Skip to main content
SpringerLink
Log in

Generalized Projective Synchronization of Vaidyanathan Chaotic System via Active and Adaptive Control

  • Chapter
  • First Online:
Book cover Advances and Applications in Nonlinear Control Systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 635))

Abstract

Generalized projective synchronization (GPS) of chaotic systems generalizes known types of synchronization schemes such as complete synchronization (CS), anti-synchronization (AS), hybrid synchronization (HS), projective synchronization (PS), etc. In this work, we have designed active and adaptive controllers for the generalized projective synchronization (GPS) of identical Vaidyanathan chaotic systems (2014). Vaidyanathan system is an eight-term chaotic system with three quadratic nonlinearities. The Lyapunov exponents of the Vaidyanathan chaotic system are obtained as \(L_1 = 6.5294, L_2 = 0\) and \(L_3 = -26.4696\). Since the maximal Lyapunov exponent of the Vaidyanathan system is \(L_1 = 6.5294\), the system exhibits highly chaotic behaviour. The Kaplan–Yorke dimension of the Vaidyanathan chaotic system is obtained as \(D_{KY} = 2.2467\). The main GPS results in this work have been established using Lyapunov stability theory. MATLAB plots have been depicted to illustrate the phase portraits of the Vaidyanathan chaotic system and also the GPS results for Vaidyanathan chaotic systems using active and adaptive controllers.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arneodo A, Coullet P, Tresser C (1981) Possible new strange attractors with spiral structure. Commun Math Phys 79(4):573–576

    Article  MathSciNet  MATH  Google Scholar 

  2. Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design, studies in computational intelligence, vol 581. Springer, Germany

    Google Scholar 

  3. Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control, studies in computational intelligence, vol 575. Springer, Germany

    Google Scholar 

  4. Basnarkov L, Duane GS, Kocarev L (2014) Generalized synchronization and coherent structures in spatially extended systems. Chaos, Solitons and Fractals 59:35–41

    Article  MathSciNet  Google Scholar 

  5. Cai G, Tan Z (2007) Chaos synchronization of a new chaotic system via nonlinear control. J Uncertain Syst 1(3):235–240

    Google Scholar 

  6. Carroll TL, Pecora LM (1991) Synchronizing chaotic circuits. IEEE Trans Circuits Syst 38(4):453–456

    Article  MATH  Google Scholar 

  7. Chen G, Ueta T (1999) Yet another chaotic attractor. Int J Bifurc Chaos 9(7):1465–1466

    Article  MathSciNet  MATH  Google Scholar 

  8. Hu A, Xu Z (2011) Multi-stable chaotic attractors in generalized synchronization. Commun Nonlinear Sci Numer Simul 16(8):3237–3244

    Article  MathSciNet  MATH  Google Scholar 

  9. Karthikeyan R, Sundarapandian V (2014) Hybrid synchronization of four-scroll systems via active control. J Electr Eng 65(2):97–103

    Google Scholar 

  10. Khalil HK (2001) Nonlinear Syst, 3rd edn. Prentice Hall, New Jersey, USA

    Google Scholar 

  11. Li D (2008) A three-scroll chaotic attractor. Phys Lett A 372(4):387–393

    Article  MathSciNet  MATH  Google Scholar 

  12. Lorenz EN (1963) Deterministic periodic flow. J Atmos Sci 20(2):130–141

    Article  Google Scholar 

  13. Lü J, Chen G (2002) A new chaotic attractor coined. Int J Bifurc Chaos 12(3):659–661

    Article  MathSciNet  MATH  Google Scholar 

  14. Pakiriswamy S, Vaidyanathan S (2012) Generalized projective synchronization of hyperchaotic Lü and hyperchaotic Cai systems via active control. Lect Notes Inst Comput Sci Soc Inf Telecommun Eng 84:53–62

    Google Scholar 

  15. Pakiriswamy S, Vaidyanathan S (2012) Generalized projective synchronization of three-scroll chaotic systems via active control. Lect Notes Inst Comput Sci Soc Inf Telecommun Eng 85:146–155

    Google Scholar 

  16. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64(8):821–824

    Article  MathSciNet  MATH  Google Scholar 

  17. Pehlivan I, Moroz IM, Vaidyanathan S (2014) Analysis, synchronization and circuit design of a novel butterfly attractor. J Sound Vib 333(20):5077–5096

    Article  Google Scholar 

  18. Pham VT, Volos CK, Vaidyanathan S, Le T, Vu V (2015) A memristor-based hyperchaotic system with hidden attractors: Dynamics, synchronization and circuital emulating. J Eng Sci Technol Rev 8(2):205–214

    Google Scholar 

  19. Rasappan S, Vaidyanathan S (2012) Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East J Math Sci 67(2):265–287

    MathSciNet  MATH  Google Scholar 

  20. Rasappan S, Vaidyanathan S (2012) Hybrid synchronization of n-scroll Chua and Lur’e chaotic systems via backstepping control with novel feedback. Arch Control Sci 22(3):343–365

    MathSciNet  MATH  Google Scholar 

  21. Rasappan S, Vaidyanathan S (2012) Synchronization of hyperchaotic Liu system via backstepping control with recursive feedback. Commun Comput Inf Sci 305:212–221

    Article  MATH  Google Scholar 

  22. Rasappan S, Vaidyanathan S (2013) Hybrid synchronization of \(n\)-scroll chaotic Chua circuits using adaptive backstepping control design with recursive feedback. Malaysian J Math Sci 7(2):219–246

    MathSciNet  Google Scholar 

  23. Rasappan S, Vaidyanathan S (2014) Global chaos synchronization of WINDMI and Coullet chaotic systems using adaptive backstepping control design. Kyungpook Math J 54(1):293–320

    Article  MathSciNet  MATH  Google Scholar 

  24. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57(5):397–398

    Article  Google Scholar 

  25. Sampath S, Vaidyanathan S, Volos CK, Pham VT (2015) An eight-term novel four-scroll chaotic System with cubic nonlinearity and its circuit simulation. J Eng Sci Technol Rev 8(2):1–6

    Google Scholar 

  26. Sarasu P, Sundarapandian V (2011) Active controller design for generalized projective synchronization of four-scroll chaotic systems. Int J Syst Signal Control Eng Appl 4(2):26–33

    Google Scholar 

  27. Sarasu P, Sundarapandian V (2011) The generalized projective synchronization of hyperchaotic Lorenz and hyperchaotic Qi systems via active control. Int J Soft Comput 6(5):216–223

    Article  Google Scholar 

  28. Sarasu P, Sundarapandian V (2012) Adaptive controller design for the generalized projective synchronization of 4-scroll systems. Int J Syst Signal Control Eng Appl 5(2):21–30

    Google Scholar 

  29. Sarasu P, Sundarapandian V (2012) Generalized projective synchronization of three-scroll chaotic systems via adaptive control. Eur J Sci Res 72(4):504–522

    Google Scholar 

  30. Sarasu P, Sundarapandian V (2012) Generalized projective synchronization of two-scroll systems via adaptive control. Int J Soft Comput 7(4):146–156

    Article  Google Scholar 

  31. Shi Y, Zhu P, Qin K (2014) Projective synchronization of different chaotic neural networks with mixed time delays based on an integral sliding mode controller. Neurocomputing 123:443–449

    Article  Google Scholar 

  32. Sprott JC (1994) Some simple chaotic flows. Phys Rev E 50(2):647–650

    Article  MathSciNet  Google Scholar 

  33. Sundarapandian V (2010) Output regulation of the Lorenz attractor. Far East J Math Sci 42(2):289–299

    MathSciNet  MATH  Google Scholar 

  34. Sundarapandian V (2013) Adaptive control and synchronization design for the Lu-Xiao chaotic system. Lect Notes Electr Eng 131:319–327

    Article  Google Scholar 

  35. Sundarapandian V (2013) Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers. J Eng Sci Technol Rev 6(4):45–52

    Google Scholar 

  36. Sundarapandian V, Karthikeyan R (2011) Anti-synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems by adaptive control. Int J Syst Signal Control Eng Appl 4(2):18–25

    Google Scholar 

  37. Sundarapandian V, Karthikeyan R (2011) Anti-synchronization of Lü and Pan chaotic systems by adaptive nonlinear control. Eur J Sci Res 64(1):94–106

    Google Scholar 

  38. Sundarapandian V, Karthikeyan R (2012) Adaptive anti-synchronization of uncertain Tigan and Li systems. J Eng Appl Sci 7(1):45–52

    Article  MATH  Google Scholar 

  39. Sundarapandian V, Karthikeyan R (2012) Hybrid synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems via active control. J Eng Appl Sci 7(3):254–264

    Article  Google Scholar 

  40. Sundarapandian V, Pehlivan I (2012) Analysis, control, synchronization, and circuit design of a novel chaotic system. Math Comput Model 55(7–8):1904–1915

    Article  MathSciNet  MATH  Google Scholar 

  41. Sundarapandian V, Sivaperumal S (2011) Sliding controller design of hybrid synchronization of four-wing chaotic systems. Int J Soft Comput 6(5):224–231

    Article  Google Scholar 

  42. Suresh R, Sundarapandian V (2013) Global chaos synchronization of a family of \(n\)-scroll hyperchaotic Chua circuits using backstepping control with recursive feedback. Far East J Math Sci 73(1):73–95

    MATH  Google Scholar 

  43. Tigan G, Opris D (2008) Analysis of a 3D chaotic system. Chaos, Solitons and Fractals 36:1315–1319

    Article  MathSciNet  MATH  Google Scholar 

  44. Vaidyanathan S (2011) Hybrid chaos synchronization of Liu and Lü systems by active nonlinear control. Commun Comput Inf Sci 204:1–10

    Article  Google Scholar 

  45. Vaidyanathan S (2011) Output regulatoin of Arneodo-Coullet chaotic system. Commun Comput Inf Sci 133:98–107

    Article  Google Scholar 

  46. Vaidyanathan S (2012) Adaptive controller and synchronizer design for the Qi-Chen chaotic system. Lect Notes Inst Comput Sci Soc Inf Telecommun Eng 85:124–133

    Google Scholar 

  47. Vaidyanathan S (2012) Analysis and synchronization of the hyperchaotic Yujun systems via sliding mode control. Adv Intell Syst Comput 176:329–337

    Article  Google Scholar 

  48. Vaidyanathan S (2012) Anti-synchronization of Sprott-L and Sprott-M chaotic systems via adaptive control. Int J Control Theory Appl 5(1):41–59

    MathSciNet  Google Scholar 

  49. Vaidyanathan S (2012) Global chaos control of hyperchaotic Liu system via sliding control method. Int J Control Theory Appl 5(2):117–123

    Google Scholar 

  50. Vaidyanathan S (2012) Output regulatoin of the Liu chaotic system. Appl Mech Mat 110–116:3982–3989

    Google Scholar 

  51. Vaidyanathan S (2012) Sliding mode control based global chaos control of Liu-Liu-Liu-Su chaotic system. Int J Control Theory Appl 5(1):15–20

    MathSciNet  Google Scholar 

  52. Vaidyanathan S (2013) A new six-term 3-D chaotic system with an exponential nonlinearity. Far East J Math Sci 79(1):135–143

    MATH  Google Scholar 

  53. Vaidyanathan S (2013) Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters. J Eng Sci Technol Rev 6(4):53–65

    MathSciNet  Google Scholar 

  54. Vaidyanathan S (2013) Analysis, control and synchronization of hyperchaotic Zhou system via adaptive control. Adv Intell Syst Comput 177:1–10

    Article  Google Scholar 

  55. Vaidyanathan S (2014) A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities. Far East J Math Sci 84(2):219–226

    MATH  Google Scholar 

  56. Vaidyanathan S (2014) Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities. Eur Phys J Spec Top 223(8):1519–1529

    Article  Google Scholar 

  57. Vaidyanathan S (2014) Analysis, control and synchronisation of a six-term novel chaotic system with three quadratic nonlinearities. Int J Model Identif Control 22(1):41–53

    Article  Google Scholar 

  58. Vaidyanathan S (2014) Generalized projective synchronisation of novel 3-D chaotic systems with an exponential non-linearity via active and adaptive control. Int J Model Identif Control 22(3):207–217

    Article  MathSciNet  Google Scholar 

  59. Vaidyanathan S (2014) Global chaos synchronization of identical Li-Wu chaotic systems via sliding mode control. Int J Model Identif Control 22(2):170–177

    Article  Google Scholar 

  60. Vaidyanathan S (2015) A 3-D novel highly chaotic system with four quadratic nonlinearities, its adaptive control and anti-synchronization with unknown parameters. J Eng Sci Technol Rev 8(2):106–115

    Google Scholar 

  61. Vaidyanathan S (2015) Analysis, properties and control of an eight-term 3-D chaotic system with an exponential nonlinearity. Int J Model Identif Control 23(2):164–172

    Article  Google Scholar 

  62. Vaidyanathan S (2015) Analysis and control of a 4-D novel hyperchaotic system. Stud Comput Intell 581:3–17

    Article  Google Scholar 

  63. Vaidyanathan S, Azar AT (2015) Analysis, control and synchronization of a nine-term 3-D novel chaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modelling and control systems design, studies in computational intelligence, vol 581. Springer, Germany, pp 19–38

    Google Scholar 

  64. Vaidyanathan S, Azar AT (2015) Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan-Madhavan chaotic systems. Stud Comput Intell 576:527–547

    Article  Google Scholar 

  65. Vaidyanathan S, Azar AT (2015) Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan chaotic systems. Stud Comput Intell 576:549–569

    Article  Google Scholar 

  66. Vaidyanathan S, Madhavan K (2013) Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system. Int J Control Theory Appl 6(2):121–137

    Google Scholar 

  67. Vaidyanathan S, Pakiriswamy S (2011) The design of active feedback controllers for the generalized projective synchronization of hyperchaotic Qi and hyperchaotic Lorenz systems. Commun Comput Inf Sci 245:231–238

    Article  Google Scholar 

  68. Vaidyanathan S, Pakiriswamy S (2012) Generalized projective synchronization of double-scroll chaotic systems using active feedback control. Lect Notes Inst Comput Sci Soc Inf Telecommun Eng 84:111–118

    Google Scholar 

  69. Vaidyanathan S, Pakiriswamy S (2013) Generalized projective synchronization of six-term Sundarapandian chaotic systems by adaptive control. Int J Control Theory Appl 6(2):153–163

    Google Scholar 

  70. Vaidyanathan S, Pakiriswamy S (2015) A 3-D novel conservative chaotic system and its generalized projective synchronization via adaptive control. J Eng Sci Technol Rev 8(2):52–60

    Google Scholar 

  71. Vaidyanathan S, Rajagopal K (2011) Anti-synchronization of Li and T chaotic systems by active nonlinear control. Commun Comput Inf Sci 198:175–184

    Article  Google Scholar 

  72. Vaidyanathan S, Rajagopal K (2011) Global chaos synchronization of hyperchaotic Pang and Wang systems by active nonlinear control. Commun Comput Inf Sci 204:84–93

    Article  Google Scholar 

  73. Vaidyanathan S, Rajagopal K (2011) Global chaos synchronization of Lü and Pan systems by adaptive nonlinear control. Commun Comput Inf Sci 205:193–202

    Article  Google Scholar 

  74. Vaidyanathan S, Rajagopal K (2011) Hybrid synchronization of hyperchaotic Wang-Chen and hyperchaotic Lorenz systems by active non-linear control. Int J Syst Signal Control Eng Appl 4(3):55–61

    Google Scholar 

  75. Vaidyanathan S, Rajagopal K (2012) Global chaos synchronization of hyperchaotic Pang and hyperchaotic Wang systems via adaptive control. Int J Soft Comput 7(1):28–37

    Article  MATH  Google Scholar 

  76. Vaidyanathan S, Rasappan S (2011) Global chaos synchronization of hyperchaotic Bao and Xu systems by active nonlinear control. Commun Comput Inf Sci 198:10–17

    Article  Google Scholar 

  77. Vaidyanathan S, Rasappan S (2011) Hybrid synchronization of hyperchaotic Qi and Lü systems by nonlinear control. Commun Comput Inf Sci 131:585–593

    Article  MATH  Google Scholar 

  78. Vaidyanathan S, Rasappan S (2014) Global chaos synchronization of \(n\)-scroll Chua circuit and Lur’e system using backstepping control design with recursive feedback. Arabian J Sci Eng 39(4):3351–3364

    Article  Google Scholar 

  79. Vaidyanathan S, Sampath S (2011) Global chaos synchronization of hyperchaotic Lorenz systems by sliding mode control. Commun Comput Inf Sci 205:156–164

    Article  Google Scholar 

  80. Vaidyanathan S, Sampath S (2012) Anti-synchronization of four-wing chaotic systems via sliding mode control. Int J Autom Comput 9(3):274–279

    Article  Google Scholar 

  81. Vaidyanathan S, Volos C, Pham VT (2014) Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation. Arch Control Sci 24(4):409–446

    MathSciNet  MATH  Google Scholar 

  82. Vaidyanathan S, Volos C, Pham VT, Madhavan K, Idowu BA (2014) Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities. Arch Control Sci 24(3):375–403

    MathSciNet  MATH  Google Scholar 

  83. Vaidyanathan S, Idowu BA, Azar AT (2015) Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. Stud Comput Intell 581:39–58

    Article  Google Scholar 

  84. Vaidyanathan S, Rajagopal K, Volos CK, Kyprianidis IM, Stouboulos IN (2015) Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW. J Eng Sci Technol Rev 8(2):130–141

    Google Scholar 

  85. Vaidyanathan S, Volos C, Pham VT, Madhavan K (2015) Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation. Arch Control Sci 25(1):135–158

    Google Scholar 

  86. Vaidyanathan S, Volos CK, Kyprianidis IM, Stouboulos IN, Pham VT (2015) Analysis, adaptive control and anti-synchronization of a six-term novel jerk chaotic system with two exponential nonlinearities and its circuit simulation. J Eng Sci Technol Rev 8(2):24–36

    Google Scholar 

  87. Vaidyanathan S, Volos CK, Pham VT (2015) Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation. J Eng Sci Technol Rev 8(2):181–191

    Google Scholar 

  88. Vaidyanathan S, Volos CK, Pham VT (2015) Analysis, control, synchronization and SPICE implementation of a novel 4-D hyperchaotic Rikitake dynamo system without equilibrium. J Eng Sci Technol Rev 8(2):232–244

    Google Scholar 

  89. Vaidyanathan S, Volos CK, Pham VT (2015) Global chaos control of a novel nine-term chaotic system via sliding mode control. In: Azar AT, Zhu Q (eds) Advances and applications in sliding mode control systems, studies in computational intelligence, vol 576. Springer, Germany, pp 571–590

    Google Scholar 

  90. Vaidyanathan S, Volos CK, Rajagopal K, Kyprianidis IM, Stouboulos IN (2015) Adaptive backstepping controller design for the anti-synchronization of identical WINDMI chaotic systems with unknown parameters and its SPICE implementation. J Eng Sci Technol Rev 8(2):74–82

    Google Scholar 

  91. Volos CK, Kyprianidis IM, Stouboulos IN, Tlelo-Cuautle E, Vaidyanathan S (2015) Memristor: A new concept in synchronization of coupled neuromorphic circuits. J Eng Sci Technol Rev 8(2):157–173

    Google Scholar 

  92. Volos CK, Pham VT, Vaidyanathan S, Kyprianidis IM, Stouboulos IN (2015) Synchronization phenomena in coupled Colpitts circuits. J Eng Sci Technol Rev 8(2):142–151

    Google Scholar 

  93. Wei Z, Yang Q (2010) Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci. Appl Math Comput 217(1):422–429

    MathSciNet  MATH  Google Scholar 

  94. Yu J, Hu C, Jiang H, Fan X (2014) Projective synchronization for fractional neural networks. Neural Netw 49:87–95

    Article  MATH  Google Scholar 

  95. Yuan Z, Xu Z, Guo L (2012) Generalized synchronization of two bidirectionally coupled discrete dynamical systems. Commun Nonlinear Sci Numer Simul 17(2):992–1002

    Article  MathSciNet  MATH  Google Scholar 

  96. Zhang D, Xu J (2010) Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller. Appl Math Comput 217(1):164–174

    MathSciNet  MATH  Google Scholar 

  97. Zhou W, Xu Y, Lu H, Pan L (2008) On dynamics analysis of a new chaotic attractor. Phys Lett A 372(36):5773–5777

    Article  MathSciNet  MATH  Google Scholar 

  98. Zhu C, Liu Y, Guo Y (2010) Theoretic and numerical study of a new chaotic system. Intell Inf Manag 2:104–109

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research and Development Centre, Vel Tech University, Avadi, Chennai, 600062, Tamil Nadu, India

    Sundarapandian Vaidyanathan

Authors
  1. Sundarapandian Vaidyanathan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sundarapandian Vaidyanathan .

Editor information

Editors and Affiliations

  1. Chennai, India

    Sundarapandian Vaidyanathan

  2. Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos Volos

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Vaidyanathan, S. (2016). Generalized Projective Synchronization of Vaidyanathan Chaotic System via Active and Adaptive Control. In: Vaidyanathan, S., Volos, C. (eds) Advances and Applications in Nonlinear Control Systems. Studies in Computational Intelligence, vol 635. Springer, Cham. https://doi.org/10.1007/978-3-319-30169-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30169-3_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30167-9

  • Online ISBN: 978-3-319-30169-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation