Electronic Design Automation of Multi-scroll Chaos Generators

Synthesis of PWL Functions

Author(s): Jesus Manuel Munoz Pacheco and Esteban Tlelo Cuautle

Pp: 45-55 (11)

DOI: 10.2174/978160805165611001010045

* (Excluding Mailing and Handling)

Abstract

The design of a nonlinear function with multi-segments is a basis for generating chaotic attractors with multidirectional orientation and with a large number of scrolls. However, it has been identified that it is quite difficult to synthesize nonlinear functions with multi-segments by using analog electronic circuits. Therefore, to cope with this problem, the electronic design automation (EDA) industry is developing design tools with a high degree of abstraction (behavioural modelling). Henceforth, the synthesis approach presented in this chapter depends on the behavioural modelling of the chaotic systems introduced in Chapter. 3. In particular, this approach is focused on designing multi-scrolls chaotic attractors with multidirectional orientation. In this manner, this chapter first presents a basic cell based on opamps to synthesize saturated nonlinear functions series. Further, a new synthesis approach is introduced to design the saturated functions in current and voltage modes by using a general scheme of connection for basic cells. Finally, at the end of this chapter, a Verilog-A model for considering no-ideal effects of the opamps is defined to execute SPICE simulations.


Keywords: Chaos, electronic design automation, dynamical systems, chaos generators, modelling and simulation.

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1518
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Trial Access" as access_base_text, CASE j.type WHEN 111 THEN true ELSE false END as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_trail` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 314378200 and j.to_ip >=314378200 )) and ((j.from_date <= 1739848484 and j.to_date >= 1739848484 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 942) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Subscribed" as access_base_text, false as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_corporate` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 314378200 and j.to_ip >=314378200 )) and ((j.from_date <= 1739848484 and j.to_date >= 1739848484 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 942) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Token Based Access" as access_base_text, false as terms_popup, "R" as content_download_type, j.uc_id as restricted_user_access_key, "v" as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_token` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 314378200 and j.to_ip >=314378200 )) and ((j.from_date <= 1739848484 and j.to_date >= 1739848484 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 942)
      680μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
      Backtrace
      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 859 and `b`.`banner_from_date` <= 1739848484 and `b`.`banner_to_date` >= 1739848484 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.09msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 859
      • 3: 1739848484
      • 4: 1739848484
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='aeeccdd7895680ea4725e2d9d18f2f33'
      220μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    5Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        KcmrIbbcIe0Z6Ff39COyaik3Zy7H5XwA4ww2byXj
        uc_cart_id
        aeeccdd7895680ea4725e2d9d18f2f33
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/942" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/942
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:16 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        []
        
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Tue, 18 Feb 2025 03:14:44 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6Ijc1blNvWmtKWFdTcTBsWUlKNDM2WlE9PSIsInZhbHVlIjoiVEdPSGVYV1Npd2JSZWpLdUFMc3BGQ1FxaUVlNXArcjcrVXBBaE5NL2d3eVAwbkRHcUc1emMwL1pHNndjU2YyZEVobmNMK0NnTTJDNTgzUHVjd0JCT083cHhDczJnbm9WeDg1Y1p5WmprS0lvNi9iSnE0WUVISGswK0RsU0Q0KysiLCJtYWMiOiJhMTdkYTFjMTNiZmNhOGJlNjdkODVjMmRjNTJlNzAxNjJiNTU1ZWMwODAwNjMxOGRiZjZlODMyNjg5NzJkODdkIiwidGFnIjoiIn0%3D; expires=Thu, 20 Feb 2025 15:14:44 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6Ijc1blNvWmtKWFdTcTBsWUlKNDM2WlE9PSIsInZhbHVlIjoiVEdPSGVYV1Npd2JSZWpLdUFMc3BGQ1FxaUVlNXArcjcrVXBBaE5NL2d3eVAwbkRHcUc1emMwL1pHNndjU2YyZEVobmNMK" 1 => "alphaeurekaselectnet_session=eyJpdiI6InN5Y3dmZmdVYVBFN3lydlE3dmpnY0E9PSIsInZhbHVlIjoibFhhRnRNOWhjWHpuQitybzNHajBIYTJmY3RxY2ZrK2lvNVREZ0l0Z21vRmcvS1RKTWVzbDBvTzJtM0VRZWkycllpOXdjUDVJKy8rOElHSmtPaWFKTWFRVlA3cmR4cmVHSmFIRjFhZmJXa3FibllhSWhGYTkybS9LM3hhcjZSV0wiLCJtYWMiOiI0NDkwNzg1MmU1OTU3YzBjNTFlNWY4OTcyZGQ4ZmQ4MjVkOGFlNmYyMmYzZjgxM2Y1OWM1YTJlMGJiZTllZTQ0IiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6InN5Y3dmZmdVYVBFN3lydlE3dmpnY0E9PSIsInZhbHVlIjoibFhhRnRNOWhjWHpuQitybzNHajBIYTJmY3RxY2ZrK2lvNVREZ0l0Z21vRmcvS1RKTWVzbDBvTzJ" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6Ijc1blNvWmtKWFdTcTBsWUlKNDM2WlE9PSIsInZhbHVlIjoiVEdPSGVYV1Npd2JSZWpLdUFMc3BGQ1FxaUVlNXArcjcrVXBBaE5NL2d3eVAwbkRHcUc1emMwL1pHNndjU2YyZEVobmNMK0NnTTJDNTgzUHVjd0JCT083cHhDczJnbm9WeDg1Y1p5WmprS0lvNi9iSnE0WUVISGswK0RsU0Q0KysiLCJtYWMiOiJhMTdkYTFjMTNiZmNhOGJlNjdkODVjMmRjNTJlNzAxNjJiNTU1ZWMwODAwNjMxOGRiZjZlODMyNjg5NzJkODdkIiwidGFnIjoiIn0%3D; expires=Thu, 20-Feb-2025 15:14:44 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6Ijc1blNvWmtKWFdTcTBsWUlKNDM2WlE9PSIsInZhbHVlIjoiVEdPSGVYV1Npd2JSZWpLdUFMc3BGQ1FxaUVlNXArcjcrVXBBaE5NL2d3eVAwbkRHcUc1emMwL1pHNndjU2YyZEVobmNMK" 1 => "alphaeurekaselectnet_session=eyJpdiI6InN5Y3dmZmdVYVBFN3lydlE3dmpnY0E9PSIsInZhbHVlIjoibFhhRnRNOWhjWHpuQitybzNHajBIYTJmY3RxY2ZrK2lvNVREZ0l0Z21vRmcvS1RKTWVzbDBvTzJtM0VRZWkycllpOXdjUDVJKy8rOElHSmtPaWFKTWFRVlA3cmR4cmVHSmFIRjFhZmJXa3FibllhSWhGYTkybS9LM3hhcjZSV0wiLCJtYWMiOiI0NDkwNzg1MmU1OTU3YzBjNTFlNWY4OTcyZGQ4ZmQ4MjVkOGFlNmYyMmYzZjgxM2Y1OWM1YTJlMGJiZTllZTQ0IiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6InN5Y3dmZmdVYVBFN3lydlE3dmpnY0E9PSIsInZhbHVlIjoibFhhRnRNOWhjWHpuQitybzNHajBIYTJmY3RxY2ZrK2lvNVREZ0l0Z21vRmcvS1RKTWVzbDBvTzJ" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "KcmrIbbcIe0Z6Ff39COyaik3Zy7H5XwA4ww2byXj" "uc_cart_id" => "aeeccdd7895680ea4725e2d9d18f2f33" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/942" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-18 03:14:44GET/chapter/942112197