Skip to main content
Log in

A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

A general theorem that guarantees the existence of an orbit connecting two hyperbolic equilibria of a parametrized autonomous differential equation in \({\mathbb {R}}^n\) near a suitable approximate connecting orbit given the invertibility of a certain explicitly given matrix is proved. Numerical implementation of the theorem is described using five examples including two Sil’nikov saddle-focus homoclinic orbits and a Sil’nikov saddle-focus heteroclinic cycle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ambrosi, D., Arioli, G., Koch, H.: A homoclinic solution for excitation waves on a contractile substratum. SIAM J. Appl. Dyn. Syst. 11(4), 1533–1542 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arioli, G., Koch, H.: Existence of Traveling Pulse Solutions of the FitzHugh–Nagumo Equation. https://www.ma.utexas.edu/mp_arc/c/13/13-91.ps.gz (2013)

  3. Arneodo, P., Coullet, P., Tresser, J.: Oscillators with chaotic behavior: an illustration of a theorem by Shilnikov. J. Stat. Phys. 27, 171–182 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beyn, W.-J.: The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10, 379–405 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beyn, W.-J.: On well-posed problems for connecting orbits in dynamical systems. In: Kloeden, P., Palmer, K. (eds.) Chaotic Dynamics, Contemporary Mathematics. Mathematical Society, vol. 172, pp. 131–168. Providence, Rhode Island (1994)

  6. Boisvert, J.J., Muir, P.H., Spiteri, R.J.: BVP\_SOLVER-2. http://cs.stmarys.ca/~muir/BVP_SOLVER_Webpage.shtml (2012)

  7. Boost \(C^{++}\) Libraries. http://www.boost.org/doc/libs/1_56_0/libs/numeric/interval/doc/interval.htm (2006)

  8. Chen, X.: Lorenz equations. Pt. I. Existence and nonexistence of homoclinic orbits. SIAM J. Math. Anal. 27, 1057–1069 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Coomes, B.A., Koçak, H., Palmer, K.J.: Rigorous computational shadowing of orbits of ordinary differential equations. Numer. Math. 69, 401–421 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coomes, B.A., Koçak, H., Palmer, K.J.: Homoclinic shadowing. J. Dyn. Diff. Equ. 17, 175–215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Coomes, B.A., Koçak, H., Palmer, K.J.: Transversal connecting orbits from shadowing. Numer. Math. 106, 427–469 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coomes, B.A., Koçak, H., Palmer, K.J.: Shadowing in ordinary differential equations. Rend. del Sem. Mat. Univ. e Politec. Torino 65, 89–113 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Davis, T.A.: Algorithm 832—UMFPACK V4.3, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196–199 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Glendinning, P., Sparrow, C.: T-points: a codimension two heteroclinic bifurcation. J. Stat. Phys. 43, 479–488 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hale, J., Koçak, H.: Dynamics and Bifurcations. Springer, New York (1991)

    Book  MATH  Google Scholar 

  16. Hassard, B., Zhang, J.: Existence of a homoclinic orbit of the Lorenz system by precise shooting. SIAM J. Math. Anal. 25, 179–196 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hastings, S.P., Troy, W.C.: A shooting approach to the Lorenz equations. Bull. Am. Math. Soc. 27, 298–303 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hastings, S.P., Troy, W.C.: A proof that the Lorenz equations have homoclinic orbits. J. Differ. Equ. 113, 166–188 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hiraoka, Y.: Rigorous numerics for symmetric homoclinic orbits in reversible dynamical systems. Kybernetika 43, 797–806 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Hiraoka, Y.: Construction of approximate solutions for rigorous numerics of symmetric homoclinic orbits. In: Workshops on Pattern Formation Problems in Dissipative Systems and Mathematical Modeling and Analysis for Nonlinear Phenomena, pp. 011–023. RIMS Kokyuroku Bessatsu B3, Research Institute for Mathematical Sciences (RIMS), Kyoto (2007)

  21. Kaplan, J.L., Yorke, J.A.: Preturbulence: a regime observed in a fluid flow model of Lorenz. Commun. Math. Phys. 67, 93–108 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leonov, G.A.: Estimation of loop-bifurcation parameters for a saddle-point separatrix of a Lorenz system. Differ. Equ. 24, 634–638 (1988)

    MATH  Google Scholar 

  23. Leonov, G.A.: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. J. Appl. Math. Mech. 65, 19–32 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Leonov, G.A.: Strange Attractors and Classical Stability Theory. St. Petersburg University Press, St. Petersburg (2008)

    MATH  Google Scholar 

  25. Lessard, J.-P., Mireles James, J., Reinhardt, C.: Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields. J. Dyn. Differ. Equ. 26, 267–313 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lessard, J.-P., Reinhardt, C.: Rigorous numerics for nonlinear differential equations using Chebyshev series. SIAM J. Numer. Anal. 52, 1–22 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. McGehee, R., Meyer, K.: Homoclinic points of area preserving diffeomorphisms. Am. J. Math. 96(3), 409–421 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. Oishi, S.: Numerical verification method of existence of connecting orbits for continuous dynamical systems. J. Univers. Comput. Sci. 4, 193–201 (1998)

    MathSciNet  MATH  Google Scholar 

  29. Palmer, K.J.: Shadowing in Dynamical Systems. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  30. Phaser (2009). www.phaser.com

  31. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. 57A, 397–398 (1976)

    Article  Google Scholar 

  32. Sil’nikov, L.P.: A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6, 163–166 (1965)

    MathSciNet  MATH  Google Scholar 

  33. Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Applied Mathematical Sciences, vol. 41. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  34. Tresser, C.: About some theorems by Sil’nikov. Ann. Inst. Henri Poincaré Sect. A 40(4), 441–461 (1984)

    MathSciNet  MATH  Google Scholar 

  35. van den Berg, J.B., Mireles-James, J.D., Lessard, J.-P., Mischaikow, K.: Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray–Scott equation. SIAM J. Math. Anal. 43(4), 1557–1594 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice-Hall, Englewood Cliffs, NJ (1963)

    MATH  Google Scholar 

Download references

Acknowledgments

The work of H.K. was supported by NSF Grant CMG-0417425 and by the National Research Council of Taiwan during several visits; the work of K.P. was supported by the National Research Council of Taiwan and by the Department of Computer Science at the University of Miami during several visits.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kenneth J. Palmer.

Additional information

This paper is dedicated to John Mallet-Paret in friendship on the occasion of his 60th birthday.

Appendices

Appendix 1: Statement of Lemma 11.7 in Palmer [2000]

For the reader’s convenience, we present here the statement of Lemma 11.7 in Palmer [29].

Lemma

Let \(E\)\(F\) be Banach spaces, \(O\subset E\) an open set and \(\mathcal{G}:O\rightarrow F\)\(C^2\) function. Suppose \(y\) is an element of \(O\) for which \(\Vert \mathcal{G}(y)\Vert \le \delta \) and \(D\mathcal{G}(y)=L\) is invertible with \(\Vert L^{-1}\Vert \le K\). Set

$$\begin{aligned} \varepsilon =2K\delta \quad \mathrm{and}\quad M=\sup \{\Vert D^2\mathcal{G}(x)\Vert :x\in O,\; \Vert x-y\Vert \le \varepsilon \}. \end{aligned}$$

Then if the closed ball of radius \(\varepsilon \) around \(y\) is in \(O\) and

$$\begin{aligned} MK\varepsilon =2MK^2\delta <1, \end{aligned}$$

there is a unique solution \(x\) of the equation \( \mathcal{G}(x)=0 \) satisfying \(\Vert x-y\Vert \le \varepsilon \).

Appendix 2: Proofs of Lemmas

Proof of Lemma 1

Since by Eq. (5), \(|x-y_k|\le \Delta _2 < R_k\), there exists \(\bar{t}\) satisfying \(0<\bar{t}\le h_k\) such that \(|\phi ^t(x,a)-\phi ^t(y,a_0)|\le R_k\) for \(0\le t\le \bar{t}\). We suppose \(\bar{t}\) is maximal with this property so that either \(\bar{t}=h_k\) or \(\bar{t}<h_k\) and \(|\phi ^{\bar{t}}(y_k,a_0)-\phi ^{\bar{t}}(y_k,a_0)|= R_k\). It follows from this that for \(0\le t\le \bar{t}\)

$$\begin{aligned} |\phi ^t(x,a)-y_k|\le |\phi ^t(x,a)-\phi ^t(y_k,a_0)|+|\phi ^t(y_k,a_0)-y_k|\le 2R_k, \end{aligned}$$

so that the norms of the derivatives of \(f_x\) and \(f_a\) at \((\phi ^t(x,a),a)\) are bounded by \(M_1\) and \(M_3\) respectively. Now

$$\begin{aligned} \phi ^t(x,a)-\phi ^t(y_k,a_0)=x-y+\int ^t_0 f(\phi ^u(x,a),a)-f(\phi ^u(y_k,a_0),a_0)du \end{aligned}$$

so that for \(0\le t\le \bar{t}\),

$$\begin{aligned} \begin{aligned}&|\phi ^t(x,a)-\phi ^t(y_k,a_0)|\\&\quad \le |x-y_k|\\&\qquad +\int ^t_0 |f(\phi ^u(x,a),a)-f(\phi ^u(x,a),a_0)| + |f(\phi ^u(x,a_0),a_0)-f(\phi ^u(y_k,a_0),a_0)|du\\&\quad \le |x-y|+\int ^t_0 M_3|a-a_0|+M_1|\phi ^u(x,a)-\phi ^u(y_k,a_0)|du. \end{aligned} \end{aligned}$$

Then by Gronwall’s lemma

$$\begin{aligned} |\phi ^t(x,a)-\phi ^t(y_k,a_0)| \le |x-y|e^{M_1t}+M_3|a-a_0|{e^{M_1t}-1\over M_1} \le [1+M_3t]e^{M_1t}\Delta \end{aligned}$$

if \(0\le t\le \bar{t}\). Then if \(\bar{t}<h_k\), it would follow that \( [1+M_3\bar{t}]e^{M_1\bar{t}}\Delta \ge R_k\), thus contradicting Eq. (5). Hence \(\bar{t}=h_k\). It follows that \(|\phi ^t(x,a)-\phi ^t(y_k,a_0)|\le R_k\) for \(0\le t\le h_k\), and hence that \( |\phi ^t(x,a)-y_k|\le 2R_k\) for \(0\le t\le h_k\), thus proving the lemma.\(\square \)

Proof of Lemma 2

From Lemma 1 we know that \(|\phi ^t(x,a)-y_k|\le 2R_k\) for \(0\le t\le h_k\), so that the norms of the derivatives \(f_x\), \(f_a\), \(f_{xx}\), \(f_{xa}\), \(f_{aa}\) at \((\phi ^t(x,a),a)\) are bounded by \(M_1\), \(M_2\), \(M_3\), \(M_4\), \(M_5\) respectively. In particular, since \(|f_x(\phi ^t(x,a),a)|\le M_1\) for \(0\le t\le h_k\) and \(Y(t)=\phi ^t_x(x,a)\) is the solution of

$$\begin{aligned} \dot{Y}=f_x(\phi ^t(x,a),a)Y,\quad Y(0)=I, \end{aligned}$$

it follows by Gronwall’s lemma that

$$\begin{aligned} | \phi ^t_x(x,a)| \le e^{M_1t},\quad 0\le t\le h_k. \end{aligned}$$

Next \(y(t)=\phi ^t_{a}(x,a)\) is the solution of

$$\begin{aligned} \dot{y}=f_x(\phi ^t(x,a),a)y+f_{a}(\phi ^t(x,a),a),\quad y(0)=0 \end{aligned}$$

so that

$$\begin{aligned} y(t)=\int ^t_0\phi ^{t-s}_x(\phi ^s(x,a),a)f_{a}(\phi ^s(x,a),a)ds. \end{aligned}$$

It follows that for \(0\le t\le h_k\), \(|y(t)|\le \int ^t_0 e^{M_1(t-s)}M_3ds\) so that for these same \(t\)

$$\begin{aligned} |\phi ^{t}_{a}(x,a)|\le M_3M^{-1}_1(e^{M_1t}-1). \end{aligned}$$

Since \(y(t)=\phi ^t_{xx}(x,a)(\xi ,\eta )\) is the solution of

$$\begin{aligned} \dot{y}=f_x(\phi ^t(x,a),a)y+f_{xx}(\phi ^t(x,a),a)(\phi ^t_x(x,a)\xi ,\phi ^t_x(x,a)\eta ),\quad y(0)=0 \end{aligned}$$

it follows that

$$\begin{aligned} y(t) =\int ^t_0\phi ^{t-s}_x(\phi ^s(x,a),a)f_{xx}(\phi ^s(x,a),a)(\phi ^s_x(x,a)\xi ,\phi ^s_x(x,a)\eta )ds \end{aligned}$$

so that

$$\begin{aligned} |y(t)|\le \int ^t_0e^{M_1(t-s)}M_2e^{2M_1s}|\xi |\,|\eta |ds =M_2e^{M_1t}\int ^t_0e^{M_1s}ds\,|\xi |\,|\eta | \end{aligned}$$

and hence for \(0\le t\le h_k\)

$$\begin{aligned} |\phi ^{t}_{xx}(x,a)|\le M_2M^{-1}_1e^{M_1t}(e^{M_1t}-1) \le M_2M^{-1}_1e^{M_1t}M_1te^{M_1t}=M_2te^{2M_1t}\le M_6te^{M_1t}.\nonumber \\ \end{aligned}$$
(36)

Since \(y(t)=\phi ^t_{xa}(x,a)\xi \) is the solution of

$$\begin{aligned} \dot{y}=f_x(\phi ^t(x,a),a)y+f_{xx}(\phi ^t_x(x,a)\xi ,\phi ^t_a(x,a))+f_{xa}\phi ^t_x(x,a)\xi ,\quad y(0)=0, \end{aligned}$$

where \(f_{xx}=f_{xx}(\phi ^t(x,a),a)\) etc., it follows that

$$\begin{aligned} y(t)=\int ^t_0\phi ^{t-s}_x(\phi ^s(x,a),a)[f_{xx}(\phi ^s_x(x,a)\xi ,\phi ^s_a(x,a))+f_{xa}\phi ^s_x(x,a)\xi ]ds \end{aligned}$$

so that

$$\begin{aligned} \begin{aligned}|y(t)|&\le \int ^t_0e^{M_1(t-s)}[M_2e^{M_1s}M_3M^{-1}_1(e^{M_1s}-1)+M_4e^{M_1s}]ds\,|\xi |\\&=\int ^t_0[M_2e^{M_1t}M_3M^{-1}_1(e^{M_1s}-1)+M_4e^{M_1t}]ds\,|\xi | \end{aligned} \end{aligned}$$

and hence for \(0\le t\le h_k\)

$$\begin{aligned} \begin{aligned} |\phi ^{t}_{xa}(x,a)|&\le M_2M_3e^{M_1t}M^{-2}_1(e^{M_1t}-M_1t-1)+tM_4e^{M_1t}\\&\le M_2M_3e^{M_1t}{1\over 2}t^2e^{M_1t}+tM_4e^{M_1t}\\&\le M_7te^{M_1t}. \end{aligned} \end{aligned}$$
(37)

Since \(y(t)=\phi ^t_{aa}(x,a)\) is the solution of

$$\begin{aligned} \dot{y}=f_x(\phi ^t(x,a),a)y+f_{xx}(\phi ^t_a(x,a),\phi ^t_a(x,a))+2f_{xa}\phi ^t_a(x,a) + f_{aa},\quad y(0)=0, \end{aligned}$$

where \(f_{xx}=f_{xx}(\phi ^t(x,a),a)\) etc., it follows that

$$\begin{aligned} y(t)=\int ^t_0\phi ^{t-s}_x(\phi ^s(x,a),a)[f_{xx}(\phi ^s_a(x,a),\phi ^s_a(x,a))+2f_{xa}\phi ^s_a(x,a)+ f_{aa}]ds \end{aligned}$$

so that

$$\begin{aligned} \begin{aligned}&|y(t)|\\&\quad \le \int ^t_0e^{M_1(t-s)}[M_2M^2_3M^{-2}_1(e^{M_1s}-1)^2+2M_4M_3M^{-1}_1(e^{M_1s}-1)+ M_5]ds\\&\quad =e^{M_1t}\int ^t_0[M_2M^2_3M^{-2}_1(2\cosh (M_1s)-2)+2M_4M_3M^{-1}_1(1-e^{-M_1s})+ M_5e^{-M_1s}]ds\\&\quad ={M_2M^2_3\over M^{2}_1}e^{M_1t}{2\sinh (M_1t)-2M_1t\over M_1}+2{M_4M_3\over M^{2}_1}e^{M_1t}(e^{-M_1t}+M_1t-1) +M_5{e^{M_1t}-1\over M_1} \end{aligned} \end{aligned}$$

and hence for \(0\le t \le h_k\)

$$\begin{aligned} \begin{aligned} |\phi ^{t}_{aa}(x,a)|&\le 2M_2M^2_3e^{M_1t}{{1\over 6}\mathrm{cosh}(M_1t)M^3_1t^3\over M^3_1}+{2M_3M_4\over M^{2}_1}e^{M_1t}{1\over 2}(M_1t)^2 +M_5{e^{M_1t}-1\over M_1}\\&= {1\over 3}M_2M^2_3e^{M_1t}\mathrm{cosh}(M_1t)t^3+M_3M_4t^2e^{M_1t}+M_5te^{M_1t}\\&\le M_{8}te^{M_1t}. \end{aligned} \end{aligned}$$
(38)

Then the lemma follows from Eqs. (36), (37) and (38).\(\square \)

Proof of Lemma 3

We break the solution of Eq. (16) into solving

$$\begin{aligned} \xi _{k+1}=e^{h_kA_+}\xi _k+\phi ^{h_k}_a(z_+,a_0),\quad k\ge N_1 \end{aligned}$$
(39)

and

$$\begin{aligned} \xi _{k+1}=e^{h_kA_+}\xi _k+g_{k}, \quad k\ge N_1. \end{aligned}$$
(40)

First we consider the difference equation Eq. (39). It is convenient to relate its solutions to those of the autonomous differential equation

$$\begin{aligned} \dot{x}=A_+x+f_a(z_+,a_0). \end{aligned}$$
(41)

Note first that \(\phi ^t_a(z_+,a_0)\) is the solution of Eq. (41) with \(x(0)=0\) so that any solution \(x(t)\) of Eq. (41) satisfies

$$\begin{aligned} x(t)=e^{(t-s)A_+}x(s)+\phi ^{t-s}_a(z_+,a_0). \end{aligned}$$
(42)

Next we define the sequence \(s_k\) for \(k\ge N_1\) by \(s_{N_1}=0\) and \(s_{k+1}=s_k+h_k\) for \(k\ge N_1\) and we claim that if \(\xi _k\) solves Eq. (39), then

$$\begin{aligned} \xi _k=x(s_k),\quad k\ge N_1 \end{aligned}$$

where \(x(t)\) is the solution of Eq. (41) with \(x(0)=\xi _{N_1}\). Clearly this is true for \(k=N_1\) and if it holds for some \(k\ge N_1\), then by Eq. (42)

$$\begin{aligned} x(s_{k+1})=e^{h_kA_+}x(s_k)+\phi ^{h_k}_a(z_+,a_0) =e^{h_kA_+}\xi _k+\phi ^{h_k}_a(z_+,a_0)=\xi _{k+1}. \end{aligned}$$

Conversely, we can show by reversing the reasoning that if \(x(t)\) is the solution of Eq. (41) with \(x(0)=\xi \), then \(\xi _k=x(s_k)\) is the solution of Eq. (39) with \(\xi _{N_1}=\xi \). Clearly \(\xi _k\) is bounded for \(k\ge N_1\) if and only if \(x(t)\) is bounded for \(t\ge 0\).

Next we claim that Eq. (41) has a unique bounded solution \(x_+(t)\) on \(t\ge 0\) with \(Q_+x(0)=0\) given by

$$\begin{aligned} x_+(t)=\int ^t_{0}e^{A_+(t-s)}Q_+f_a(z_+,a_0)ds -\int ^{\infty }_te^{A_+(t-s)}(I-Q_+)f_a(z_+,a_0)ds. \end{aligned}$$

Note \(x_+(t)\) is well-defined and bounded since, using Eq. (3), the integrals of the norms of the integrand are bounded by

$$\begin{aligned} \left[ \int ^t_{0}K_+e^{-\alpha _+(t-s)}ds+\int ^{\infty }_tK_+e^{-\beta _+(s-t)}ds\right] |f_a(z_+,a_0)| \end{aligned}$$

so that

$$\begin{aligned} |x_+(t)|\le K_+(\alpha _+^{-1}+\beta ^{-1}_+)|f_a(z_+,a_0)|. \end{aligned}$$
(43)

That \(x_+(t)\) is a solution of Eq. (41) follows by direct differentiation. It is unique because the difference between any two such solutions would be a bounded solution \(x(t)\) of \(\dot{x}=A_+x\) with \(Q_+x(0)=0\) and so must be \(0\).

Then it follows from the correspondence between the solutions of Eqs. (39) and (41) that \(\xi _k=x_+(s_k)\) is the unique solution of Eq. (39) which is bounded for \(k\ge N_1\) and satisfies \(Q_+\xi _{N_1}=0\).

Next we consider Eq. (40). Given \(\eta \) in the range of \(Q_+\), we claim that the unique solution \(\bar{\xi }_k\) of Eq. (40) bounded on \(k\ge N_1\) such that \(Q_+\bar{\xi }_{N_1}=\eta \) is given by

$$\begin{aligned} \bar{\xi }_k =e^{(s_k-s_{N_1})A_+}\eta +\sum ^k_{\ell =N_1+1}e^{(s_k-s_{\ell })A_+}Q_+g_{\ell -1} -\sum ^{\infty }_{\ell =k+1}e^{(s_k-s_{\ell })A_+}(I-Q_+)g_{\ell -1}. \end{aligned}$$

That the infinite sum is well-defined and that \(\bar{\xi }_k\) is bounded is shown below; that it is a solution can be verified by direct substitution; it is unique because the difference \(\xi _k\) between any two such solutions would be bounded on \(k\ge N_1\) and equal \(e^{(s_k-s_{N_1})A_+}(I-Q_+)\xi _{N_1}\) and therefore must be zero.

Then it follows by superposition that for each \(\eta \) in the range of \(Q_+\), Eq. (16) has a unique solution bounded on \(k\ge N_1\) such that \(Q_+\xi _{N_1}=\eta \) given by

$$\begin{aligned} \xi _k(\eta ,b)=bx_+(s_k)+\bar{\xi }_k,\quad k\ge N_1, \end{aligned}$$
(44)

where \(s_k=h_{N_1}+\cdots +h_{k-1}\). Note also, using Eq. (3) again, that

$$\begin{aligned}&|\bar{\xi }_k| \le K_+|\eta | + \sum ^k_{\ell =N_1+1}K_+e^{-\alpha _+(s_k-s_{\ell })}h_{\ell -1}|\bar{g}_{\ell -1}| +\sum ^{\infty }_{\ell =k+1}K_+e^{-\beta _+(s_{\ell }-s_k)}h_{\ell -1}|\bar{g}_{\ell -1}|\nonumber \\&\quad \le K_+|\eta |+ K_+\Vert \bar{g}\Vert _{\infty }\left[ \sum ^k_{\ell =N_1+1}e^{-\alpha _+(s_k- s_{\ell })}(s_{\ell }-s_{\ell -1}) +\sum ^{\infty }_{\ell =k+1}e^{-\beta _+(s_{\ell }-s_k)}(s_{\ell }-s_{\ell -1})\right] \nonumber \\&\quad \le K_+|\eta |\nonumber \\&\qquad +\, K_+\Vert \bar{g}\Vert _{\infty }\left[ \sum ^k_{\ell =N_1+1}e^{\alpha _+h_{\ell -1}}e^{-\alpha _+(s_k- s_{\ell -1})}(s_{\ell }-s_{\ell -1}) +\sum ^{\infty }_{\ell =k+1}e^{-\beta _+(s_{\ell }-s_k)}(s_{\ell }-s_{\ell -1})\right] \nonumber \\&\quad \le K_+|\eta |+K_+\Vert \bar{g}\Vert _{\infty }\left[ e^{\alpha _+h_{\mathrm{max}}}\int ^{s_k}_{s_{N_1}}e^{-\alpha _+(s_k-t)}dt +\int ^{\infty }_{s_k}e^{-\beta _+(t-s_k)}dt\right] \nonumber \\&\qquad \le K_+|\eta |+K_+(\alpha ^{-1}_+e^{\alpha _+h_{\mathrm{max}}} + \beta ^{-1}_+)\Vert \bar{g}\Vert _{\infty }. \end{aligned}$$
(45)

Then the inequality (17) for \(|\xi _k(\eta ,b)|\) follows from Eqs. (43), (44) and (45). Finally note that we have Eq. (18) where

$$\begin{aligned} \tilde{q}=-\sum ^{\infty }_{\ell =N_1+1}e^{(s_{N_1}-s_{\ell })A_+}(I-Q_+)g_{\ell -1} \end{aligned}$$

so that \(Q_+\tilde{q}=0\) and

$$\begin{aligned} \begin{aligned} v_+&=\int ^{\infty }_0e^{-tA_+}(I-Q_+)f_a(z_+,a_0)dt\\&=-A_+^{-1}\int ^{\infty }_0{d\over dt}e^{-tA_+}(I-Q_+)f_a(z_+,a_0)dt\\&=-A_+^{-1}[e^{-tA_+}(I-Q_+)f_a(z_+,a_0)]^{\infty }_0\\&= A_+^{-1}(I-Q_+)f_a(z_+,a_0)\\&=H_+(I-P_r)v. \end{aligned} \end{aligned}$$

Noting that the inequality for \(|\tilde{q}|\) in Eq. (19) follows as in Eq. (45), the proof of the lemma is complete.\(\square \)

Proof of Lemma 4

We break the solution of Eq. (20) into solving

$$\begin{aligned} \xi _{k+1}=e^{h_kA_-}\xi _k+\phi ^{h_k}_a(z_-,a_0),\quad k<-N_2 \end{aligned}$$
(46)

and

$$\begin{aligned} \xi _{k+1}=e^{h_kA_-}\xi _k+g_{k}, \quad k<-N_2. \end{aligned}$$
(47)

Note first that \(\phi ^t_a(z_-,a_0)\) is the solution of the autonomous equation

$$\begin{aligned} \dot{x}=A_-x+f_a(z_-,a_0) \end{aligned}$$
(48)

with \(x(0)=0\) so that any solution \(x(t)\) of Eq. (48) satisfies

$$\begin{aligned} x(t)=e^{(t-s)A_-}x(s)+\phi ^{t-s}_a(z_-,a_0). \end{aligned}$$
(49)

Next we define the sequence \(s_k\) for \(k\le -N_2\) by \(s_{-{N_2}}=0\) and \(s_{k+1}=s_k+h_k\) for \(k\le -N_2-1\). Then we claim that if \(\xi _k\) solves Eq. (46), then

$$\begin{aligned} \xi _k=x(s_k),\quad k\le -N_2 \end{aligned}$$

where \(x(t)\) is the solution of Eq. (48) with \(x(0)=\xi _{-N_2}\). Clearly this is true for \(k=-N_2\) and if it holds for some \(k\le -N_2\), then by Eq. (49)

$$\begin{aligned} \xi _k=x(s_{k})=e^{h_{k-1}A_-}x(s_{k-1})+\phi ^{h_{k-1}}_a(z_-,a_0) \end{aligned}$$

so that

$$\begin{aligned} \begin{aligned}x(s_{k-1})&=e^{-h_{k-1}A_-}\xi _k-e^{-h_{k-1}A_-}\phi ^{h_{k-1}}_a(z_-,a_0)\\&=e^{-h_{k-1}A_-}(e^{h_{k-1}A_-}\xi _{k-1}+\phi ^{h_{k-1}}_a(z_-,a_0))-e^{-h_{k-1}A_-}\phi ^{h_{k-1}}_a(z_-,a_0))\\&=\xi _{k-1}. \end{aligned} \end{aligned}$$

Conversely, we can show by reversing the reasoning that if \(x(t)\) is the solution of Eq. (48) with \(x(0)=\xi \), then \(\xi _k=x(s_k)\) is the solution of Eq. (46) with \(\xi _{-N_2}=\xi \). Clearly \(\xi _k\) is bounded if and only if \(x(t)\) is bounded.

Next note that Eq. (48) has a unique bounded solution \(x_-(t)\) on \(t\le 0\) with \((I-Q_-)x(0)=0\) given by

$$\begin{aligned} x_-(t)=\int ^t_{-\infty }e^{A_-(t-s)}Q_-f_a(z_-,a_0)ds -\int ^{0}_te^{A_-(t-s)}(I-Q_-)f_a(z_-,a_0)ds \end{aligned}$$

and, using Eq. (3), we obtain the estimate

$$\begin{aligned} |x_-(t)|\le K_-(\alpha ^{-1}_-+\beta ^{-1}_-)|f_a(z_-,a_0)|. \end{aligned}$$
(50)

Next for each \(\omega \) in the nullspace of \(Q_-\), Eq. (47) has a unique solution bounded on \(k\le -N_2\) such that \((I-Q_-)\xi _{-N_2}=\omega \) given by

$$\begin{aligned} \bar{\xi }_k =e^{(s_k-s_{-N_2})A_-}\omega +\sum ^k_{\ell =-\infty }e^{(s_k-s_{\ell })A_-}Q_-g_{\ell -1} -\sum ^{-N_2}_{\ell =k+1}e^{(s_k-s_{\ell })A_-}(I-Q_-)g_{\ell -1}. \end{aligned}$$

That the infinite sum is well-defined and that \(\bar{\xi }_k\) is bounded is shown below; that it is a solution can be verified by direct substitution; it is unique because the difference \(\xi _k\) between any two such solutions would be bounded on \(k\le -N_2\) and equal \(e^{(s_k-s_{-N_2})A_-}Q_-\xi _{-N_2}\) and therefore must be zero.

By the superposition principle it follows that the unique solution of Eq. (20) bounded on \(k\le -N_2\) such that \((I-Q_-)\xi _{-N_2}=\omega \) is given by

$$\begin{aligned} \xi _k(\omega ,b)=bx_-(s_k)+\bar{\xi }_k,\quad k\le -N_2. \end{aligned}$$
(51)

Also using Eq. (3) again

$$\begin{aligned} \begin{aligned} |\bar{\xi }_k|&\le K_-|\omega | + \sum ^k_{\ell =-\infty }K_-e^{-\alpha _-(s_k-s_{\ell })}h_{\ell -1}|\bar{g}_{\ell -1}| +\sum ^{-N_2}_{\ell =k+1}K_-e^{-\beta _-(s_{\ell }-s_k)}h_{\ell -1}|\bar{g}_{\ell -1}|\\&\le K_-|\omega |\\&+ K_-\Vert \bar{g}\Vert _{\infty }\left[ \sum ^k_{\ell =-\infty }e^{-\alpha _-(s_k- s_{\ell })}(s_{\ell }-s_{\ell -1}) +\sum ^{-N_2}_{\ell =k+1}e^{-\beta _-(s_{\ell }-s_k)}(s_{\ell }-s_{\ell -1})\right] \\&\le K_-|\omega |\\&+ K_-\Vert \bar{g}\Vert _{\infty }\left[ \sum ^k_{\ell =-\infty }e^{\alpha _-h_{\ell -1}} e^{-\alpha _-(s_k- s_{\ell -1})}(s_{\ell }-s_{\ell -1}) +\sum ^{-N_2}_{\ell =k+1}e^{-\beta _-(s_{\ell }-s_k)}(s_{\ell }-s_{\ell -1})\right] \\&\le K_-|\omega |+K_-\Vert \bar{g}\Vert _{\infty }\left[ e^{\alpha _-h_{\mathrm{max}}}\int ^{s_k}_{-\infty }e^{-\alpha _-(s_k-t)}dt +\int ^{s_{-N_2}}_{s_k}e^{-\beta _-(t-s_k)}dt\right] \\&\le K_-|\omega |+K_-(\alpha ^{-1}_-e^{\alpha _-h_{\mathrm{max}}}+\beta ^{-1}_-)\Vert \bar{g}\Vert _{\infty }. \end{aligned} \end{aligned}$$
(52)

Then inequality Eq. (21) for \(|\xi _k(\omega ,b)|\) follows from Eqs. (50), (51) and (52). Finally note that Eq. (22) holds with

$$\begin{aligned} \tilde{r}=\sum ^{-N_2}_{\ell =-\infty }e^{(s_{-N_2}-s_{\ell })A_-}Q_-g_{\ell -1} \end{aligned}$$

so that \((I-Q_-)\tilde{r}=0\) and

$$\begin{aligned} \begin{aligned} v_-&= -\int ^{\infty }_0e^{tA_-}Q_-f_a(z_-,a_0)dt\\&= -A^{-1}_-\int ^{\infty }_0{d\over dt}e^{tA_-}Q_-f_a(z_-,a_0)dt\\&= -A^{-1}_-[e^{tA_-}Q_-f_a(z_-,a_0)]^{\infty }_0\\&=A^{-1}_-Q_-f_a(z_-,a_0)\\&=H_-P_rv. \end{aligned} \end{aligned}$$

Noting that the inequality for \(|\tilde{r}|\) in Eq. (23) follows as in Eq. (52), the proof of the lemma is complete.\(\square \)

Proof of Lemma 5

We just prove the lemma for \(z_+\), as the proof for \(z_-\) is similar. If we write \(x=z_+ + y\), the equation \(\dot{x}=f(x,a)\) becomes

$$\begin{aligned} \dot{y}=A_+y+g(y,a), \end{aligned}$$
(53)

where

$$\begin{aligned} g(y,a)=f(z_++y,a)-f(z_+,a_0)-f_x(z_+,a_0)y. \end{aligned}$$

Note that if \(|y|\le \mu \), \(|a-a_0|\le \Delta _1\)

$$\begin{aligned} |g_y(y,a)|\!\le \! |f_x(z_++y,a)\!-\!f_x(z_+,a)|\!+\!|f_x(z_+,a)-f_x(z_+,a_0)| \le M_2|y|+M_4|a-a_0| \end{aligned}$$

and \(|g(0,a)|\le M_3|a-a_0|\) so that

$$\begin{aligned} |g(y,a)|\le [M_2|y|+M_4|a-a_0|]|y|+M_3|a-a_0|. \end{aligned}$$

Note we can use the bounds \(M_i\) on the norms of the derivatives of \(f\) because \(\mu \le \Delta _0\) and the ball of radius \(\Delta _0\) with centre \(z_+\) is in \(U\). If \(y(t)\) is a solution of Eq. (53) with \(|y(t)|\le \mu \) for all \(t\), then

$$\begin{aligned} y(t)=\int ^t_{-\infty }e^{(t-u)A_+}Q_+g(y(u),a)du -\int ^{\infty }_te^{-(u-t)A_+}(I-Q_+)g(y(u),a)du. \end{aligned}$$
(54)

That is, \(y\) is a fixed point of the operator \(S\) defined by the right side of Eq. (54). \(S\) is defined on the complete metric space \(X\) consisting of continuous \({\mathbb {R}}^n-\)valued functions \(y(t)\) with \(|y(t)|\le \mu \) for all \(t\), the metric being given by the supremum norm \(\Vert \cdot \Vert _{\infty }\). \(S\) maps \(X\) into itself since, using Eq. (3),

$$\begin{aligned} \begin{aligned} |(Sy)(t)|&\le \int ^t_{-\infty }K_+e^{-\alpha _+(t-u)}[(M_2\mu +M_4|a-a_0|)\mu +M_3|a-a_0|]du\\&\quad +\int ^{\infty }_tK_+e^{-\beta _+(u-t)}[(M_2\mu +M_4|a-a_0|)\mu +M_3|a-a_0|]du \end{aligned} \end{aligned}$$

so that by Eq. (4)

$$\begin{aligned} \Vert Sy\Vert _{\infty } \le K_+(\alpha _+^{-1}+\beta _+^{-1})[(M_2\mu +M_4|a-a_0|)\mu +M_3|a-a_0|]\le \mu . \end{aligned}$$

Also by similar arguments, if \(y_1\) and \(y_2\) are in \(X\),

$$\begin{aligned} \Vert Sy_1-Sy_2\Vert _{\infty } \le K_+(\alpha _+^{-1}+\beta _+^{-1})(M_2\mu +M_4|a-a_0|)\Vert y_1-y_2\Vert _{\infty }\le {1\over 2}\Vert y_1-y_2\Vert _{\infty }. \end{aligned}$$

Then if \(y(t)\) is the unique fixed point of \(S\), \(x_+(t,a) =z_++y(t)\) is the unique solution of \(\dot{x}=f(x,a)\) such that \(|x_+(t,a)-z_+|\le \mu \) for all \(t\). This completes the proof of the lemma and the paper.\(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coomes, B.A., Koçak, H. & Palmer, K.J. A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics. J Dyn Diff Equat 28, 1081–1114 (2016). https://doi.org/10.1007/s10884-015-9437-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-015-9437-y

Keywords

Mathematics Subject Classification

Navigation