By finding bounds(upper & lower), perhaps we may use completeness axioms.

@Eka Acharya: The main problem is to find those bounds.

A standard approach consists of using energy estimates techniques. The latter can be used to prove dissipation for a broad class of ODEs (and PDEs).

The example of the Lorenz model (63) is standard and gives already some insights on how energy estimates operate. For this model, a proof of the boundedness of the trajectories can be found for instance in the book of Roger Temam (Chapter I, sec. 2.3) "Infinite-Dimensional Dynamical Systems in Mechanics and Physics."

I hope this will help you.

Best,

Mickael

HI Zeraoulia, I took the liberty to insert your question to our database and the following first page results came out. Check these, may be useful.

1.

Boundedness Theorems for the Relaxation Method.

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By: Amaldi, Edoardo; Hauser, Raphael. Mathematics of Operations Research. Nov2005, Vol. 30 Issue 4, p939-955. 17p. DOI: 10.1287/moor.1050.0164. , Database: Business Source Complete

Subjects: LINEAR models (Statistics); MATHEMATICS; INEQUALITIES (Mathematics); RELAXATION methods (Mathematics); NUMERICAL analysis

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2.

UNIFORM BOUNDEDNESS OF A PRECONDITIONED NORMAL MATRIX USED IN INTERIOR-POINT METHODS.

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By: Monteiro, Renato D. C.; O'Neal, Jerome W.; Tsuchiya, Takashi. SIAM Journal on Optimization. 2004, Vol. 15 Issue 1, p96-100. 5p. DOI: 10.1137/S1052623403426398. , Database: Business Source Complete

Subjects: LINEAR programming; ALGORITHMS; MATRICES; INTERIOR-point methods; ITERATIVE methods (Mathematics)

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3.

Global convergence of an SQP method without boundedness assumptions on any of the iterative sequences.

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By: Solodov, Mikhail V. Mathematical Programming. Apr2009, Vol. 118 Issue 1, p1-12. 12p. DOI: 10.1007/s10107-007-0180-y. , Database: Business Source Complete

Subjects: STOCHASTIC convergence; ALGORITHMS; MATHEMATICS; LOGIC; QUADRATIC programming; SEQUENCES (Mathematics); A priori

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4.

On boundedness of (quasi-)convex integer optimization problems.

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By: Obuchowska, Wiesława T. Mathematical Methods of Operations Research. 2008, Vol. 68 Issue 3, p445-467. 23p. DOI: 10.1007/s00186-007-0196-3. , Database: Business Source Complete

Subjects: INTEGER programming; MATHEMATICAL programming; MATHEMATICAL optimization; FUNCTIONS of bounded variation; POLYNOMIALS; INEQUALITIES (Mathematics)

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5.

Uniform boundedness of the inverse of a Jacobian matrix arising in regularized interior-point methods.

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By: Armand, Paul; Benoist, Joël. Mathematical Programming. Feb2013, Vol. 137 Issue 1/2, p587-592. 6p. 1 Chart. DOI: 10.1007/s10107-011-0498-3. , Database: Business Source Complete

Subjects: NONLINEAR programming; STOCHASTIC convergence; JACOBIAN matrices; INVERSE problems (Differential equations); REGULARIZATION (Mathematics); INTERIOR-point methods; CONSTRAINED optimization

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6.

ON THE BOUNDEDNESS OF THE FEASIBLE SET WITHOUT CONVEXITY ASSUMPTIONS.

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By: Hurwicz, Leonid; Reiter, Stanley. International Economic Review. Oct73, Vol. 14 Issue 3, p580. 7p. 2 Graphs. , Database: Business Source Complete

Subjects: ECONOMICS -- Statistical methods; EQUILIBRIUM (Economics); CONVEX domains

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7.

ON THE CONVERGENCE OF AUGMENTED LAGRANGIAN METHODS FOR CONSTRAINED GLOBAL OPTIMIZATION.

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By: Luo, H. Z.; Sun, X. L.; Li, D. SIAM Journal on Optimization. 2007, Vol. 18 Issue 4, p1209-1230. 22p. DOI: 10.1137/060667086. , Database: Business Source Complete

Subjects: MATHEMATICAL optimization; MATHEMATICS; MATHEMATICAL analysis; LAGRANGIAN functions; CALCULUS of variations; DYNAMICS

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8.

Advances in the Continuous Dynamic Network Loading Problem.

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By: Xu, Y. W.; Wu, J. H.; Florian, M.; Marcotte, P.; Zhu, D. L. Transportation Science. Nov99, Vol. 33 Issue 4, p341. 13p. 1 Diagram, 6 Graphs. , Database: Business Source Complete

Subjects: LOADING & unloading; SIMULATION methods & models; MATHEMATICAL models; ALGORITHMS; ELECTRIC networks; TRAFFIC flow; ORIGIN & destination traffic surveys; NONLINEAR models (Statistics); MODULAR arithmetic; COMPUTATIONAL complexity

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9.

A survey of the implications of the behavior of the central path for the duality theory of...

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By: Guler, O.; Roos, C. Management Science. Dec1995, Vol. 41 Issue 12, p1922. 13p. , Database: Business Source Complete

Subjects: LINEAR programming; MATHEMATICAL programming; ALGORITHMS; MATHEMATICAL analysis; MATHEMATICAL models; MATHEMATICS; DUALITY theory (Mathematics); LOGARITHMS; ALGEBRA; CALCULUS; FUNCTIONS (Mathematics)

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10.

Modeling and Verification of Distributed Systems by Petri Nets.

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By: Yamada, Tomohiko; Tsuji, Kohkichi; Kumagai, Sadatoshi; Kodama, Shinzo. Electronics & Communications in Japan, Part 1: Communications. Jul88, Vol. 71 Issue 7, p22-32. 11p. , Database: Business Source Complete

Subjects: DISTRIBUTED computing; JAPAN; PETRI nets; GRAPH theory; CABLE structures

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Salam brother Hussin Hejase and thank you very much for the list of references.

Dear Zeraoulia

I like to know if the list was of use? Maybe I can check more for you.

Regards