An Approach for the Construction of Systems That Self-Generate Chaotic Solitons

Affiliation(s)

Department of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, P.R. China.

Department of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, P.R. China.

ABSTRACT

This paper proposes a method for constructing partial differential equation (PDE) systems with chaotic solitons by using truncated normal forms of an ordinary differential equation (ODE). The construction is based mainly on the fact that the existence of a soliton in a PDE system is equal to that of a homoclinic orbit in a related ODE system, and that chaos of ?i’lnikov homoclinic type in the ODE imply that the soliton in the PDE changes its profile chaotically along propagation direction. It is guaranteed that the constructed systems can self-generate chaotic solitons without any external perturbation but with constrained wave velocities in a rigorously mathematical sense.

This paper proposes a method for constructing partial differential equation (PDE) systems with chaotic solitons by using truncated normal forms of an ordinary differential equation (ODE). The construction is based mainly on the fact that the existence of a soliton in a PDE system is equal to that of a homoclinic orbit in a related ODE system, and that chaos of ?i’lnikov homoclinic type in the ODE imply that the soliton in the PDE changes its profile chaotically along propagation direction. It is guaranteed that the constructed systems can self-generate chaotic solitons without any external perturbation but with constrained wave velocities in a rigorously mathematical sense.

Cite this paper

B. Chen, "An Approach for the Construction of Systems That Self-Generate Chaotic Solitons,"*Applied Mathematics*, Vol. 3 No. 7, 2012, pp. 755-759. doi: 10.4236/am.2012.37112.

B. Chen, "An Approach for the Construction of Systems That Self-Generate Chaotic Solitons,"

References

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[6] S. V. Dmitriev and T. Shigenari, “Short-Lived TwoSoliton Bound States in Weakly Perturbed Nonlinear Schr?dinger Equation,” Chaos, Vol. 12, No. 2, 2002, pp. 324-332. doi:10.1063/1.1476951

[7] F. Benvenuto, et al., “Manifestations of Classical and Quantum Chaos in Nonlinear Wave Propagation,” Physical Review A, Vol. 44, No. 6, 1991, R3423. doi:10.1103/PhysRevA.44.R3423

[8] N. Akhmediev, et al., “Pulsating Solitons, Chaotic Solitons, Period Doubling, and Pulse Coexistence in ModeLocking Lasers: Complex Ginzburg-Landau Equation Approach,” Physical Review E, Vol. 63, No. 5, 2001, 056602. doi:10.1103/PhysRevE.63.056602

[9] C. L. Zheng, “Coherent Soliton Structures with Chaotic and Fractal Behaviors in a Generalized (2+1)-Dimensional Korteweg de-Vries System,” Chinese Journal of Physics, Vol. 41, 2003, pp. 442-456.

[10] S. P. Novikov and A. P. Veselov, “Two-Dimensional Schr?dinger Operator: Inverse Scattering Transform and Evolutional Equations,” Physica D: Nonlinear Phenomena, Vol. 18, No. 1-3, 1986, pp. 267-273. doi:10.1016/0167-2789(86)90187-9

[11] F. C. P. Silva, “?i'l'nikov Theorem—A Tutorial,” IEEE Transactions on Circuits and Systems, Vol. 40, 1993, pp. 675-682.

[12] V. I. Arnold, “Lectures in Bifurcation in Versal Families,” Russian Mathematical Surveys, Vol. 27, No. 5, 1972, pp. 54-123. doi:10.1070/RM1972v027n05ABEH001385

[13] A. Arneodo, et al., “Asymptotic Chaos,” Physica D: Nonlinear Phenomena, Vol. 14, No. 3, 1985, pp. 327-347. doi:10.1016/0167-2789(85)90093-4

[14] P. H. Coullet and E. A. Spiegel, “Amplitude Equations for Systems with Competing Instabilities,” SIAM Journal on Applied Mathematics, Vol. 43, No. 4, 1983, pp. 776-821. doi:10.1137/0143052

[15] A. Arneodo, et al., “Oscillators with Chaotic Behavior: An Illustration of a Theorem by Shilnikov,” Journal of Statistical Physics, Vol. 27, 1982, pp. 171-182. doi:10.1007/BF01011745

[16] S. Wiggins, “Global Bifurcation and Chaos,” SpringerVerlag, New York, 1988. doi:10.1007/978-1-4612-1042-9

[1] T. H. Lee, “Electrical Solitons Come of Age,” Nature, Vol. 440, 2006, pp. 36-37. doi:10.1038/440036a

[2] A. Argyris, et al, “Chaos-Based Communications at High Bit Rates Using Commercial Fibre-Optic Links,” Nature, Vol. 438, 2005, pp. 343-346. doi:10.1038/nature04275

[3] M. Z. Wu, B. A. Kalinikos and C. E. Patton, “Self-Generation of Chaotic Solitary Spin Wave Pulses in Magnetic Film Active Feedback Rings,” Physical Review Letters, Vol. 95, No. 23, 2005, 237202. doi:10.1103/PhysRevLett.95.237202

[4] D. S. Riketts, X. Li and D. Ham, “Electrical Soliton Oscillator,” IEEE Transactions on Microwave Theory and Techniques, Vol. 54, No. 1, 2006, pp. 373-382. doi:10.1109/TMTT.2005.861652

[5] S. V. Dmitriev, et al, “Chaotic Character of Two-Soliton Collisions in the Weakly Perturbed Nonlinear Schr?dinger Equation,” Physical Review E, Vol. 66, No. 4, 2002, 046609. doi:10.1103/PhysRevE.66.046609

[6] S. V. Dmitriev and T. Shigenari, “Short-Lived TwoSoliton Bound States in Weakly Perturbed Nonlinear Schr?dinger Equation,” Chaos, Vol. 12, No. 2, 2002, pp. 324-332. doi:10.1063/1.1476951

[7] F. Benvenuto, et al., “Manifestations of Classical and Quantum Chaos in Nonlinear Wave Propagation,” Physical Review A, Vol. 44, No. 6, 1991, R3423. doi:10.1103/PhysRevA.44.R3423

[8] N. Akhmediev, et al., “Pulsating Solitons, Chaotic Solitons, Period Doubling, and Pulse Coexistence in ModeLocking Lasers: Complex Ginzburg-Landau Equation Approach,” Physical Review E, Vol. 63, No. 5, 2001, 056602. doi:10.1103/PhysRevE.63.056602

[9] C. L. Zheng, “Coherent Soliton Structures with Chaotic and Fractal Behaviors in a Generalized (2+1)-Dimensional Korteweg de-Vries System,” Chinese Journal of Physics, Vol. 41, 2003, pp. 442-456.

[10] S. P. Novikov and A. P. Veselov, “Two-Dimensional Schr?dinger Operator: Inverse Scattering Transform and Evolutional Equations,” Physica D: Nonlinear Phenomena, Vol. 18, No. 1-3, 1986, pp. 267-273. doi:10.1016/0167-2789(86)90187-9

[11] F. C. P. Silva, “?i'l'nikov Theorem—A Tutorial,” IEEE Transactions on Circuits and Systems, Vol. 40, 1993, pp. 675-682.

[12] V. I. Arnold, “Lectures in Bifurcation in Versal Families,” Russian Mathematical Surveys, Vol. 27, No. 5, 1972, pp. 54-123. doi:10.1070/RM1972v027n05ABEH001385

[13] A. Arneodo, et al., “Asymptotic Chaos,” Physica D: Nonlinear Phenomena, Vol. 14, No. 3, 1985, pp. 327-347. doi:10.1016/0167-2789(85)90093-4

[14] P. H. Coullet and E. A. Spiegel, “Amplitude Equations for Systems with Competing Instabilities,” SIAM Journal on Applied Mathematics, Vol. 43, No. 4, 1983, pp. 776-821. doi:10.1137/0143052

[15] A. Arneodo, et al., “Oscillators with Chaotic Behavior: An Illustration of a Theorem by Shilnikov,” Journal of Statistical Physics, Vol. 27, 1982, pp. 171-182. doi:10.1007/BF01011745

[16] S. Wiggins, “Global Bifurcation and Chaos,” SpringerVerlag, New York, 1988. doi:10.1007/978-1-4612-1042-9