t)\) has an exponential dichotomy with the projections to the stable and unstable subspaces, \(P^*_s(t)\) and \(P^*_u(t)\ ,\) being adjoint operators of \(P_s(t)\) and \(P_u(t)\) respectively. Solutions on the unstable (stable) subspaces of \(T^*(s,t)\) decay exponentially if solved forward (backward) in time (Palmer 1984, Lin 1986).
Assume that \(T(t,s)\) has exponential dichotomies on \((-\infty, 0]\) and \([0,\infty)\ ,\) and \[ \text{dim}\mathcal{R}P_u(0-)=\text{dim}\mathcal{R}P_u(0+)=d^+.\] \[\mathcal{R}P_u(0-)\cap \mathcal{R}P_s(0+)= \text{span}\{\phi(0)\}, \] where \(\phi(t)(=\dot q(t))\) is the only bounded solution (up to constant multiple) to \(\dot x=A(t)x\ .\) Then the adjoint equation also has a unique bounded solution \( \psi(t)\) where \( \psi(0) \in \mathcal{R}P_u^*(0+) \cap \mathcal{R} P_s^*(0-) = (\mathcal{R}P_u(0-)+\mathcal{R}P_s(0+))^\perp \ .\)
Lemma 5.1: (Lemma 2.3, Lin 1990) For a given \(f\in C[a,b]\) and \((\phi_s,\phi_u) \in (\mathcal{R}P_s(a), \mathcal{R}P_u(b))\ ,\) consider the nonhomogeneous boundary value problem: \[\tag{6} \dot x -A(t)x = f(t),\quad a \leq t \leq b,\; a<0<b,\]
\[P_s(a)x(a)= \phi_s,\quad P_u(b) x(b)=\phi_u.\] The system has a unique \(C^1\) solution \(x(t)\) with \(x(0)\perp \phi(0)\) if and only if \[\tag{7} \int_{a}^{b} \langle\psi(t), f(t)\rangle dt +\langle\psi(a),\phi_s\rangle-\langle\psi(b),\phi_u\rangle=0. \]
If (7) does not hold, then let the left hand side be \(G\). There exists a unique piecewise \(C^1\) solution \(x\in C^1[a,0]\cap C^1[0,b]\) for (6) with \(x(0\pm)\perp \phi(0)\) such that \[x(0-) - x(0+) = G\; \psi(0). \quad (\text{Assume }|\psi(0)|=1). \] \[ |G| \leq C(\|f\| + e^{-\alpha|a|}|\phi_s| + e^{-\alpha|b|}|\phi_u|). \] Moreover, the solution \(x(t)\) is bounded by \((\phi_s, \phi_u,f)\) in some exponentially weighted norms.
Remark: In the spirit of Palmer (Palmer 1984), consider the linear mapping defined by (6), \[\mathcal{F}: C^1[a,b] \to (\mathcal{R}P_s(a), \mathcal{R}P_u(b), C^0[a,b]), \quad \mathcal{F}u = (\phi_s, \phi_u,f). \] Then \(\mathcal{F}\) is Fredholm with index zero. The kernel of \(\mathcal{F}\) is spanned by \(\phi(t)\ .\) The range of \(\mathcal{F}\) is of codimension one and satisfies (7). To solve (6) one usually adds \(\xi g(t)\) to \(f(t)\ ,\) where \((0,0,g)\) is complementary to \(\mathcal{R}\mathcal{F}\) and the parameter \(\xi\) is undetermined. This can be an alternative implementation of the LSH. The introduction of a gap \(\xi \psi(0)\) makes the method more intuitive and corresponds to allowing \(g(t)=\delta(0) \psi(0)\) to be a \(\delta\) type function.
Evolution of the method
Following the original work of Chow, Hale and Mallet-Paret (1980), many people have helped to develop a function space approach to homoclinic/heteroclinic bifurcation problems. Palmer (1984) proved the Fredholm property of the linear variational problem around a homoclinic orbit and used the shadowing lemma to study solutions near a homoclinic orbit. The definition of exponential dichotomies for semiflows is due to Henry (1981). Hale and Lin (1986) used exponential trichotomies to study bifurcations of heteroclinic orbits connecting periodic orbits for delay equations. The Lyapunov-Schmidt reduction was used by Hale and Sakamoto (1988) to study singular perturbation problems, and by Chow, Lin and Mallet-Paret (1989} to study a singularly perturbed delay equation. Lin (1990) combined previous results to treat heteroclinic chains for systems of ODEs and delay equations.
The method has since been generalized and popularized by Fiedler, Vanderbauwhede, Sandstede, and many others as Lin's method, although it is in fact due to the work of many people. Vanderbauwhede and Fiedler (1992) studied reversible and conservative systems that did not satisfy some generic conditions. Sandstede (1993, 1997) generalized the method to higher codimensional cases and to PDEs with sectorial linear part. Sandstede (1997), Oldeman, Champneys, Krauskopf and Riess applied the method to numerical computations (Oldeman et al. 2003, Krauskopf et al. 2008). Lin and Vivancos (2002) studied impulsive periodic orbits that occur in reduced slow manifolds of singular perturbation problems. Sandstede, Jones and Alexander (1997) studied the stability of traveling waves. Knobloch (2000) and others generalized the method to discrete systems. Peterhof, Sandstede and Scheel (1997) generalized it to elliptic PDEs. Lin (1994b, 1996c, 1996d) studied parabolic PDEs with time dependent spatial layers. Mallet-Paret (1997), Harterich, Sandstede and Scheel (2002), Hupkes and Verduyn-Lunel (2008) studied mixed type functional differential equations. Mallet-Paret (1999) and Georgi (2008) studied spatially discrete dynamical systems (lattice differential equations). The method has been adapted to cycles involving periodic orbits by Rademacher (2005), Riess (2008), and Krauskopf and Riess (2008). The list is by no means complete.
References
S.-N. Chow, J. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), 351-373.
J. Hale and K. Sakamoto, Existence and stability of transition layers, Japan Journal lf Appl. math. 5 (1988), 367-405.
P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana 5 (1960), 220-241.
X. B. Lin, Shadowing lemma and singularly perturbed boundary value problems, SIAM J. Appl. Math. 49 (1989), 26-54,
X. B. Lin, Using Melnikov's method to solve Silnikov's problems, Proc. Roy. Soc. Edinburgh 116A (1990), 295-325.
J. Mallet-Paret, The Fredholm Alternative for functional differential equations of mixed type, J. Dyn. and Diff. Eq. 11 (1997), 1-47.
J. Mallet-Paret, The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems, Journal of Dynamics and Differential Equations, 11 (1999), 49-127.
K. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225-256.
B. Sandstede, Verzweigungstheorie homokliner Verdopplungen, Ph.D. thesis, University of Stuttgart, 1993.
B. Sandstede, Stability of multi-pulse solutions, Trans. Amer. math. Soc. 350 (1998), 429-472.
B. Sandstede, C.K.R.T. Jones and J. Alexander, Existence and stability of N-Pulses on optical fibres with phase-sensitive amplifiers, Physica D 106 (1997), 167-206.
L. P. Sil'nikov, On the generalization of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSR Sb. 6 (1968), 427-437.
L. P. Sil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR. Sb. 10 (1970), 91-102.
A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z Angew Math Phys 43 (1992), 292-318.
E. Yanagida, Branching of double pulse solutions from single pulse solutions in nerve axon equation, J. Differential Equations 66 (1987), 243-262.
Additional references
S.-N. Chow and X. B. Lin, Bifurcation of homoclinic orbits asymptotic to a saddle-node equilibrium, Differential and Integral Equations 3 (1990), 435--466
S.-N. Chow, X. B. Lin and J. Mallet-Paret, Transition layers for singularly perturbed delay-differential equations with monotone nonlinearities, J. Dyn. Differential Equations 1 (1989), 3-43.
F. Geng, X. B. Lin and X. Liu, Chaotic Traveling Wave Solutions in Coupled Chua’s Circuits, J Dyn Diff Equat, https://doi.org/10.1007/s10884-017-9631-1 (2017).
H. Dankowicz, F. Schilder and M. Saghafi, Continuation of connecting orbits with Lin’s method using COCO, ENOC2011, 24-29 July 2011, Rome, Italy.
B. Deng, The Sil'nikov problem, exponential expansion, strong \(\lambda\)-lemma, \(C^1\) linearization and homoclinic bifurcation, J. Differential Eq. 79 (1989), 189-231.
M. Georgi, Homoclinic period blow-up and the detection of multi-pulse solutions in lattice differential equations, preprint 2008.
M. Georgi, Bifurcations from homoclinic orbits to non-hyperbolic equilibria in reversible lattice differential equations. Nonlinearity 21 (2008), 735-763.
J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, Applied mathematical Sciences 42, Springer 1983, 1986.
J. Hale and X. B. Lin, Heteroclinic Orbits for Retarded Functional Differential Equations J. Differential Equations, 65 (1986), 175-202.
J. Harterich, B. Sandstede, A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Ind. Univ. Math. J. 51, pp. 1081-1109, 2002.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, 1981.
D. Henry, Dan Henry's Manuscripts, "http://www.ime.usp.br/~map/dhenry/", 2006.
A. J. Homburg, J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Amer. Math. Soc. 358 (2006), 1715-1740.
Ale Jan Homburg, Alice C. Jukes, Juergen Knobloch and Jeroen S.W. Lamb, Saddle-nodes and period-doublings of Smale horseshoes: a case study near resonant homoclinic bellows. Bull. Belg. Math. Soc. Simon Stevin, in press. Preprint
Ale Jan Homburg, Alice C. Jukes, Juergen Knobloch and Jeroen S.W. Lamb, Bifurcation from codimension one relative homoclinic cycles. Preprint
H. Hupkes and S. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433–2487.
J. Knobloch, Lin`s method for discrete dynamical systems Journal of Difference Equations and Applications, 6 (2000), 577-623
J. Knobloch, Lin's method for discrete and continuous dynamical systems and applications, thesis, TU Ilmenau, 2004.
J. Knobloch and T. Wagenknecht, Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems, Physica D, 206 (1-2) ((2005), 82-93
B. Krauskopf and T. Riess, A Lin's method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity 21 (2008), 1655-1690.
J. Knobloch and T. Rieß, Lin's method for heteroclinic chains involving periodic orbits, Nonlinearity 23(1)(2010), 23-54.
J. Knobloch, J. Lamb and K. Webster, Using Lin's method to solve Bykov's problems, J. Differential Equations, Available online July 2014, 1-64.
H. Krishnan, Uniqueness of rapidly oscillating periodic solutions to a singularly perturbed differential-delay equation, Electronic Journal of Differential Equations, 2000 (2000), No. 56, 1-18.
X. B. Lin, Exponential Dichotomies and Homoclinic Orbits in Functional Differential Equations, J. Differential Equations 63 (1986), 227-254.
X. B. Lin, Exponential dichotomies and stability of long period solutions in predator-prey models with diffusion, in Partial Differential Equations, Proceedings, J. Wiener & J. Hale (ed), Pitman Research Notes in Math. Series 273, Longman (1992), 101-105.
X. B. Lin, Spatially nonhomogeneous patterns generated by homoclinic/equilibrium bifurcations, Tatra Mountains Math. Publ. 4 (1994), 147-152.
X. B. Lin, Exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator-prey model, J. Differential Equations 108 (1994), 36-63.
X. B. Lin, Twisted and nontwisted bifurcations induced by diffusion, (received 1993), J. Dyn. Differential Equations 8 (1996), 325-372.
X. B. Lin, Homoclinic bifurcations with weakly expanding center manifolds, Dynamics Reported, 5, 99-189, Expositions in Dynamical Systems, C. Jones, U. Kirchgraber & H. O. Walther eds., Springer-Verlag, 1996.
X. B. Lin, Shadowing matching errors for wave-front-like solutions, J. Differential Equations 129 (1996), 403-457.
X. B. Lin, Local and global existence of multiple waves near formal approximations, Progr. Nonlinear Differential Equations Appl. 19 (1996), 385-404.
X. B. Lin and I. B. Vivancos, Heteroclinic and periodic cycles in a perturbed convection model, J. Differential Equations, 182 (2002), 219-265.
J. Mallet-Paret, Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations, J. Diff. Eq., to appear.
Vahagn Manukian, Nicola Costanzino, Christopher K. R. T. Jones and Björn Sandstede, Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations, J Dyn Diff Equat 21 (2009), 607–622
B. Oldeman, A. Champneys and B. Krauskopf, Homoclinic branch switching: a numerical implementation of Lin's method, International Journal of Bifurcation and Chaos 13 (2003), 2977-2999, World Scientific.
D. Peterhof, B. Sandstede and A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders. J. Differential Equations 140 (1997), 266--308.
J. Rademacher, Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit, J. Diff. Eqns. 218 (2005), 390-443.
B. Sandstede, Convergence estimates for the numerical approximation of homoclinic solutions, IMA Journal of Numerical Analysis 17 (1997), 437-462.
B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two J. Dyn. Dif. Eqns. 9 (1997), 269-288.
T. Riess, A Lin's method approach to heteroclinic connections involving periodic orbits - analysis and numerics, Thesis, TU Ilmenau, 2008.
S. van Gils, M. Krupa and V. Tchistiakov, Homoclinic twist bifurcation in a system of two coupled oscillators, J. Dynam. Differential Equations 12 (2000), 733–806
S. Velummylum, Internal transition layers in singularly perturbed boundary Value Problems, Thesis, NCSU, 1998.
T. Wagenknecht, About a homoclinic pitchfork bifurcation in reversible systems with additional 2-symmetry, Nonlinearity, 15 (2002), 2097-2119.
A. Yew, Multipulses of nonlinearly coupled Schrödinger equations. J. Differential Equations 173 (2001), 92-137.
Internal references
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- Ian Gladwell (2008) Boundary value problem. Scholarpedia, 3(1):2853.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Alan R. Champneys and Mario di Bernardo (2008) Piecewise smooth dynamical systems. Scholarpedia, 3(9):4041.
- Ernest Barreto (2008) Shadowing. Scholarpedia, 3(1):2243.
- Leonid Pavlovich Shilnikov and Andrey Shilnikov (2007) Shilnikov bifurcation. Scholarpedia, 2(8):1891.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
See also
Bifurcations, Lyapunov-Schmidt Reduction, Shilnikov bifurcation, Shadowing lemma for flow
